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ISSN 0002-9920 (print) ISSN 1088-9477 (online) Notices ofof the American MathematicalMathematical Society May 2020 Volume 67, Number 5

The cover design is based on imagery from “Higgs Bundles—Recent Applications,” page 625. THE NEXT GENERATION SUPPORT FUND Photo by T. Christine Stevens T. Photo by

The Next Generation Fund is a new endowment at the AMS that exclusively supports programs for doctoral and postdoctoral scholars. It will assist rising mathematicians each year at modest but impactful levels, with funding for travel grants, collaboration support, mentoring, and more.

To learn more or make a gift, go to www.ams.org/nextgen

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Want to learn more? Visit www.ams.org/giving Or contact [email protected] 401.455.4111 A WORD FROM... Louise Jakobson, AMS Development Officer1

The world is full of opportunities and problems big and small, whether at the planetary level, in far-flung regions of the world, or right on our doorstep. The array of environ- mental, social, educational, scientific, and medical issues that exist is endless. But, formi- dable as these issues are, human beings work collectively to make things better. They find solutions and implement them through government, industry, or business. At individual and group levels, they also make progress by giving and volunteering via nonprofits and other organizations. Focusing on the United States, Americans have the reputation for being generous, and the data reflect that. The World Giving Index report for 2018 (https://www.cafonline .org/about-us/publications/2018-publications/caf-world-giving -index-2018) ranks the United States fourth worldwide (the top three countries are In-

Courtesy of Billy Durvin. Courtesy donesia, Australia, and New Zealand; Ireland, the UK, and Singapore are close behind the US). According to this study, 72% of Americans helped a stranger, 61% donated money, and 39% volunteered. How much do people donate and what are they supporting? Giving USA reports that Americans gave $427.71 billion to nonprofits in 2018. Individuals account for 77% of that number (including bequests); foundations gave 18%, and corporations 5%. Where did donors direct their generosity? Religion (29%) received the largest segment, followed by Education (14%); Human Services (12%); Foundations (12%); Health (10%); Public-Society Benefit (7%); International Affairs (5%); Arts, Culture, and Humanities (5%); Environment/Animals (3%); and Individuals (2%). There is a multitude of organizations to give to, and everyone has their own reasons for what they choose to support. At the AMS, donors regularly let us know why they are making their gift, large or small. In fact, one of the joys of working in fundraising is listening to donors share their stories! Often, they want to give back to their profession and support the younger generation, make opportunities available to others that they did not have themselves, spur important research in their by supporting a prize, or honor a mentor or give in memory of an esteemed colleague or great mathematician. Each organization offers something different to the mathematics community. In many ways, the AMS is uniquely positioned to make an impact because of its longev- ity, financial stability, national and international reach, and active community of mathematicians in governance and committees. There are many examples of mathematicians and people in other fields striving to advance a particular area to help solve some of the mathematical world’s challenges. Paul Sally’s passion for education drove him to establish the Arnold Ross Lectures endowed fund to bring top scholars to talented high school students; Don and Jill Knuth’s contributions over recent years have helped Mathematical Reviews in many ways, including indexing, author dis- ambiguation, and supporting native scripts; Joan and Joseph Birman created a Fellowship for Women Scholars to nurture outstanding research by mid-career women mathematicians; and many AMS donors have donated to the Epsilon Fund to benefit high school summer math camps. Others prefer to give unrestricted funds, allowing the AMS flexibility to direct resources to the area of greatest need, including advocacy efforts for mathematics in Washington, DC; meetings and conferences; or additional funds to the MathSciNet for Developing Countries Program to ensure access to the mathematical literature. On average, approximately half of AMS annual donors choose to make their gift unrestricted.

Louise Jakobson is a development officer at the AMS. Her email address is [email protected]. 1The opinions expressed here are not necessarily those of the Notices or the AMS.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 609 As a donor, you may also want to think about when you want your gift to be used. If you want to effect change now, you can choose to have the AMS utilize your contribution immediately. If support for future needs and challenges is what is important to you, you can direct your gift to an existing endowment (meaning that the funds are invested and provide funding for a specified purpose for the years to come), or talk to development staff about creating a new one. The AMS has nineteen endowed funds supporting prizes; donors may also establish named endowed funds to support broad areas of need, including early career mathematicians, education, diversity, and advocacy. One such example is the Mark Green and Kathryn Kert Green Fund for Inclusion and Diversity; another is the Next Generation Fund, which was recently created to support the needs of current and future generations of early career mathematicians, an ongoing priority of the AMS. I was interested to learn that endowments have a much longer history than one might imagine. The earliest known endowed chairs were established by the Roman emperor (and Stoic philosopher) Marcus Aurelius in Athens in AD 176 for the four major schools of philosophy, and the earliest known surviving endowed professorships were created by Lady Margaret Beaufort in Oxford and Cambridge in 1502 (these two funds are still in existence today!). Finally, a number of donors also give through charitable estate planning, taking a longer-term view of their charitable giving. For example, Franklin Peterson, Cathleen Synge Morawetz, and mathematical couple Steven Schot and Joanna Wood Schot made unrestricted bequests to the AMS, while Edmund and Nancy Tomastik have declared their bequest intention to establish a prize in differential equations. All of these individuals demonstrate the wish to impact math- ematics beyond their lifetimes. It can be very gratifying to know that you will be building mathematical research and scholarship for the future. I know from experience that donors care very much what happens in the organizations they support. The annual AMS Contributors List for the year 2019 will be published in next month’s issue of Notices. Approximately 1,400 donors a year support the AMS; as a development officer, I’ve had the privilege of talking with a number of these generous individuals. I can report that our donors care and make thoughtful and often deeply personal choices to invest in mathematics, its community, and its future.

610 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 May 2020 Cover Credit: The image used in the cover design appears in “Higgs Bundles—Recent Applications,” p. 625, and is by Laura P. Schaposnik.

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The Covering Method for Exponential Sums and Some Applications ...... 616 Ivelisse M. Rubio Higgs Bundles—Recent Applications ...... 625 Laura P. Schaposnik Can You Pave the Plane with Identical Tiles? ...... 635 Chuanming Zong Finding Solitons ...... 647 Jorge Lauret

A Word from... Louise Jakobson ...... 609 Book Review: American Mathematics 1890–1913: Catching Up to Europe ...... 701 Letters to the Editor ...... 614 Reviewed by Deborah Kent Early Career: Research ...... 659 Bookshelf ...... 706 How to Balance Research with Everything Else We Have to Do ...... 659 AMS Bookshelf ...... 707 David Zureick-Brown How to Read a Research Paper ...... 660 2020 Sectional Sampler: Mathematics of Cellular Matt Baker Evolution and Some Biomedical Applications ...... 708 Changing Focus ...... 662 Natalia L. Komarova Mark Andrea de Cataldo Keystones: Probabilistic Inference for Trust ...... 710 Mentoring Undergraduate Research: Andres Molina-Markham and Joseph J. Rushanan Advanced Planning Tools and Tips ...... 663 Courtney R. Gibbons AMS Communication: GROWing a Graduate Cohort ...... 724 Memorial Tribute: Ola Bratteli and His Diagrams ..... 665 Evelyn Lamb Tone Bratteli, Trond Digernes, George A. Elliott, David E. Evans, Palle E. T. Jorgensen, Aki Kishimoto, Magnus B. Landstad, News: Mathematics People ...... 730 Derek W. Robinson, and Erling Størmer News: Community Updates ...... 737 Memorial Tribute: Memories of Goro Shimura ...... 677 Don Blasius, Toni Bluher, Haruzo Hida, Kamal Khuri-Makdisi, News: Mathematics Opportunities ...... 738 Kenneth Ribet, Alice Silverberg, and Hiroyuki Yoshida Classi ed Advertising ...... 741 History: The Legacy of Józef Marcinkiewicz: Four Hallmarks of Genius ...... 690 New Books Offered by the AMS ...... 744 Nikolay Kuznetsov Meetings & Conferences of the AMS ...... 751 FROM THE INVITATIONS AMS SECRETARY FROM THE AMS

Calls for Nominations ...... 712 Call for Nominations: 2021 Frank and Brennie Levi L. Conant Prize ...... 712 Morgan AMS-MAA-SIAM Prize ...... 716 Cole Prize in Algebra ...... 712 Mary P. Dolciani Prize for Excellence Position Available: Associate Executive Director for in Research ...... 712 Meetings and Professional Services ...inside back cover Ulf Grenander Prize in Stochastic Theory and Modeling ...... 713 Bertrand Russell Prize of the AMS ...... 713 Ruth Lyttle Satter Prize in Mathematics ...... 713 Albert Leon Whiteman Memorial Prize ...... 714 Mathematics Programs that Make a Difference ....714 Birkhoff Prize in Applied Mathematics (AMS-SIAM Prize) ...... 714 Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student ...... 715 2020 Award for an Exemplary Program or Achievement in a Mathematics Department ...... 717 2020 Award for Impact on the Teaching and Learning of Mathematics ...... 720 2020 Mathematics Programs That Make a Difference Award ...... 722

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Letter to the Editor Where does “mathematical making” Dear Colleagues, fit in our community? Many thanks for a very interesting article, “How to At the end of a fantastic semester of Illustrating Mathemat- Keep Your Secrets in a Post-Quantum World,” published ics at the Institute for Computational and Experimental Re- in the January 2020 issue of Notices of the AMS. This article search in Mathematics (ICERM), many of the participants describes ideas for “post-quantum cryptosystems that are gathered to discuss the future of what we see as a growing not currently known to be breakable in polynomial time movement. Where can we publish scholarly articles about by a full-scale quantum computer.” These are all great mathematical visualization if the theorems alone might not ideas, but readers who are not very familiar with this topic justify publication? How does the mathematical commu- should be informed that already in the 1980s, researchers nity value the creation of new ways to see and communicate had developed quantum cryptography schemes—such as mathematics? The extraordinary creativity sparked by our the 1984 Bennetts’ and Brassard’s Quantum Key Distribu- being brought together makes us confident that more math- tion scheme—which are not breakable even by a quantum ematicians will delight in taking up this enterprise. Those computer. These are not just purely theoretical schemes: of us who have signed the Mathematical Makers’ Manifesto according to the Wikipedia page on quantum cryptography, below urge the mathematical community to support efforts several companies already manufacture such communica- in the same way ICERM so generously supported us this fall. tion schemes, and they are actively used—in particular, for communications over hundreds of kilometers. Of course, —Frank A. Farris this does not mean that the problem is fully solved: the ex- Santa Clara University isting quantum communication schemes have limitations, e.g., limitations on communication speed; from this view- point, it would be great to have faster alternative schemes, e.g., schemes described in the Notices article.

—Vladik Kreinovich and Luc Longpre Department of Computer Science University of Texas at El Paso

(Received December 20, 2019) Photo by Frank A. Farris Photo by *We invite readers to submit letters to the editor at notices-letters A human-scale model of the Weaire-Phelan foam. @ams.org. Mathematical installation by Glen Whitney.

614 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Letters to the Editor

Mathematical Makers’ Manifesto We are mathematical makers. We are makers because we make things, by which we might mean literal objects, such as sculptures, paintings, or fabrics, but our making includes creation of digital images, software, and even performance arts. We are mathematical makers because our creations require mathematical knowledge as a key ingredient. Why do we make these things? Our reasons are diverse, including education, outreach, and experimentation to investigate and create new mathematical understanding; we are also inspired to create works of art and useful crafts. We work to include mathematicians of many different backgrounds in our making, from beginning students to researchers in the farthest branches of mathematics. As the ultimate interdis- Now available for ciplinary subfield of mathematics, mathematical making deserves support from universities, museums, governments, your local library: and corporations around the world. The AMS | MAA Press Signed by Aaron Abrams, Washington and Lee University Archive eBook collection. Silviana Amethyst, University of Wisconsin–Eau Claire Roger Antonsen, University of Oslo, Norway The AMS/MAA Press Archive eBook Collection Pierre Arnoux, Université d’Aix-Marseille contains AMS/MAA Press titles published Jayadev Athreya, University of Washington, Seattle between 1925 and 2018, many available as David Bachman, Pitzer College eBooks for the first time. Dina Buric, University of Victoria J. Scott Carter, University of South Alabama This valuable archive contains over 300 titles, Arnaud Chéritat, CNRS/Université de Toulouse ranging from undergraduate textbooks to Rémi Coulon, CNRS/Université de Rennes 1 Gabriel Dorfsman-Hopkins, University of California, Berkeley research monographs. David Dumas, University of Illinois at Chicago Subscribers to the collection will receive Bernat Espigulé, Universitat de Barcelona Frank A. Farris, Santa Clara University access to all of the series listed below. Herbert Gangl, Durham University • Anneli Lax New Mathematical Library Edmund Harriss, University of Arkansas Alexander E. Holroyd, University of Bristol, UK • The Carus Mathematical Monographs Oliver Labs, MO-Labs • Classroom Resource Materials Daniel Lautzenheiser, University of Nevada, Las Vegas Samuel Lelièvre, Université Paris-Saclay • Dolciani Mathematical Expositions Stepan Paul, • AMS/MAA Textbooks Alba Málaga Sabogal, ICERM, Brown University Tashrika Sharma, University of Vienna • Problem Books Martin Skrodzki, Semester Postdoc at ICERM, Brown University • Spectrum Katherine E. Stange, University of Colorado, Boulder Laura Taalman, James Madison University Mikael Vejdemo-Johansson, CUNY College of Staten Island / For more information, including a CUNY Graduate Center Glen Whitney, StudioInfinity.org complete list of volumes included Carolyn Yackel, Mercer University in the collection, visit (Received January 23, 2020) www.ams.org/ebooks.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 615 The Covering Method for Exponential Sums and Some Applications

p = 3

1 4 19

10 13

−2 −3 7 6 16 15

18

9 0

11 12 2 3 20 21

17 14

−1 8 5

Ivelisse M. Rubio

Introduction elementary and intuitive way to estimate or compute the 푝- Exponential sums over finite fields are an important tool divisibility of exponential sums, which is particularly con- for solving mathematical problems and have applications venient in the applications. The covering method allows to many other areas. However, some of the methods and us to determine solvability of systems of polynomial equa- proofs of the results are nonelementary. The main pur- tions, improve the search for balanced Boolean functions, pose of this article is to present the covering method, an give better estimates for covering radius of codes, and has many other applications. Ivelisse M. Rubio is a professor of mathematics in the Department of Computer Solvability of systems of polynomial equations. One of Sciences at the University of Puerto Rico, R´ıo Piedras. Her email address is the prominent problems in mathematics is to determine [email protected]. if a polynomial equation has solutions. In 1935 Artin Communicated by Notices Associate Editor Emilie Purvine. conjectured that a homogeneous polynomial over a finite For permission to reprint this article, please contact: field has a nontrivial zero if the number of variables is [email protected]. larger than the degree. Chevalley obtained almost imme-

DOI: https://doi.org/10.1090/noti2073 diately a slightly better result changing the hypothesis of

616 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 homogeneity to the weaker one of the polynomial having of 푥. If 푣푝(푥) ≠ ∞, we say that 푣푝(푥) is the exact 푝- no constant term. Note that the homogeneous and the no- divisibility of 푥. constant-term conditions imply that the polynomial has Example 1. Consider 푥 = 36 = (32)(4) ∈ ℤ. Note that we the trivial zero. The theorem guarantees additional zeros. can also represent 푥 as 푥 = 32 + 33 ∈ ℚ . This implies that Warning improved Chevalley’s result by proving that if 3 the exact 3-divisibility of 36 is 2. That is, 푣 (36) = 2. the number of variables is larger than the sum of the de- 3 grees of a system of polynomials, then 푝, the characteristic The set of 푝-adic integers is the local ring ℤ푝 = of the field, divides the number of common zeros. This {푥 ∈ ℚ푝 | 푣푝(푥) ≥ 0} with maximal ideal 푝ℤ푝 and residue classical result is known as the Chevalley–Warning theo- field ℤ푝/푝ℤ푝 ≅ 픽푝. One can use this valuation to define −푣푝(푥) rem and has an elementary proof [12, 20]. By elementary the 푝-adic absolute value of 푥 by |푥|푝 = 푝 if 푥 ≠ 0 we mean that it uses only elementary results from number and |0|푝 = 0. theory. Note that the number of zeros could be 0, but if The 푝-adic field ℚ푝 is a completion of the rationals ℚ, the system has the trivial zero, Chevalley–Warning guaran- and its construction is similar to the construction of the tees nontrivial solutions. real numbers ℝ from ℚ but using the 푝-adic absolute value. There are many results improving Chevalley–Warning’s So, ℚ푝 is the completion of ℚ with respect to |.|푝. The 푝- theorem. The results presented by Ax [2], Katz adic numbers offer a different perspective to study prob- [11], Adolphson–Sperber [1], Moreno–Moreno [15], and lems, and these methods can be helpful to understand con- Moreno et al. [17] have proofs that are nonelementary cepts and prove properties that may be difficult without or semielementary. Other results presented by Moreno– them. For an accessible introduction to the beautiful the- Moreno [13], Wan [19], and Castro et al. [6] have entirely ory of 푝-adic numbers we refer the reader to [9]. elementary proofs. As in the Chevalley–Warning theorem, For a polynomial 퐹 ∈ 픽푝[퐗], where 퐗 = (푋1, 푋2, … , 푋푛), 푎 solvability is not guaranteed; nontrivial solutions exist if and 휁 a primitive 푝th root of unity over ℚ푝, let 휁 = the system has the trivial zero. 휁푎 mod 푝, and define the exponential sum associated to The covering method to study the 푝-divisibility of expo- 퐹 as nential sums is an elementary method introduced in [6] 퐹(퐱) 푆(퐹) = ∑ 휁 ∈ ℤ푝. that lets us determine sufficient conditions to guarantee 푛 퐱∈(픽푝) solvability and allows us to construct general families of The explicit evaluation of the exponential sum of a poly- solvable systems of polynomial equations [4,5]. nomial might be a difficult task, but for many applications Applications to cryptography and coding theory. The di- it is enough to have estimates for 푣 (푆(퐹)). For simplicity, visibility of exponential sums has been used to character- 푝 in some of the results in this article we consider only one ize and prove properties in coding theory and cryptogra- polynomial, but the results can be extended to systems of phy [3, 7, 18]. The computation of bounds or the exact 2- polynomials 퐹 , 퐹 , … , 퐹 ∈ 픽 [퐗] by adding 푡 extra variables divisibility of exponential sums of Boolean functions pro- 1 2 푡 푝 푌 , … , 푌 (one per polynomial) and constructing a new vides information on the Hamming weight of the function 1 푡 polynomial 푃 = 푌 퐹 + 푌 퐹 + ⋯ + 푌 퐹 . The exponential and can be used to obtain information on the covering ra- 1 1 2 2 푡 푡 sum associated to the system of polynomials 퐹 , 퐹 , … , 퐹 dius and the weight distribution of certain codes. These 1 2 푡 is the exponential sum associated to 푃. The relation be- properties are important for the analysis of decoding algo- 푛 tween the number 풩 of elements (푥 , … , 푥 ) ∈ (픽 ) that rithms and are also related to cryptography, as they can 1 푛 푝 are common zeros of the system and the exponential sum be used to study nonlinearity and to search for balanced associated to the system is given by the following lemma: Boolean functions. Lemma 1. Let 풩 be the number of common zeros of Exponential Sums Associated to Polynomials 퐹1, 퐹2, … , 퐹푡 ∈ 픽푝[퐗]. Then We will restrict our exposition to exponential sums asso- 풩 = 푝−푡 ∑ 휁푦1퐹1(퐱)+푦2퐹2(퐱)+⋯+푦푡퐹푡(퐱). ciated to polynomials in 픽 [푋 , … , 푋 ], where 푝 is a prime 푝 1 푛 푛 푡 퐱∈(픽 ) ,퐲∈(픽 ) number and 픽푝 is the finite field with 푝 elements. The defi- 푝 푝 nition of these exponential sums depends on 휁, a 푝th root −푡 −푡 Since 풩 = 푝 푆(푌1퐹1 + 푌2퐹2 + ⋯ + 푌푡퐹푡) = 푝 푆(푃), of unity over the 푝-adic field computing the exact value of 풩 depends on the compu- tation of 푆(푃), which is not easy. However, if we can get 푟 푟+1 ℚ푝 = {푎푟푝 + 푎푟+1푝 + ⋯ | 푎푖 ∈ {0, … , 푝 − 1}, 푟 ∈ ℤ} . the exact 푝-divisibility of 푆(푃), 푣푝(풩) < ∞, we know that 푝푣푝(풩)+1 ∤ 풩. This implies that 풩 ≠ 0 and the system 푟 For 푎푟 ≠ 0, define the 푝-adic valuation of 푥 = 푎푟푝 + is solvable. Therefore, being able to compute the exact 푝- 푟+1 푎푟+1푝 + ⋯ ∈ ℚ푝 as 푣푝(푥) = 푟, the highest power of 푝 divisibility of an exponential sum of a system of polyno- dividing 푥, 푣푝(0) = ∞. We also call 푣푝(푥) the 푝-divisibility mials gives a criterion for solvability of the system.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 617 2-Divisibility of Exponential Sums However, a bound does not guarantee that 풩 ≠ 0. To de- of Boolean Functions termine solvability one needs to obtain exact 2-divisibility. Most of the applications of exponential sums to coding Theorem 1 is general and tight in the sense that there theory and cryptography consider Boolean functions 푓 ∶ are polynomials that attain the bound and have exact 2- 푛 divisibility |퐶|. This implies that to determine if a polyno- (픽2) ⟶ 픽2. Any Boolean function 푓 can be identified 퐞 mial has exact 2-divisibility or to improve the bound, we with a unique Boolean polynomial 퐹 = ∑퐞∈Supp(퐹) 퐗 , where Supp(퐹) is the set of exponents of the nonzero terms, need to impose additional conditions. The next theorem 푛 퐞 푒1 푒2 푒푛 has simple conditions that are sufficient to obtain exact 2- 퐞 = (푒1, … , 푒푛) ∈ (픽2) , and 퐗 = 푋1 푋2 ⋯ 푋푛 . This polynomial is known as the algebraic normal form of the divisibility.

Boolean function. The exponential sum of a Boolean poly- Theorem 2 ([7]). Let 퐹 ∈ 픽2[퐗], and let 퐶1, … , 퐶푐 be all the nomial 퐹 ∈ 픽2[퐗] is minimal coverings of 퐹. If, for each 1 ≤ 푖 ≤ 푐, each monomial 푆(퐹) = ∑ (−1)퐹(퐱). in 퐶푖 has at least two variables that are not present in the other 푛 monomials of 퐶 , then 푣 (푆(퐹)) = |퐶 | if 푐 is odd, and other- 퐱∈(픽2) 푖 2 푖 wise 푣2 (푆(퐹)) ≥ |퐶푖| + 1, where |퐶푖| is the size of a minimal The covering method for 2-divisibility. In [14], Moreno– covering. Moreno introduced the covering method, which provides an elementary way to give a bound on the 2-divisibility With the given conditions, the above theorem refines of exponential sums of Boolean functions. Using this Moreno–Moreno’s Theorem 1. method, they gave an improvement to Ax’s theorem in Example 3. The polynomial in Example 2 has exactly two [2] for the binary case. However, the result does not give minimal coverings. Moreno–Moreno’s Theorem 1 implies exact 2-divisibility and cannot be used to determine solv- that 푣2(퐹) ≥ 3, but Theorem 2 guarantees that 푣2(퐹) ≥ 4. ability or to find nonbalanced Boolean functions. Addi- This might seem a small improvement, but in the applica- tional conditions have to be imposed to determine exact tions even small improvements are important. 2-divisibility. We now assume that any polynomial 퐹 is not a polynomial in some proper subset of the variables The next example shows that even though different 푋1, … , 푋푛. Boolean functions might have the same unique minimal covering, and hence the same 2-divisibility, there is an am- Definition 1. A set 퐶 of monomials 퐹 , … , 퐹 of a polyno- 푖1 푖푟 ple spectrum for the exact value of 푆(퐹). mial 퐹 = 퐹1 + ⋯ + 퐹푚 ∈ 픽2[퐗] is called a covering of 퐹 퐹 = 푋 푋 푋 푋 +푋 푋 푋 푋 +푋 푋 푋 if every variable 푋푖 is in at least one monomial of 퐶. The Example 4. Consider 1 2 3 4 4 5 6 7 7 8 9 ′ size of a covering 퐶 is its cardinality |퐶|. A set 퐶 is called and 퐹 = 푋1푋2푋3푋4 +푋4푋5푋6푋7 +푋7푋8푋9 +푋1 +푋2 +⋯+푋9 a minimal covering of 퐹 if there is no other covering of 퐹 in 픽2 [푋1, … , 푋9]. It can be verified that 푆(퐹) = 8 ⋅ 3 ⋅ 13 and ′ of smaller size. 푆(퐹 ) = 8. 푎 Although, in general, it is not an easy task to find all the Note that since, for 푎 ≠ 0, 푋 = 푋 over 픽2, if we take the minimal coverings of a given polynomial, one can easily product of the monomials 퐹푖1 ⋯ 퐹푖푟 in 퐶 we get 푋1푋2 ⋯ 푋푛, the monomial with all the variables. This fact will be use- construct polynomials for which one knows all the mini- ful in the generalization of the covering to any characteris- mal coverings and hence knows the exact 2-divisibility. For tic 푝. example, to obtain unique minimal coverings it is enough to construct systems of polynomials with lead monomials Example 2. Let 퐹 = 푋1 + 푋2 + ⋯ + 푋8 + 푋1푋2푋3푋4 + of degree at least 2 and of disjoint support that cover all 푋3푋5푋6 + 푋2푋7푋8 + 푋4푋7푋8 ∈ 픽2[푋1, … , 푋8]. Then 퐶1 = the variables. {푋1, 푋2, … , 푋8}, 퐶2 = {푋1푋2푋3푋4, 푋3푋5푋6, 푋2푋7푋8}, and 퐶 = {푋 푋 푋 푋 , 푋 푋 푋 , 푋 푋 푋 } are coverings of 퐹, but Example 5. Consider the following system of polynomials 3 1 2 3 4 3 5 6 4 7 8 3 퐶2, 퐶3 are the only minimal coverings of 퐹. in 13 variables, where (훼1, 훼2, 훼3) ∈ (픽2) : Moreno–Moreno used minimal coverings of a Boolean 퐹1 + 퐺1 = 푋1푋2푋3푋4푋5 + ∑ 푋푖 − 훼1, function 퐹 to obtain a bound on the 2-divisibility of the 푖 exponential sum of 퐹. 퐹2 + 퐺2 = 푋6푋7푋8푋9 + ∑ 푋푖푋푗 − 훼2, 푖<푗 Theorem 1 ([14]). Let 퐶 be a minimal covering of 퐹 ∈ 픽2[퐗]. 퐹3 + 퐺3 = 푋10푋11푋12푋13 + ∑ 푋푖푋푗푋푘 − 훼3, Then 푖<푗<푘 푣2(푆(퐹)) ≥ |퐶|. where 퐹1 = 푋1푋2푋3푋4푋5, 퐹2 = 푋6푋7푋8푋9, and 퐹3 = One can use Theorem 1 and Lemma 1 to give a bound 푋10푋11푋12푋13. Note that 퐶 = {푌1퐹1, 푌2퐹2, 푌3퐹3} is the on the 2-divisibility of the number of solutions 풩 of 퐹. unique minimal covering of the associated polynomial

618 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 푃 = 푌1 (퐹1 + 퐺1) + 푌2 (퐹2 + 퐺2) + 푌3 (퐹3 + 퐺3). This implies Example 8. Consider the system that 푆(푃) has exact 2-divisibility 푣2 (푆(푃)) = 3. 푋1푋2푋3푋4푋5 + ∑ 푋푖 = 훼1, 푖 ′ ′ ′ ′ ′ ′ Note that any system 퐹1 + 퐺1, … , 퐹푡 + 퐺푡, where 퐹1 , … , 퐹푡 푋 푋 푋 푋 + ∑ 푋 푋 = 훼 , ′ ′ 6 7 8 9 푖 푗 2 have disjoint support and deg (퐺푖 ) < min푖 {deg (퐹푖 )}, will 푖<푗 also have an associated polynomial 푃 with unique mini- 푋 푋 푋 푋 + ∑ 푋 푋 푋 = 훼 . mal covering, and hence 푆(푃) will have exact 2-divisibility 10 11 12 13 푖 푗 푘 3 푖<푗<푘 푣푝 (푆(푃)) = 푡. One can determine other conditions so that families of “deformations” 퐹 + 퐺푖 of a polynomial The solutions of this system are the zeros of the system 퐹 have the same minimal coverings as 퐹. This provides of polynomials 퐹1 +퐺1, 퐹2 +퐺2, 퐹3 +퐺3 in Example 5. Since a way to obtain the 2-divisibility of exponential sums of 푣2 (푆(푃)) = 3, 푣2 (풩) = 0 and 2 ∤ 풩. This implies that 풩 ≠ 3 polynomial deformations 퐹 + 퐺푖 from the 2-divisibility of 0 and the system is solvable for any (훼1, 훼2, 훼3) ∈ (픽2) . the exponential sum of the polynomial 퐹. Other applications. Other important applications of ex- ponential sums are to coding theory and cryptography. Theorem 3 ([7]). Let 퐹, 퐺 ∈ 픽2[퐗]. Suppose that the mini- Error-correcting codes are used to protect digital informa- mal coverings of 퐹 are the minimal coverings of 퐹 +퐺 and each tion from accidental errors that might occur during trans- monomial in each minimal covering 퐶퐹 has at least two vari- mission or storage; the aim is for the receiver to be able ables that are not present in the other monomials of 퐶퐹 . Then to detect and correct errors that were introduced acciden- |퐶 |+1 푆(퐹 + 퐺) ≡ 푆(퐹) (mod 2 퐹 ). Moreover, if the number of tally and retrieve the original message that was sent. On minimal coverings is odd, then 푣2 (푆(퐹 + 퐺)) = 푣2 (푆(퐹)) = the other hand, cryptography is used to hide information |퐶퐹 |. from intruders; the information transmitted should be un- derstood only by its intended receiver. Coding theory and Example 6. Consider 퐹 = 푋1푋2푋3 + 푋4푋5푋6 ∈ cryptography serve different purposes, but they both share 픽2[푋1, … , 푋6] and let 퐹 + 퐺 be any polynomial in some theoretical concepts and methods. 픽2[푋1, … , 푋6] with deg(퐺) ≤ 2. Then 퐶 = {푋1푋2푋3, 푋4푋5푋6} In the coding process an encoder adds redundancy to a is the unique minimal covering of 퐹 and 퐹 + 퐺, and each block of symbols of length 푘 that represents the message 퐦 monomial in 퐶 has three variables that are not present in to transform it into a codeword 퐜 of block length 푛 so that the other monomial. This implies that 푆(퐹) and 푆(퐹 + 퐺) when received, the decoder can detect and correct errors. have exact 2-divisibility 푣2 (푆(퐹 + 퐺))=푣2 (푆(퐹))=|퐶|=2. The code 풞 is the set of all codewords. One can identify the messages with 푘-tuples of symbols from a finite field 픽푞 and give the code the structure of a vector space of dimension Example 7. Consider 퐹 = 푋1푋2푋3 + 푋4푋5푋6 + 푋1푋4푋5 + 푘 over 픽 . The encoder is then a one-to-one linear map 푋2푋4푋6 + 푋3푋5푋6 ∈ 픽2[푋1, … , 푋6] and let 퐹 + 퐺 be any 푞 푘 푛 polynomial in 픽2[푋1, … , 푋6] where deg(퐺) ≤ 2. Again, 퐶 = ℰ풞 ∶ (픽푞) ⟶ (픽푞) , and the linear code 풞 = Im (ℰ풞). {푋1푋2푋3, 푋4푋5푋6} is the unique minimal covering of 퐹 and At first one might think that the decoder could justbe 퐹 + 퐺 and 푣2 (푆(퐹 + 퐺)) = 푣2 (푆(퐹)) = 2. the inverse of the encoding function. But the problem is that after the codeword 퐜 = ℰ풞(퐦) is transmitted, the re- Examples 6 and 7 provide families of polynomials ceived word is 퐫 = 퐜 + 퐞, where 퐞 is an error vector. So, −1 whose exponential sums have exact 2-divisibility. The in- ℰ풞 (퐫) ≠ 퐜 if 퐞 ≠ 0. Hence we need “good” codes, coding tuitive and simple condition of 퐹 + 퐺 and 퐹 having the and decoding algorithms that allow us to detect and cor- same minimal coverings allows us to easily construct fam- rect errors. The main problem in coding theory is to find 푘 ilies of deformations with exact 2-divisibility. We will see codes with large rate of information symbols 푘 per to- 푛 later that this has useful applications to the determination tal number of symbols 푛 that can correct “enough” errors, of nonbalanced Boolean functions. where “large” and “enough” will depend on the transmis- Solvability. As mentioned above, one of the main appli- sion channel for which the code is designed. cations of 푝-divisibility of exponential sums is to obtain From now on we will consider binary linear codes, information about the number of solutions of systems of that is, linear codes over 픽2. The Hamming weight of equations. Lemma 1 gives the relation between exponen- a vector 퐱, 푤퐻 (퐱), is the number of entries of 퐱 that tial sums and the number of solutions 풩 of a system of are nonzero. The Hamming distance between two vec- polynomial equations 퐹1 = ⋯ = 퐹푡 = 0. Using Theorem 2 tors 퐱, 퐲, 푑퐻 (퐱, 퐲) defines a metric and is the number one could determine if 풩 has exact 2-divisibility 푣2(풩). If of places on which the vectors disagree; this is equiva- this happens, 2푣2(풩)+1 does not divide 풩, 풩 ≠ 0, and the lent to the Hamming weight of 퐱 + 퐲. The minimum dis- system is solvable. tance 푑 of a code 풞 is the minimum (Hamming) distance

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 619 between any two codewords, 푑 = min {푑퐻 (퐱, 퐲) |퐱, 퐲 ∈ 풞}, The Reed–Muller code of order 1, 푅(1, 푛), is the set of and, since we are considering linear codes, this is equal to Boolean polynomials in 푛 variables with degree less than the minimum of the Hamming weight of the codewords or equal to 1. The nonlinearity of a Boolean function 퐹 is 푑 = min {푤퐻 (퐱) | 퐱 ∈ 풞}. the Hamming distance from 퐹 to 푅(1, 푛). Let 푑 ≥ 2푡 + 1 be the minimum distance of a code 풞 Exponential sums, Hamming weights, and nonlinearity. and suppose that a codeword 퐜 has been transmitted and The Hamming weight associated to a Boolean function 퐫 = 퐜 + 퐞 has been received, where 퐞 is an error vector with 퐹, 푤퐻 (퐹), is the Hamming weight of its truth table. This is 푛 푤퐻 (퐞) ≤ 푡. Then, 퐜 is the only codeword with distance the number of 퐱 ∈ (픽2) such that 퐹(퐱) = 1. If 푤0(퐹) 푛 from 퐫 less than or equal to 푡. Hence a possible decoding is the number of 퐱 ∈ (픽2) such that 퐹(퐱) = 0, then 푛 퐹(퐱) algorithm for a received word 퐫 is to look for the codeword 2 = 푤 (퐹) + 푤 (퐹). Also, 푆(퐹) = ∑ 푛 (−1) = 퐻 0 퐱∈(픽2) that is closest in Hamming distance to 퐫. In practice this 0 1 푤0(퐹)(−1) + 푤퐻 (퐹)(−1) = 푤0(퐹) − 푤퐻 (퐹). This implies would be too inefficient but guarantees that if 푡 or fewer 푛−1 1 that 푤퐻 (퐹) = 2 − 푆(퐹) and gives a correspondence be- errors occur, we can always correct them. This is why a lin- 2 tween results on exponential sums and Hamming weights ear block code with minimum distance 푑 ≥ 2푡 + 1, block of Boolean functions. Hence, any result for exponential length 푛, and dimension 푘 is called a 푡-error-correcting sums of a Boolean function also gives a corresponding re- code with parameters (푛, 푘, 푡). The goal of research in cod- sult about the Hamming weight of the function. ing theory is to construct codes that have large 푑 for the 푘 Defining the Hamming distance of a Boolean function given rate and to design efficient algorithms to encode 푛 퐹 to a vector 퐱 as the Hamming weight of the sum of 퐱 and decode them. with the truth table of 퐹, one can define the Hamming A cryptographic system is a set of transformations of distance from 퐹 to a code 풞 as min퐜∈풞 {푤퐻 (퐹 + 퐜)}. This the set of all messages into another space with certain prop- lets us define a measure for the principle of confusion in erties. The message is enciphered into the ciphertext us- the cryptographic system, a sense of “how far is a Boolean ing a particular key that defines an injective mapping. Two function 퐹 from being linear.” The nonlinearity of 퐹 is important principles in the design of cryptographic sys- 푁푙(퐹) = 푤퐻 (퐹 + 푅(1, 푛)), that is, the minimum Hamming tems are confusion and diffusion. The principle of diffusion distance between 퐹 and all the codewords in 푅(1, 푛). This makes different messages equally likely to occur; one way can be defined in terms of the exponential sums of cosets to measure diffusion is to determine if the function used of 푅(1, 푛), to cipher is balanced. The principle of confusion measures 푛−1 1 the complexity of the decryption process; the nonlinearity 푁푙(퐹) = min퐜∈푅(1,푛) {2 − 푆(퐹 + 퐜)} , of the functions used in the system gives a measure for con- 2 fusion. Functions that are balanced and have large nonlin- and we can use results on exponential sums of deforma- earity are desired. tions of Boolean functions to study the nonlinearity of a Reed–Muller codes are some of the oldest and most- Boolean function 퐹. studied codes; nevertheless, there are still many open prob- Covering radius of a code. The covering radius 휌(풞) is lems related to them that are important in both coding and another important parameter of a code 풞: cryptographic applications. To define a Reed–Muller code 푛 of size 2푛, given a fixed ordering of (픽 ) , one associates a 휌(풞) = max {min {푤 (퐱 + 퐜)}} . 2 푛 퐻 퐱∈(픽2) 퐜∈풞 Boolean polynomial 퐹 ∈ 픽2[푋1, … , 푋푛] with the vector of 푛 size 2 consisting of all the values of 퐹(퐱) as 퐱 varies accord- This measure gives the maximum weight of a correctable er- ing to the ordering. This is also called the truth table of 퐹. ror and can be used for the design of decoding algorithms. 3 For example, if (퐹2) is ordered in lexicographic order with A code of minimum distance 2푡 + 1 is called perfect if 푋1 > 푋2 > 푋3, the truth table for 퐹 (푋1, 푋2, 푋3) = 푋1 +푋2푋3 휌(풞) = 푡 and quasi-perfect if 휌(풞) = 푡 + 1. 8 is (0, 0, 0, 1, 1, 1, 1, 0) ∈ (픽2) . When convenient, we will The covering radius of the Reed–Muller code of order 푛 work with this representation of 퐹 in (픽2) instead of its 1, 휌 (푅(1, 푛)), is the maximum Hamming distance of all polynomial representation; we will also alternate between 푛-variate Boolean polynomials to 푅(1, 푛). We then have calling 퐹 a function or a polynomial. The 푘th order Reed– 푁푙(퐹) ≤ 휌 (푅(1, 푛)). The covering radius of 푅(1, 푛) gives Muller code of length 2푛, 푅(푘, 푛), is the set of truth tables a point of comparison for the nonlinearity of a Boolean of all the Boolean polynomials in 푛 variables and degree polynomial and hence a sense of “how good” the function less than or equal to 푘. That is, 푅(푘, 푛) can be identified could be for cryptographic applications. with the set of Boolean polynomials in 푛 variables and de- Results on the 2-divisibility of exponential sums have gree less than or equal to 푘. been used in several papers [16] to give elementary direct proofs of the covering radius of certain cyclic codes and to prove that families of cyclic codes are quasi-perfect.

620 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Weight distribution. The weight distribution of a code 2-divisibility. Therefore all the polynomials in these cosets 풞 counts how many codewords of each weight there are. are nonbalanced. Much work has been done studying the weight distribu- Hou presented representatives for each of the different tion of Reed–Muller codes, but (as mentioned earlier) orbits in 푅(푘, 푛)/푅(푘 − 1, 푛) for 푘 = 3, 푛 = 6, 7, 8. Exam- many problems remain open. Many of the properties of a ple 9 shows two of the six cosets of 푅(3, 6)/푅(2, 6). Cu- Boolean function 퐹 that are important to cryptography can sick and Cheon noticed in [8] the uneven distribution of be related to the weight distribution of the coset 퐹 +푅(1, 푛) the balanced functions in the table of balanced functions and can be studied using exponential sums. Canteaut in the cosets of 푅(3, 6)/푅(2, 6). Two of the six cosets, the [3] obtained a result that is a refinement of the Ham- two cosets of Examples 6, 7, and 9, have zero balanced ming weight version of Katz’s theorem for the Boolean case functions compared to more than 1.5 million in each of and used it to study the weight distribution of cosets of the other four cosets. The covering method gives a simple first-order Reed–Muller codes. Her result is tight for the explanation for this phenomenon: as was seen in the ex- Boolean case and can be improved only by imposing ad- amples, for any 퐺 ∈ 푅(2, 6), 퐹 and 퐹 + 퐺 have the same ditional conditions. The additional conditions to the cov- unique minimal covering {푋 푋 푋 , 푋 푋 푋 }, where each ering method in Theorem 3 allowed us to obtain an im- 1 2 3 4 5 6 monomial has three variables not contained in the other provement of her results [7]. monomial, and hence 퐹 + 퐺 is not balanced. Balanced functions. A Boolean function 퐹 is said to be It is not difficult to find sufficient conditions thatcan balanced if the function is equal to 1 in half of the val- 푛 be used to determine a priori cosets of Reed–Muller codes ues of 퐱 ∈ (픽 ) . Equivalently, an 푛-variate Boolean func- 2 that do not contain any balanced function, saving com- tion 퐹 is balanced if the Hamming weight of its truth ta- putational time. For example, we can use the covering ble, 푤 (퐹), is 2푛−1. This property is important in cryp- 퐻 method to identify by inspection 15 coset representatives tographic applications because it follows the principle of (out of 32) in 푅(3, 8)/푅(2, 8) for which at least half of the diffusion: the function has no bias towards a value. The functions in each coset are not balanced and provide con- search for balanced Boolean functions and the develop- structions for these nonbalanced functions [7, 10]. This ment of new methods for constructing them are active ar- can be used to determine a priori types of polynomials to eas of research. avoid in the search for balanced functions. It is easy to see that a Boolean function 퐹 is balanced if and only if 푆(퐹) = 0. If 푆(퐹) has exact 2-divisibility, then 푝-Divisibility of Exponential Sums 푣 (푆(퐹))+1 푝 푝 ∤ 푆(퐹), 푆(퐹) ≠ 0, and 퐹 is not balanced. Hence, As mentioned above, the proof of most of the improve- if one can describe families of Boolean functions with ments and extensions of the Chevalley–Warning theorem exact 2-divisibility, one is describing families of Boolean are nonelementary. Ax and Katz used estimates on the 푝- functions that are not balanced, and this can reduce the divisibility of exponential sums to improve the Chevalley– search for balanced Boolean functions. Warning theorem. Katz [11] obtained that 풩, the number In [10] Hou used the action of the group GL(푛, 2) on of common zeros of polynomials 퐹1, … , 퐹푡 in 픽푞[푋1, … , 푋푛] quotients of Reed–Muller codes 푅(푘, 푛)/푅(푘−1, 푛) to count 휇 of degree 푑1, … , 푑푡, is divisible by 푞 , where 휇 is the small- the number of balanced polynomials in the cosets of est nonnegative integer 휇 ≥ 휇0: 푅(푘−1, 푛). Note that the number of balanced polynomials 푡 in a coset of 푅(푘 − 1, 푛) is included in the weight distribu- 푛 − ∑푖=1 푑푖 휇0 = . tion of the coset. Cosets of 푅(푘 − 1, 푛) belonging to the max {푑푖} same orbit under this action have the same weight distri- Note that the theorem gives information on the 푝- bution and hence the same number of balanced polyno- divisibility of 풩 only if there are “enough variables” 푛; mials. This implies that to know the number of balanced 푡 if 푛 < ∑푖=1 푑푖, 휇 = 0, the conclusion is that 1|풩, and polynomials of all the cosets in an orbit, it is enough to the theorem does not give any information. Adolphson– study a coset representative for the orbit. Cosets of Reed– Sperber [1] improved Katz’s result using a Newton polyhe- Muller codes 퐹 +푅(푘−1, 푛) are sets of deformations of the dra approach, and Moreno–Moreno gave an improvement polynomial 퐹, and one can use Theorem 3 to determine by using the 푝-weight degree of the polynomials instead of nonbalanced polynomials a priori and improve the search their regular degree. The 푝-weight degree of a polynomial for balanced functions. 퐹, 푤푝(퐹) is the maximal 푝-weight degree of its monomials. 퐞 푒1 푒푛 The 푝-weight degree of the monomial 퐗 = 푋1 ⋯ 푋푛 is Example 9. Consider the cosets of 푅(3, 6)/푅(2, 6), 푋1푋2푋3+ 퐞 푤푝 (퐗 ) = 휎푝(푒1) + ⋯ + 휎푝(푒푛), 푋4푋5푋6 +푅(2, 6), and 푋1푋2푋3 +푋4푋5푋6 +푋1푋4푋5 +푋2푋4푋6 + 푟 푟 푋3푋5푋6 + 푅(2, 6). The polynomials in these cosets satisfy where for 푎 = 푎0 + 푎1푝 + ⋯ + 푎푟푝 , 휎푝(푎) = ∑푖=0 푎푖. For the conditions in Examples 6 and 7 and hence have exact 푞 = 푝푓, Moreno–Moreno [13] found that 풩 is divisible by

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 621 휇 푝 , where 휇 is the smallest nonnegative integer 휇 ≥ 휇0, for 휆1, … , 휆푛 ≥ 1, and such that 푠1 + ⋯ + 푠푚 is as small 푠1 푠2 푠푚 푡 as possible [4]. If 푝 = 2, then 퐶 = {퐹1 , 퐹2 , … , 퐹푚 } is a 푛 − ∑ 푤푝(퐹푖) 푖=1 covering for 퐹, as some of the 푠푖 could be zero. This is the 휇0 = 푓 . max {푤푝(퐹푖)} motivation for the definition of a minimal (푝−1)-covering below. Note that the solutions do not depend on the co- Again, note that the theorem gives information on the 푝- efficients of the polynomial 퐹. divisibility of 풩 only if there are “enough variables”; if the number of variables is less than or equal to the sum Definition 2. Let 퐹(퐗) = 푎1퐹1 +푎2퐹2 +⋯+푎푚퐹푚. A set 퐶 = 푠1 푠푚 of the 푝-weight degree of the polynomials, 휇 = 0 and the {퐹1 , … , 퐹푚 } of powers of the monomials in 퐹 is a minimal theorem does not give any information. A tight bound 푠1 푠푚 휆1(푝−1) 휆푛(푝−1) (푝 − 1)-covering of 퐹 if 퐹1 ⋯ 퐹푚 = 푋1 ⋯ 푋푛 for the 푝-divisibility of exponential sums was given by 푚 with 휆푖 ≥ 1 and its size, ∑ 푠푖, is minimal. Moreno et al. in [17]. That and Adolphson–Sperber’s re- 푖=1 sults use the exponents 퐞1, … , 퐞푚 of all the monomials in A (푝−1)-covering need not use all the 퐹푖’s, and therefore 푚 푒1푖 푒푛푖 some of the 푠푖’s could be equal to 0. the polynomial 퐹(퐗) = ∑푖=1 푎푖푋1 ⋯ 푋푛 in contrast to the Chevalley–Warning, Ax–Katz, and Moreno–Moreno re- 2 3 2 3 Example 10. Let 퐹(퐗) = 푋1 푋2 + 푋1 + 푋2 ∈ 픽7 [푋1, 푋2]. sults that use only the degree or the 푝-weight degree of the 2 3 6 2 3 3 2 polynomial. In this sense the result in [17] resembles the Then 퐶1 = {(푋1 푋2 ) } , 퐶2 = {(푋1 ) , (푋2 ) }, and 퐶3 = 2 3 2 2 1 covering method for Boolean polynomials. None of these {(푋1 푋2 ) , (푋1 ) } are 6-coverings of 퐹, and 퐶3 is the unique results can be used to determine if the exponential sum has minimal 6-covering of 퐹 (of size 3). exact 푝-divisibility, and solvability cannot be determined. The covering method for 푝-divisibility. In [6], Castro et A minimal (푝 − 1)-covering of a polynomial 퐹 might al. introduced a generalization to any prime field of the not be unique, and the concept is independent of the coef- covering method introduced in [14] for characteristic 2. ficients of 퐹. However, the exact 푝-divisibility of 푆(퐹) and With it they proved the prime field case of the theorem the solvability of equations involving 퐹 depend on both on the 푝-divisibility of exponential sums presented in [17]. the minimal (푝 − 1)-coverings and the relation among the The new proof was entirely elementary, and, as a conse- coefficients. Also, if there are powers of the monomials quence, elementary proofs and improvements of previous in 퐹 that cover some (but not all) of the variables and are results on the 푝-divisibility of exponential sums were ob- minimal in some sense, it is very hard to determine the tained. exact 푝-divisibility. In Theorem 4 we avoid polynomials For the case 푞 = 푝, the tight bound in [17] relies on find- with this type of minimal partial (푝 − 1)-covering. ing a minimal solution (푠1, … , 푠푚) to a system of 푛 modular Definition 3. Let 퐹(퐗) = 푎1퐹1 +푎2퐹2 +⋯+푎푚퐹푚. A set 퐶 = 푠 푠 equations 푒푗1푠1 + 푒푗2푠2 + ⋯ + 푒푗푚푠푚 ≡ 0 (mod 푝 − 1), asso- 1 푚 {퐹1 , … , 퐹푚 } of powers of the monomials in 퐹 is a partial ciated to the exponents of each variable in the polynomial 푠1 푠푚 휆1(푝−1) 휆푛(푝−1) 푚 푒1푖 푒푛푖 (푝 − 1)-covering of 퐹 if 퐹1 ⋯ 퐹푚 = 푋1 ⋯ 푋푛 퐹(퐗) = ∑푖=1 푎푖푋1 ⋯ 푋푛 . That is, one needs to find solu- with 휆푖 ≥ 0. The set 퐶 is a minimal partial (푝−1)-covering tions to a system 푚 of 퐹 if its size ∑푖=1 푠푖 + 푠(푝 − 1), where 푠 is the number of variables missing, is the size of a minimal (푝 − 1)-covering ⎧푒11푠1 + 푒12푠2 + ⋯ + 푒1푚푠푚 = 휆1(푝 − 1) ⋮ ⋮ of 퐹. ⎨ ⎩푒푛1푠1 + 푒푛2푠2 + ⋯ + 푒푛푚푠푚 = 휆푛(푝 − 1), Note that instead of requiring each exponent 휆푖(푝 − 1) of 푋푖 to be a positive multiple of 푝 − 1, in the definition of 휆푖 ∈ ℕ, where each column corresponds to a term and each 푚 a partial (푝−1)-covering, 휆푖 could be equal to 0, and there- row to a variable, that are minimal in terms of 퐿 = ∑ 푠 . 푖=1 푖 fore some variables could be missing. If 푠 = 0, there are 퐿 In this case, 푣푝(푆(퐹)) ≥ . 푝−1 no variables missing, and we have the previous definition If the system is rewritten as of the (푝 − 1)-covering. 2 3 2 3 푒11 푒1푚 휆1 Example 11. Let 퐹(퐗) = 푋1 푋2 +푋1 +푋2 ∈ 픽7 [푋1, 푋2] be the ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 3 2 푒21 푒2푚 휆2 polynomial of Example 10. Then 퐶4 = {(푋2 ) } is a partial ⎜ ⎟ 푠1 + ⋯ + ⎜ ⎟ 푠푚 = ⎜ ⎟ (푝 − 1), (1) ⎜ ⋮ ⎟ ⎜ ⋮ ⎟ ⎜ ⋮ ⎟ 6-covering of 퐹 of size 2 + 6 = 8. In Example 10 we saw 푒 푒 휆 ⎝ 푛1 ⎠ ⎝ 푛푚 ⎠ ⎝ 푛 ⎠ that the minimal 6-coverings have size 3, and therefore 퐶4 one sees that the solutions that one is looking for are ex- is not a minimal partial 6-covering of 퐹. 푒1푖 푒푛푖 ponents 푠푖 for the monomials 퐹푖 = 푋1 ⋯ 푋푛 in 퐹 such By avoiding minimal partial (푝 − 1)-coverings we can that improve previous results by computing exact 푝-divisibility 푠1 푠2 푠푚 휆1(푝−1) 휆푛(푝−1) 퐹1 퐹2 ⋯ 퐹푚 = 푋1 ⋯ 푋푛 , of exponential sums or improving previous bounds. The

622 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 next result [4] is a generalization of Theorem 2 to any char- for 푃 is acteristic. 푝−1 푝−1 2 4 2 6 2 2 퐶 = {(푌1푋1 푋2 ) , (푌1푋3 푋4 ) , Theorem 4. Let 퐶1, … , 퐶푐 be all the minimal (푝 − 1)- 퐿 퐹 = 푎 퐹 + ⋯ + 푎 퐹 푝−1 푝−1 coverings of size of a polynomial 1 1 푚 푚, 2 2 2 6 2 2 푠푖1 푠푖푚 (푌2푋5 푋6 ) , (푌2푋7 푋8 ) } , 퐶푖 = {퐹1 , … , 퐹푚 }, and suppose that any minimal partial 퐿 (푝 − 1)-covering is one of the 퐶푖’s. Then, 푣푝 (푆(퐹)) = there are no minimal partial (푝−1)-coverings with missing 푝−1 푠1 푠4 푠푖1 푠푖푚 푎1 ⋯푎4 푐 푎1 ⋯푎푚 variables, and ≢ 0 (mod 푝). This implies that the if ∑푖=1 ≢ 0 (mod 푝), and otherwise 푣푝 (푆(퐹)) ≥ 푠1!⋯푠4! 푠푖1!⋯푠푖푚! 푝−1 퐿 exact 푝-divisibility of 풩 is 푣푝(풩) = 4 ( ) /(푝−1)−2 = 0. + 1. 2 푝−1 Therefore, 푝 ∤ 풩, 풩 ≠ 0, and the system is solvable for any ∗ 푠푖1 푠푖푚 훼, 훽 ∈ 픽 . 푐 푎1 ⋯푎푚 푝 The condition ∑푖=1 ≢ 0 (mod 푝) is the gener- 푠푖1!⋯푠푖푚! By imposing conditions on polynomials 퐺 so that 퐹 +퐺 alization of the number 푐 of minimal coverings being odd and 퐹 have the same minimal (푝 − 1)-coverings, one can in the case of 푝 = 2. The condition of the minimal par- extend known results on the 푝-divisibility of 푆(퐹) to results tial (푝 − 1)-coverings being one of the 퐶 ’s implies that any 푖 on 푆(퐹 + 퐺) for deformations of 퐹. minimal partial (푝 − 1)-covering does not have a missing variable. Similarly to the case where 푝 = 2, a sufficient Example 13. Let 풩 be the number of solutions of the sys- condition for not having minimal partial (푝−1)-coverings tem with missing variables is to require that for each of the min- 푝−1 푝−1 푎푋1 + ⋯ + 푎푋푝 + 퐺 = 0, imal (푝− 1)-coverings 퐶 , each monomial in 퐶 has at least 푖 푖 푏 푋 + ⋯ + 푏 푋 + 훼 = 0, two variables that are not present in the other monomials 1 1 푝 푝 (2) ∗ of 퐶푖. This simple condition provides families of polyno- where 푎, 푏푖 ∈ 픽푝 , 훼 ∈ 픽푝, 퐺 ∈ 픽푝[퐗], and deg 퐺 < 푝 − 1. mials for which the “greater than or equal to” relation ob- This system has 푝 variables, and sum of the degree and tained in the classical results on 푝-divisibility is replaced of the 푝-weight degree of the polynomials is also 푝. Hence by either equality or strict inequality. Ax–Katz and Moreno–Moreno’s theorems do not give any Theorem 4, when applied to the number of solutions information on solvability or the 푝-divisibility of 풩. There of systems of polynomial equations, gives refinements to are 푝 different minimal (푝 − 1)-coverings with form the prime field case of many of the known results by giving 푝−1 {푌 푋푝−1, … , 푌 푋푝−1, (푌 푋 ) } precise conditions for when the system is solvable or the 1 푖1 1 푖푝−1 2 푖푝 푝−1 bound on the 푝-divisibility of the number of solutions 풩 푝 푎푝−1푏 푝 and size 퐿 = 2(푝 − 1). Since ∑ 푖 = , we is improved. Moreover, it can also give information on 푖=1 (푝−1)! (푝−1)! the solvability or 푝-divisibility of 풩 for cases that are not 2(푝−1) have 푣푝(풩) = −2 + 푣푝 (푆(푃)) > −2 + = 0. The result covered by previous theorems. 푝−1 does not give information about solvability but gives some Example 12. Let 푝 ≠ 2 and consider the system information about the 푝-divisibility of 풩.

2 4 6 2 푋1 푋2 + 푋3 푋4 + 푋1 + 푋2 + 푋6 + 푋7 = 훼, Conclusions 2 2 6 2 푋5 푋6 + 푋7 푋8 + 푋3 + 푋5 + 푋8 = 훽 The covering method is an elementary method to obtain information about the 푝-divisibility of exponential sums. 픽∗ over 푝. Note that the system has 8 variables and the sum It provides an intuitive approach to the computation of of the degree of the polynomials is 16; hence Ax–Katz’s exact 푝-divisibility that can be exploited in applications. theorem does not give any information on solvability nor It also gives a simple way to construct families of systems 푝 풩 푝 = 3, 5 푝 > 5 -divisibility of . For and the sum of of polynomial equations that are solvable, determine 푝- 푝 8, 10 the -weight degree of the polynomials is , and 16, divisibility of the number of solutions of systems for cases respectively; hence Moreno–Moreno’s theorem does not where previous results do not give information, and can give any information either. be applied to answer questions in coding theory and cryp- To use the covering method, we first compute the poly- tography. nomial associated to this system:

2 4 6 2 푃 = 푌1 (푋1 푋2 + 푋3 푋4 + 푋1 + 푋2 + 푋6 + 푋7 − 훼)

2 2 6 2 + 푌2 (푋5 푋6 + 푋7 푋8 + 푋3 + 푋5 + 푋8 − 훽) . It is easy to see that the unique minimal (푝 − 1)-covering

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624 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Higgs Bundles— Recent Applications

Laura P. Schaposnik

Introduction • Via the nonabelian Hodge correspondence de- This note is dedicated to introducing Higgs bundles and veloped by Corlette, Donaldson, Simpson, and the Hitchin fibration, with a view towards their appearance Hitchin and in the spirit of Uhlenbeck–Yau’s work within different branches of mathematics and physics, fo- for compact groups, the is analyti- cusing in particular on the role played by the integrable cally isomorphic as a real manifold to the de Rahm system structure carried by their moduli spaces. On a com- moduli space ℳ푑푅 of flat connections on a smooth pact Σ of genus 푔 ≥ 2, Higgs bundles are complex bundle. pairs (퐸, Φ) where • Via the Riemann–Hilbert correspondence there is a complex analytic isomorphism between the de • 퐸 is a holomorphic vector bundle on Σ, and Rham space and the Betti moduli space ℳ of sur- • the Higgs field Φ ∶ 퐸 → 퐸 ⊗ 퐾 is a holomorphic 퐵 face group representations 휋 (Σ) → 퐺 . map for 퐾 ≔ 푇∗Σ. 1 ℂ Since their origin in the late 1980s in work of Hitchin Some prominent examples where these moduli spaces ap- and Simpson, Higgs bundles manifest as fundamental ob- pear in mathematics and physics are: jects that are ubiquitous in contemporary mathematics • Through the Hitchin fibration, ℳ퐺ℂ gives exam- and closely related to theoretical physics. For 퐺ℂ a com- ples of hyperkähler manifolds that are integrable plex semisimple Lie group, the Dolbeault moduli space of systems, leading to remarkable applications in

퐺ℂ-Higgs bundles ℳ퐺ℂ has a hyperkähler structure, and physics, which we shall discuss later on. via different complex structures it can be seen as different • Building on the work of Hausel and Thaddeus re- moduli spaces: lating Higgs bundles to Langlands duality, Donagi

and Pantev presented ℳ퐺ℂ as a fundamental ex- Laura P. Schaposnik is a professor of mathematics at the University of Illinois ample of mirror symmetry. [email protected] at Chicago. Her email address is . • Within the work of Kapustin and Witten, Higgs The author was supported in part by the NSF grant DMS-1509693 and the NSF bundles were used to obtain a physical derivation CAREER Award DMS-1749013. This material is based upon work supported by the NSF under Grant No. DMS1440140 while the author was in residence of the geometric Langlands correspondence through at the Mathematical Sciences Research Institute in Berkeley, California, during mirror symmetry. Soon after, Ngô found Higgs the Fall 2019 semester. bundles to be key ingredients when proving the Communicated by Notices Associate Editor Chikako Mese. fundamental lemma of the Langlands program, For permission to reprint this article, please contact: which led him to the Fields Medal a decade ago. [email protected]. Higgs bundles and the corresponding Hitchin inte- DOI: https://doi.org/10.1090/noti2074 grable systems have been an increasingly vibrant area, and

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 625 thus there are several expository articles, some of which we SL(2, ℂ)-Higgs bundles (퐸, Φ푎) parametrized by quadratic shall refer to: from the Notices article “What is... a Higgs differentials 푎 ∈ 퐻0(Σ, 퐾2) is given by bundle?” [3], to several graduate notes on Higgs bundles (e.g., the author’s recent [18]), to more advanced reviews 0 푎 (퐸 = 퐾1/2 ⊕ 퐾−1/2,Φ = ( )) . (3) such as Ngô’s 2010 ICM Proceedings article [4]. Hoping 푎 1 0 to avoid repeating material nicely covered in other reviews whilst still attempting to inspire the reader to learn more One may also consider 퐺-Higgs bundles for 퐺 a real about the subject, we shall take this opportunity to focus form of 퐺ℂ, which in turn correspond to the Betti moduli on some of the recent work done by leading young mem- space of representations 휋1(Σ) → 퐺. For example, SL(2, ℝ)- 1 bers of the community. Higgs bundles are pairs (퐸 = 퐿 ⊕ 퐿∗, Φ) for 퐿 a line bundle and Φ off diagonal, a family of which is described in Ex- Higgs Bundles ample 1. Higgs bundles arise as solutions to self-dual Yang–Mills In order to define a Hausdorff moduli space of Higgs equations, a nonabelian generalization of Maxwell’s equa- bundles, one needs to incorporate the notion of stability. tions that recurs through much of modern physics. Solu- For this, recall that holomorphic vector bundles 퐸 on Σ tions to Yang–Mills self-duality equations in Euclidean 4D are topologically classified by their rank 푟푘(퐸) and their space are called , and when these equations are degree deg(퐸), through which one may define their slope reduced to Euclidean 3D space by imposing translational as 휇(퐸) ≔ deg(퐸)/푟푘(퐸). Then, a vector bundle 퐸 is stable invariance in one dimension, one obtains monopoles as (or semistable) if for any proper subbundle 퐹 ⊂ 퐸 one has solutions. Higgs bundles were introduced by Hitchin in that 휇(퐹) < 휇(퐸) (or 휇(퐹) ≤ 휇(퐸)). It is polystable if it is a [10] as solutions of the so-called Hitchin equations, the 2- direct sum of stable bundles whose slope is 휇(퐸). dimensional reduction of the Yang–Mills self-duality equa- One can generalize the stability condition to Higgs bun- tions given by dles (퐸, Φ) by considering Φ-invariant subbundles 퐹 of 퐸, vector subbundles 퐹 ⊂ 퐸 for which Φ(퐹) ⊂ 퐹 ⊗ 퐾. A Higgs 퐹 + [Φ, Φ∗] = 0, 휕 Φ = 0, (1) 퐴 퐴 bundle (퐸, Φ) is said to be stable (semistable) if for each where 퐹퐴 is the curvature of a unitary connection ∇퐴 = proper Φ-invariant 퐹 ⊂ 퐸 one has 휇(퐹) < 휇(퐸) (equiv. ≤). 휕퐴 + 휕퐴 associated to a Dolbeault operator 휕퐴 on a holo- Then, by imposing stability conditions, one can construct ℳ 퐺 morphic principal 퐺ℂ bundle 푃. The equations give a flat the moduli space 퐺ℂ of stable ℂ-Higgs bundles up to connection holomorphic automorphisms of the pairs (also denoted ℳ ). Going back to Hitchin’s equations, one of the ∗ 퐷표푙 ∇퐴 + Φ + Φ (2) most important characterizations of stable Higgs bundles and express the harmonicity condition for a metric in the is given in the work of Hitchin and Simpson and which carries through to more general settings: If a Higgs bun- resulting flat bundle. Concretely, principal 퐺ℂ-Higgs bun- dles are pairs (푃, Φ) where dle (퐸, Φ) is stable and deg 퐸 = 0, then there is a unique unitary connection ∇ on 퐸, compatible with the holo- • 푃 is a principal 퐺 -bundle, and 퐴 ℂ morphic structure, satisfying (1). • Φ is a holomorphic section of ad(푃) ⊗ 퐾. Finally, Hitchin showed that the underlying smooth We shall refer to classical Higgs bundles as those described manifold of solutions to (1) is a hyperkähler manifold, in the introduction and consider 퐺ℂ-Higgs bundles in their with a natural symplectic form 휔 defined on the infinitesi- vector bundle representation: seen as classical Higgs bun- mal deformations (퐴,̇ Φ)̇ of a Higgs bundle (퐸, Φ) by dles satisfying some extra conditions reflecting the nature of 퐺ℂ, dictated by the need for the (projectively) flat con- 휔((퐴̇ , Φ̇ ), (퐴̇ , Φ̇ )) = ∫ (퐴̇ Φ̇ − 퐴̇ Φ̇ ), nection to have holonomy in 퐺ℂ. For instance, when 1 1 2 2 tr 1 2 2 1 (4) Σ 퐺ℂ = SL(푛, ℂ), a 퐺ℂ-Higgs bundle (퐸, Φ) is composed of a holomorphic rank 푛 vector bundle 퐸 with trivial deter- 퐴̇ ∈ Ω0,1( 퐸) Φ̇ ∈ Ω1,0( 퐸) minant Λ푛퐸 ≅ 풪 and a Higgs field satisfying Tr(Φ) = 0, where End0 and End0 . Moreover, 0 he presented a natural way of studying the moduli spaces for which we shall write Φ ∈ 퐻 (Σ, End0(퐸) ⊗ 퐾). ℳ퐺ℂ of 퐺ℂ-Higgs bundles through what is now called the Example 1. Choosing a square root of 퐾, consider the Hitchin fibration, which we shall consider next. vector bundle 퐸 = 퐾1/2 ⊕ 퐾−1/2. Then, a family of Integrable Systems {푝 , … , 푝 } 1As in other similar reviews, the number of references is limited to twenty, and Given a homogeneous basis 1 푘 for the ring of in- thus we shall refer the reader mostly to survey articles where precise references variant polynomials on the Lie algebra 픤ℂ of 퐺ℂ, we denote can be found. by 푑푖 the degree of 푝푖. The Hitchin fibration, introduced in

626 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 [11], is then given by the line bundle reflecting the nature of 퐺ℂ. This ap- 푘 proach originates in the work of Hitchin and of Beauville, ℎ ∶ ℳ ⟶ 풜 ≔ 퐻0(Σ, 퐾푑푖 ), Narasimhan, and Ramanan (see [18] for references), and 퐺ℂ 퐺ℂ ⨁ 푖=1 we shall describe here an example to illustrate the setting. For SL(푛, ℂ)-Higgs bundles, the linear term in (5) vanishes (퐸, Φ) ↦ (푝1(Φ), … , 푝푘(Φ)). since Tr(Φ) = 0, and the generic fibres are isomorphic to ℎ The map is referred to as the Hitchin map: it is a proper Prym varieties Prym(푆, Σ) since Λ푛퐸 ≅ 풪. map for any choice of basis and makes the moduli space into an whose base and fibres have di- Example 2. For rank two Higgs bundles, we return to Ex- ample 1 in which the Hitchin fibration is over 퐻0(Σ, 퐾2) mension dim(ℳ퐺ℂ )/2. In what follows we shall restrict our attention to GL(푛, ℂ)-Higgs bundles, which are those and the Hitchin map is ℎ ∶ (퐸, Φ) ↦ −det(Φ). The family Higgs bundles introduced in the first paragraph of these (퐸, Φ푎) gives a section of the Hitchin fibration: a smooth notes and whose Hitchin fibration in low dimension is de- map from the Hitchin base to the fibres, known as the picted in Figure 1. Hitchin section. Moreover, this comprises a whole compo- nent of the moduli space of real SL(2, ℝ)-Higgs bundles in-

side ℳ퐺ℂ , which Hitchin identified with Teichmüller space and which is now referred to as a Hitchin component or Te- ichmüller component. Recall that the Teichmüller space 풯(푆) of the underlying surface 푆 of Σ is the space of marked con- formal classes of Riemannian metrics on 푆. In the early 1990s Hitchin showed that for any split group 퐺, e.g., for the split form SL(푛, ℝ) of SL(푛, ℂ), the above components are homeomorphic to a vector space of dimension dim(퐺)(2푔 − 2) and conjectured that they should parametrize geometric structures. These spaces pre- Figure 1. An example of a Hitchin fibration. sented the first family of higher Teichmüller spaces within the Betti moduli space of reductive surface group representa- ℳ (퐺) The generic or regular fibre of the Hitchin fibration— tions 퐵 , which leads us to applications of Higgs bun- 퐺 퐺 appearing in violet in Figure 1—is an abelian variety, dles within higher Teichmüller theory for real forms of ℂ. leading to what is referred to as the abelianization of the Higher Teichmüller Theory moduli space of Higgs bundles and which can be seen ge- The moduli spaces of 퐺-Higgs bundles have several con- ometrically by considering eigenvalues and eigenspaces of nected components. For a split real form 퐺 of 퐺 , the the Higgs field. Indeed, a Higgs bundle (퐸, Φ) defines a ℂ Hitchin component of 퐺-Higgs bundles, or equivalently ramified cover 휋 ∶ 푆 → Σ of the Riemann surface given of surface group representations, can be defined as the by its eigenvalues and obtained through its characteristic connected component of the Betti moduli space ℳ (퐺) equation, 퐵 containing Fuchsian representations in 퐺, which are rep- 푆 = {det(Φ − 휂) = 0} ⊂ Tot퐾, (5) resentations obtained by composing a discrete and faith- for 휂 the tautological section of 휋∗퐾. This cover allows one ful representation 휌 ∶ 휋1(Σ) → SL(2, ℝ) (classically called to construct the spectral data associated to generic (퐸, Φ) Fuchsian) with the unique (up to conjugation) irreducible given by: representation SL(2, ℝ) → 퐺. Moreover, as mentioned be- • the spectral curve 푆 from (5), generically smooth, fore, these representations, called Hitchin representations, defining a generic point in the Hitchin base, since are considered the first example of higher Teichmüller the coefficients of {det(Φ − 휂) = 0} give a basis of space for surfaces: a component of the set of representa- invariant polynomials, and tions of discrete groups into Lie groups of higher ranks • a line bundle on 푆, defining a point in the Hitchin consisting entirely of discrete and faithful elements. In fibre and obtained as the eigenspace of Φ. order to give a geometric description of Hitchin represen- tations and motivated by dynamical properties, Labourie For classical Higgs bundles, the smooth fibres are Jacobian introduced the notion of Anosov representations, which can varieties Jac(푆), and one recovers (퐸, Φ) up to isomorphism be thought of as a generalization of convex-cocompact rep- from the data (푆, 퐿 ∈ Jac(푆)) by taking the direct images resentations to Lie groups 퐺 of higher real rank.2 퐸 = 휋∗퐿 and Φ = 휋∗휂. When considering 퐺ℂ-Higgs bundles, one has to re- 2For example, for representations in SL(2, ℂ), these are quasi-Fuchsian quire appropriate conditions on the spectral curve and representations.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 627 As beautifully described in Wienhard’s ICM Proceed- coordinate charts on 푀 with values in 푋 such that the tran- ings article [20], building on Labourie’s work, higher sition maps are given by elements of 퐺. Higgs bundles Teichmüller theory recently emerged as a new field in have played a key role in describing the closed manifold mathematics, closely related to Higgs bundles (see also on which (퐺, 푋)-structures live. For example, Higgs bun- [5, 13]). There are two known families of higher Teich- dles were used to show that maximal representations to müller spaces, giving the only known examples of compo- 푃푂(2, 3) give rise to (퐺, 푋)-manifolds when 푋 is the space nents that consist entirely of Anosov representations for of null geodesics (photons) in particular Einstein mani- surfaces: folds and when 푋 = ℙ(ℝ5) (e.g., see [5]). For an excellent (I) the space of Hitchin representations into a real review of geometric structures, see Kassel’s ICM Proceed- split simple Lie group 퐺 and ings [13]. (II) the space of maximal representations into a Her- Harmonic Metrics mitian Lie group 퐺. Equivariant harmonic maps play an important role in A representation 휌 ∶ 휋 (Σ) → 퐺 is maximal if it maximizes 1 the nonabelian Hodge correspondence mentioned before the Toledo invariant 푇(휌), a topological invariant defined (and beautifully reviewed in [3]), and thus we shall devote for any simple Lie group 퐺 of Hermitian type as this section to some of the advances made in this direc- 1 ∫ 푓∗휔 (6) tion. In our setting, from the work of Corlette and Don- 2휋 Σ aldson, any reductive representation 휌 ∶ 휋1(Σ) → 퐺ℂ has for 휔 the invariant Kähler form on the Riemannian sym- associated a 휌-equivariant harmonic map 푓 from the uni- metric space, and 푓 ∶ Σ̃ → 푋 any 휌-equivariant smooth versal cover Σ̃ of Σ to the corresponding symmetric space map. of 퐺ℂ, which in turn defines a Higgs bundle (퐸, Φ). Re- call that a map 푓 ∶ Σ̃ → 푀 is called 휌-equivariant if Example 3. The Toledo invariant can be expressed in 푓(훾 ⋅ 푥) = 휌(훾) ⋅ 푓(푥) for all 푥 ∈ Σ̃ and 훾 ∈ 휋1(Σ). Moreover, terms of Higgs bundles. For example, for SL(2, ℝ)-Higgs Σ ∗ through a choice of metric on , one may define the energy bundles (퐿 ⊕ 퐿 , Φ), the Toledo invariant is 2 deg(퐿) and density satisfies 0 ≤ |2 deg(퐿)| ≤ 2푔 − 2. Hence, the family (퐸, Φ푎) from Example 1 is maximal. 1 푒(푓) = ⟨푑푓, 푑푓⟩ ∶ Σ̃ → ℝ, (7) 2 The existence of spaces other than those in (I) and (II) with similar properties to Teichmüller space is a topic of which is 휌-invariant and descends to Σ. Then, the energy of significant investigation. Expected candidates are spaces 푓 is defined as of 휃-positive representations conjectured by Guichard– Wienhard, some of which were shown to exist via Higgs 퐸(푓) = ∫ 푒(푓)푑Vol. (8) bundles [1]. Σ Whilst Anosov representations give a clear link be- It depends only on the conformal class and is finite since tween discrete and faithful representations and geometric Σ is compact. The map 푓 is harmonic if it is a critical point structures, there is no known Higgs bundle characteriza- of the energy functional 퐸(푓) in (8). tion of Anosov representations, and very little is known Conversely, through the work of Hitchin and Simpson, about which explicit geometric structures correspond to a polystable Higgs bundle admits a Hermitian metric ℎ these spaces. For instance, work of Choi and Goldman on the bundle such that the associated Chern connection shows that the holonomy representations of convex pro- 퐴 solves the Hitchin equations (1), and such a metric is jective structures are the Hitchin representations when 퐺 = called harmonic. Moreover, the harmonic metric induces PSL(3, ℝ). a completely reducible representation 휌 ∶ 휋 (Σ) → 퐺 Whilst there is no Higgs bundle characterization of 1 ℂ and a 휌-equivariant harmonic map into the correspond- Anosov representations,3 Higgs bundles have been an ef- ing symmetric space. These two directions together give fective tool for describing these structures. This brings the celebrated nonabelian Hodge correspondence. us to one of the fundamental problems in modern ge- Understanding the geometric and analytic properties of ometry: the classification of geometric structures admit- the harmonic maps arising from Hitchin’s equations (1) ted by a manifold 푀. Recall that a model geometry is a is of significant importance. For instance, one may ask pair (퐺, 푋) where 푋 is a manifold (model space) and 퐺 is how these metrics behave at the boundaries of the mod- a Lie group acting transitively on 푋 (group of symmetries). uli space or how the energy densities of the corresponding A (퐺, 푋)-structure on a manifold 푀 is a maximal atlas of harmonic maps at different points of the Hitchin fibration 3Anosov representations are holonomy representations of geometric structures relate to each other (the reader may be interested in the on certain closed manifolds. reviews [15] and [7] and references therein).

628 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 From Hitchin’s work, the moduli space of Higgs bun- underlying surface 푆 in which the 휌-equivariant harmonic dles has a natural ℂ∗-action 휆 ⋅ (퐸, Φ) = (퐸, 휆Φ), whose metric is a minimal immersion. In particular, Labourie fixed point sets allow one to study different aspects ofthe showed that for all Anosov representations such a con- topology and the geometry of the space, as done in [11] formal structure exists, but the difficulty lies in proving (see also [5, 16]). Moreover, as shown by Simpson, the uniqueness; the conjecture has been established only for fixed points of this action are complex variations ofHodge Lie groups of rank two ([5, 14]). To understand this prob- structure (VHS). Recall that a VHS is a ℂ∞ vector bundle lem, one may consider the study of deformations of con- 푝,푞 푉 with a Hodge decomposition 푉 = ⨁푝+푞 푉 , a ratio- formal structures on surfaces and the corresponding har- nal structure, and a flat connection satisfying the axioms monic metric. of Griffiths transversality and existence of a polarization. Some of these deformations can be seen through the hy- From the above, one may ask how the energy density of perkähler structure of the moduli space, by virtue of which harmonic maps changes along the ℂ∗-flow on the moduli it has a ℂℙ1-worth of complex structures labelled by a pa- space of Higgs bundles. Whilst this remains a challenging rameter 휉. Indeed, we can think of a hyperkähler mani- open question in the area, a better understanding might fold as a manifold whose tangent space admits an action come from the following conjectural picture of Dai–Li de- of three complex structures 퐼, 퐽, and 퐾 satisfying the quater- scribed in Figure 2 and through which the harmonic map nionic equations and compatible with a single metric. In ∗ our case, 퐼 arises from the complex structure on the Rie- of a fixed point set of the ℂ action on ℳ퐺ℂ gives rise to two other related harmonic maps. mann surface Σ, while 퐽 is from the complex structure on the group 퐺ℂ. In this setting, one has the following moduli spaces: • for 휉 = 0 the space of Higgs bundles, • for 휉 ∈ ℂ× the space of flat connections4 −1 ̄ ∗ ∇휉 = 휉 Φ + 휕퐴 + 휕퐴 + 휉Φ , (10) • for 휉 = ∞ the space of “anti-Higgs bundles.” The hyperkähler metric on Hitchin moduli space is ex- pected to be of type “quasi-ALG,” which is some expected generalization of ALG. A far-reaching open question is the understanding of the behavior of the metrics at the bound- Figure 2. The nilpotent cone in red over the 0 and the points 퐴, 퐵, and 퐶 lying over the ℂ∗-flow and over the Hitchin section, aries of the space, for instance along a path in the Hitchin respectively. base via the limit ̄ lim(휕퐴, 푡Φ). 푡→∞ A point 퐴 within the Hitchin fibration naturally deter- Almost a decade ago Gaiotto–Moore–Neitzke gave a mines two other points: the point 퐵, which is the limit of conjectural description of the hyperkähler metric on ℳ퐺ℂ the ℂ∗-flow 휆 ⋅ 퐴 as 휆 → 0 in the nilpotent cone, and the near infinity, which surprisingly suggests that much of the point 퐶, which is the intersection point of the Hitchin fi- asymptotic geometry of the moduli space can be derived bre containing 퐴 and the Hitchin section. Then Dai–Li’s from the abelian spectral data. Recent progress has been conjecture states that the energy densities defined as in (7) made by Mazzeo–Swoboda–Weiss–Witt, Dumas–Neitzke, of the corresponding harmonic maps 푓퐴, 푓퐵, 푓퐶 satisfy and Fredrickson, but the global picture remains open (see [7]). 푒(푓 ) < 푒(푓 ) < 푒(푓 ). (9) 퐵 퐴 퐶 Finally, one further type of limiting structure we would As evidence for the above conjecture, one can consider the like to mention is that of opers, appearing as certain limits integral version (through (8)) for which Hitchin showed of Higgs bundles in the Hitchin components. To see this, that 퐸(푓퐵) < 퐸(푓퐴), but where the other corresponding note that for a solution of (1) in the SL(푛, ℂ)-Hitchin sec- inequality in (9) remains open. tion, one can add a real parameter 푅 > 0 to (10) to obtain a natural family of connections5 Limiting Structures ∗ ∇(휉, 푅) ≔ 휉−1푅Φ + 휕̄ + 휕 + 휉푅Φ . (11) The study of 휌-equivariant harmonic metrics and higher 퐴 퐴 Teichmüller theory through Higgs bundles has received Gaiotto conjectured that the space of opers (a gener- much attention in recent years and brings us to one of alization of projective structures that, like the Hitchin the most important conjectures in the area. This conjec- 4In particular, for 휉 = 1 we recover (2). ture, due to Labourie, states that for each Hitchin repre- 5 ∗ Note that 휕퐴 and Φ depend on the scaling parameter 푅. Indeed, one needs to sentation 휌 there is a unique conformal structure 푋휌 on the use the metric solving the Hitchin equation for 푅Φ to get a flat connection.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 629 section, is parametrized by the Hitchin base) should be obtained as the ℏ-conformal limit of the Hitchin section: taking 푅 → 0 and 휉 → 0 simultaneously while holding the ratio ℏ = 휉/푅 fixed. The con- jecture was recently established for general simple Lie groups by Dumistrescu–Fredrickson–Kydonakis–Mazzeo– Mulase–Neitzke, who also conjectured that this oper is the quantum curve in the sense of Dumitrescu–Mulase, a quantization of the spectral curve 푆 of the corresponding Higgs bundle by topological recursion (see [6]). More re- Figure 3. A star-shaped quiver. cently, Collier–Wentworth showed that the above confor- mal limit exists in much more generality and gives a cor- hyperpolygons as in Figure 3 may be identified with a mod- respondence between (Lagrangian) strata for every stable uli space of certain rank 푟 parabolic Higgs bundles on ℙ1. VHS—and not only the Hitchin components. Specifically, In this setting, one has to puncture ℙ1 along a positive they constructed a generalization of the Hitchin section divisor 퐷 and then regard the Higgs field as being valued in 0 by considering stable manifolds 풲 (퐸0,Φ0) arising from 풪(푞) = 퐾⊗풪(퐷), with poles along 퐷 and satisfying certain each VHS (퐸0,Φ0) given by conditions on its residues at the poles. This takes us to a generalization of Higgs bundles on higher genus surfaces {(퐸, 휙) ∈ ℳ퐺ℂ | lim 푡 ⋅ (퐸, Φ) = (퐸0,Φ0)}. (12) 푡→0 obtained by allowing the Higgs field to have poles, leading The analog of the Hitchin section is then obtained by pa- to the moduli spaces of tame or parabolic Higgs bundles 0 rameterizing 풲 (퐸0,Φ0) with a slice in the space of Higgs (for logarithmic singularities) initiated by Simpson [19] or bundles through a global slice theorem, analogous to the of wild Higgs bundles (for higher order poles) initiated by definition of the Hitchin section. Boalch and Biquard; see references in [2] to learn more about these other settings. Understanding the more gen- Correspondences eral appearance of parabolic (and wild) Higgs bundles on The appearance of Higgs bundles as parameter spaces for higher genus Riemann surfaces in terms of hyperpolygons geometric structures is an example of the study of corre- remains an open question. spondences between solutions to Hitchin’s equations (1) In a different direction, given a fixed Riemann sur- and different mathematical objects. In what follows we face Σ and a homomorphism between two Lie groups ′ shall restrict our attention to a few correspondences be- Ψ ∶ 퐺ℂ → 퐺ℂ, there is a naturally induced map between tween Higgs bundles and two classes of mathematical ob- the Betti moduli spaces of representations jects: quiver varieties and hyperpolygons (e.g., see refer- Ψ ∶ ℳ (퐺 ) → ℳ (퐺′ ). ences in [12, 16]). 퐵 ℂ 퐵 ℂ Recall that a quiver 푄 = (푉, 퐴, ℎ, 푡) is an oriented graph It follows that there must be a corresponding induced map consisting of a finite vertex set 푉, a finite arrow set 퐴, and between the Higgs bundle moduli spaces, but this does head and tail maps ℎ, 푡 ∶ 퐴 → 푉. A Nakajima repre- not transfer readily to the induced map on the Hitchin fi- sentation of a quiver 푄 can be written as families 푊 ≔ brations, in particular since the image might be over the ((푊푣), 휙푎, 휓푎) for 푎 ∈ 퐴 and 푣 ∈ 푉, where 푊푣 is a finite- singular locus of the base. When the maps arise through dimensional vector space, the map 휙푎 ∶ 푊푡(푎) → 푊ℎ(푎) is a isogenies, together with Bradlow and Branco, the author linear map for all 푎 ∈ 퐴, and 휓푎 is in the cotangent space obtained a description of the map for spectral data in terms to Hom(푊푡(푎), 푊ℎ(푎)) at 휙푎. In particular, a hyperpolygon is of fibre products of spectral curves [18], but of much in- a representation of the star-shaped quiver, an example of terest is the understanding of other maps arising in this which appears in Figure 3. manner. For the star-shaped quiver in Figure 3, for which the di- mensions of 푊푣 are indicated in each vertex, the cotangent space 푇∗Rep(푄) of representations of 푄 is 푛 푇∗ ( Hom(ℂ, ℂ푟)) = 푇∗ (Hom(ℂ푛,ℂ푟)) . Figure 4. A degeneration of the Riemann surface. ⨁ 푖=1 Konno showed that hyperpolygon spaces are hyper- Finally, when considering compactifications of the kähler analogs of polygon spaces, which are representa- moduli space, one may ask how the moduli spaces trans- tion spaces of the star-shaped quivers with simple arrows. form when the base Riemann surface Σ changes (for in- Moreover, through the work of Fisher–Rayan, the space of stance, when degenerating the surface Σ as in Figure 4), a

630 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 question closely related to the relation between Higgs bun- • Abelianization—zero-dimensional intersection. dles and singular geometry, which we shan’t touch upon When 퐺 is a split real form, the author showed here; see [2] for a survey and open problems in this direc- that the (퐵, 퐴, 퐴)-brane intersects the Hitchin fibra- tion. tion in torsion two points. • Abelianization—positive dimensional intersection. Mirror Symmetry and Branes Moreover, we can also show that for other real One of the most interesting correspondences of Hitchin groups such as SU(푛, 푛), the intersection has pos- 퐿 systems arises through mirror symmetry. For 퐺ℂ the Lang- itive dimension but may still be described solely lands dual group of 퐺ℂ, there is an identification of the in terms of abelian data. 퐿 Hitchin bases 풜퐺ℂ ≃ 풜 퐺ℂ . The two moduli spaces ℳ퐺ℂ • Cayley/Langlands type correspondences. 퐿 and ℳ 퐺ℂ are then torus fibrations over a common base, Surprisingly, many spaces of Higgs bundles cor- and through the famous SYZ conjecture, mirror symmetry responding to nonabelian real gauge theories should manifest as a duality between the spaces of Higgs do admit abelian parametrizations via auxiliary bundles for Langlands dual groups fibred over the same spectral curves, as shown with Baraglia through base via the Hitchin fibration. As first observed by Hausel– Cayley/Langlands type correspondences for the Thaddeus for SL(푛, ℂ) and PGL(푛, ℂ)-Higgs bundles and groups 퐺 = SO(푝 + 푞, 푝) and 퐺 = Sp(2푝 + 2푞, 2푝). shown by Donagi and Pantev for general pairs of Lang- • Nonabelianization. lands dual reductive groups, the nonsingular fibres are in- Together with Hitchin we initiated the study deed dual abelian varieties. Kapustin and Witten gave a of branes that don’t intersect the regular lo- physical interpretation of this in terms of S-duality, using it cus through the nonabelianization of Higgs bun- as the basis for their approach to the geometric Langlands dles, which characterized the branes for 퐺 = program. SL(푛, ℍ), SO(푛, ℍ), and Sp(푛, 푛) in terms of non- The appearance of Higgs bundles (and flat connections) abelian data given by spaces of rank two vector within and the geometric Langlands program bundles on the spectral curves. has led researchers to study the derived category of coherent Moreover, it has been conjectured (Baraglia–Schapos- sheaves (퐵-model) and the Fukaya category (퐴-model) of nik) that the Langlands dual in (13) to the above (퐵, 퐴, 퐴)- these moduli spaces. It then becomes fundamental to un- branes should correspond to the (퐵, 퐵, 퐵)-branes of Higgs derstand Lagrangian submanifolds of ℳ (the support of 퐺ℂ bundles with structure group the Nadler group [17]. More 퐴-branes), and their dual objects (the support of 퐵-branes). generally, branes of Higgs bundles have proven notori- By considering the support of branes, we shall refer to a ously difficult to compute in practice, and very few broad submanifold of a hyperkähler manifold as being of type 퐴 classes of examples are known; e.g., see [18] for a partial or 퐵 with respect to each of the complex structures (퐼, 퐽, 퐾). list of examples. We shall next describe a family of branes Hence one may study branes of the four possible types: defined by the author and Baraglia, obtained by imposing (퐵, 퐵, 퐵), (퐵, 퐴, 퐴), (퐴, 퐵, 퐴), and (퐴, 퐴, 퐵), whose dual part- symmetries to the solutions of (1); see [17] and references ner is predicted by Kontsevich’s homological mirror sym- therein. metry to be Higgs Bundles and 3-Manifolds (퐵, 퐴, 퐴) ⟷ (퐵, 퐵, 퐵), (13) By considering actions on the Riemann surface Σ and on (퐴, 퐴, 퐵) ⟷ (퐴, 퐴, 퐵), (14) the Lie group 퐺ℂ, one can induce actions on the moduli (퐴, 퐵, 퐴) ⟷ (퐴, 퐵, 퐴). (15) space of Higgs bundles and on the Hitchin fibration and study their fixed point sets. Indeed, for 휌 the compact struc- In view of the SYZ conjecture, it is crucial to obtain ture of 퐺 and 휎 a real form of 퐺 , together with Baraglia the duality between branes within the Hitchin fibration. ℂ ℂ we defined the following: In particular, understanding the correspondence between branes contained completely within the irregular fibres has (i) Through the Cartan involution 휃 of a real form 퐺 ̄ ̄ remained a very fruitful direction of research for decades. of 퐺ℂ one obtains 푖1(휕퐴, Φ) = (휃(휕퐴), −휃(Φ)). In 2006 Gukov, Kapustin, and Witten introduced the first (ii) A real structure 푓 ∶ Σ → Σ on Σ induces ̄ ∗ ̄ ∗ studies of branes of Higgs bundles in relation to the geo- 푖2(휕퐴, Φ) = (푓 (휌(휕퐴)), −푓 (휌(Φ))). metric Langlands program and electromagnetic duality (iii) Lastly, by looking at 푖3 = 푖1 ∘ 푖2, one obtains ̄ ∗ ̄ ∗ where the (퐵, 퐴, 퐴)-branes of real 퐺-Higgs bundles were 푖3(휕퐴, Φ) = (푓 휎(휕퐴), 푓 휎(Φ)). considered. These branes, which correspond to surface The fixed point sets of 푖1, 푖2, 푖3 are branes of group representations into the real Lie group 퐺, may in- type (퐵, 퐴, 퐴), (퐴, 퐵, 퐴), and (퐴, 퐴, 퐵), respectively. The tersect the regular fibres of the Hitchin fibration invery topological invariants can be described using 퐾푂, 퐾푅, and different ways (see [17,18] for references): equivariant 퐾-theory [17]. In particular, the fixed points of

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 631 푖1 give the (퐵, 퐴, 퐴)-brane of 퐺-Higgs bundles mentioned the 3-manifolds in the previous section, an example of which appears in Σ × [−1, 1] 푀 = , (16) Figure 5 and which one can study via the monodromy ac- (푥, 푡) ↦ (푓(푥), −푡) tion on the Hitchin fibration (e.g., see [18]). for which 휕푀 = Σ (e.g., a handle body). In this setting, together with Baraglia, we were able to show that a connec- tion solving the Hitchin equations (1) on Σ extends over 푀 given in (16) as a flat connection if and only if the Higgs ̃0 bundle (퐸, Φ) is fixed by 푖2 and the class [퐸] ∈ 퐾ℤ2 (Σ) in reduced equivariant 퐾-theory is trivial. That is, the Higgs bundles that will extend are only those whose vector bun- dle is preserved by the lift of the involution 푖2 and for Figure 5. A real slice fixed by 푖1 of the moduli space of SL(2, ℂ)-Higgs bundles, from two different angles, obtained which the action of 푖2 over the fibres of 퐸 is trivial when through Hausel’s 3D prints of slices of ℳ퐺ℂ . restricted to each fixed circle. Global Topology The fixed points of 푖 are pseudo-real Higgs bundles. To 3 The computation of topological invariants of Higgs bundle describe the fixed points of the involution 푖 , note that a 2 moduli spaces has received vast attention from researchers real structure (or anticonformal map) on a compact con- who have tackled this problem with a diverse set of mathe- nected Riemann surface Σ is an antiholomorphic involu- matical tools (see references in [9,16]). One of the central tion 푓 ∶ Σ → Σ. The classification of real structures on questions considered for Higgs bundles and their general- compact Riemann surfaces is a classical result of Klein, izations is what the Poincar´epolynomial of the space is. A who showed that all such involutions on Σ may be char- useful fact is that the total space of the Hitchin fibration de- acterized by two integer invariants (푛, 푎): the number 푛 of formation retracts onto the nilpotent cone ℎ−1(0) via the disjoint union of copies of the circle embedded in Σ fixed gradient flow of the moment map ofthe ℂ∗-action intro- by 푓 and 푎 ∈ ℤ determining whether the complement of 2 duced in the harmonic metrics section. The cohomology the fixed point set has one (푎 = 1) or two (푎 = 0) compo- ring localizes to the fixed-point locus inside ℎ−1(0): as first nents; e.g., see Figure 6. seen by Morse-theoretic methods in the work of Hitchin, the Poincar´eseries that generates the Betti numbers of the • rational cohomology 퐻 (ℳ퐺ℂ , ℚ) is a weighted sum of the Poincar´eseries of the connected components of the fixed- point locus.

Figure 6. A genus 2 Riemann surface and its fixed point sets Example 4. As shown by Hitchin, for the family of under an antiholomorphic involution with invariants SL(2, ℂ)-Higgs bundles in Example 1, when the genus of (푛, 푎) = (3, 0). Σ is 푔 = 2, the Poincar´eseries is 1 + 푡2 + 4푡3 + 2푡4 + 34푡5 + 2푡6. (17) A real structure 푓 on the Riemann surface Σ induces in- volutions on the moduli space of representations 휋 (Σ) → Using Morse theory, it has been possible to compute 1 Poincar´epolynomials only for low rank groups, and ex- 퐺ℂ of flat connections and of 퐺ℂ-Higgs bundles on Σ, and the fixed point sets define the (퐴, 퐵, 퐴)-branes of Higgs tending this to higher rank has been a challenging open bundles in (ii). These branes can be shown to be real in- problem for some time. More recently, interesting alterna- tegrable systems, given by (possibly singular) Lagrangian tive techniques have been used to access the higher-rank fibrations. Poincar´epolynomials by Mozgovoy, Schiffmann, Mellit, From a representation theoretic point of view, one may and others. One may further ask about the structure of the ring 퐻•(ℳ , ℚ) itself: for instance, Heinloth recently ask which interesting representations these branes corre- 퐺ℂ spond to, a question closely related to the understanding proved that the intersection pairing in the middle dimen- of which representations of 휋 (Σ) extend to 휋 (푀) for 푀 a 3- sion for the smooth moduli space vanishes in all dimen- 1 1 퐺 = (푛, ℂ) manifold whose boundary is Σ. Whilst this question in its sions for ℂ PGL , and Cliff–Nevins–Shen proved full generality remains an important open problem, we that the Kirwan map from the cohomology of the mod- 퐺 can consider some particular cases in which the answer be- uli stack of -bundles to the moduli stack of semistable 퐺 comes clear from the perspective of Higgs bundles. For -Higgs bundles fails to be surjective. this, as seen in [17] and references therein, we considered One of the most important cohomological conjec- tures in the area is de Cataldo–Hausel–Migliorini’s P=W conjecture, which gives a correspondence between the

632 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 weight filtration and the perverse filtration on the coho- References mology of ℳ퐵 and ℳ퐷표푙, respectively, obtained via non- [1] Marta Aparicio-Arroyo, Steven Bradlow, Brian Col- abelian Hodge theory. Only certain special cases are lier, Oscar Garc´ıa-Prada, Peter B. Gothen, and Andr´e known, e.g., for rank two Higgs bundles, shown by de Oliveira, SO(푝, 푞)-Higgs bundles and higher Teichmüller com- Cataldo–Hausel–Migliorini (see [9]), and for certain mod- ponents, Invent. Math. 218 (2019), no. 1, 197–299, DOI uli spaces of wild Higgs bundles, proven recently by Shen– 10.1007/s00222-019-00885-2. MR3994589 [2] Lara B. Anderson, Mboyo Esole, Laura Fredrickson, and Zhang and Szabo. Laura P. Schaposnik, Singular geometry and Higgs bun- Inspired by the SYZ conjecture mentioned before, dles in string theory, SIGMA Symmetry Integrability Geom. Hausel–Thaddeus conjectured that mirror moduli spaces Methods Appl. 14 (2018), Paper No. 037, 27 pp., DOI of Higgs bundles present an agreement of appropriately 10.3842/SIGMA.2018.037. MR3787793 defined Hodge numbers: [3] Steven B. Bradlow, Oscar Garc´ıa-Prada, and Peter B. Gothen, What is… a Higgs bundle?, Notices Amer. Math. 푝,푞 푝,푞 퐿 ℎ (ℳ퐺ℂ ) = ℎ (ℳ 퐺ℂ ). (18) Soc. 54 (2007), no. 8, 980–981. MR2343296 [4] Ngô Bao Châu, Endoscopy theory of automorphic forms, Very recently, the first proof of this conjecture was estab- Proceedings of the International Congress of Mathemati- lished for the moduli spaces of SL(푛, ℂ) and PGL(푛, ℂ)- cians. Volume I, Hindustan Book Agency, New Delhi, 2010, Higgs bundles by Groechenig–Wyss–Ziegler in [8], where pp. 210–237. MR2827891 they established the equality of stringy Hodge numbers us- [5] Brian Collier, Studying deformations of Fuchsian represen- tations with Higgs bundles, SIGMA Symmetry Integrability ing 푝-adic integration relative to the fibres of the Hitchin Geom. Methods Appl. 15 (2019), Paper No. 010, 32 pp., fibration and interpreted canonical present on these DOI 10.3842/SIGMA.2019.010. MR3910865 moduli spaces as characters on the Hitchin fibres. [6] Olivia Dumitrescu, A journey from the Hitchin section to the

Further combinatorial properties of ℳ퐺ℂ can be oper moduli, String-Math 2016, Proc. Sympos. Pure Math., glimpsed through their twisted version, consisting of vol. 98, Amer. Math. Soc., Providence, RI, 2018, pp. 107– Higgs bundles (퐸, Φ) on Σ with Φ ∶ 퐸 → 퐸 ⊗ ℒ, where 138. MR3821751 Σ now has any genus, 퐿 is a line bundle with deg(퐿) > [7] Laura Fredrickson, Perspectives on the asymptotic geometry deg(퐾), but without any punctures or residues being fixed. of the Hitchin moduli space, SIGMA Symmetry Integrability The corresponding moduli spaces carry a natural ℂ∗-action Geom. Methods Appl. 15 (2019), Paper No. 018, 20 pp., DOI 10.3842/SIGMA.2019.018. MR3921918 but are not hyperkähler, and there is no immediate rela- [8] Michael Groechenig, Dimitri Wyss, and Paul Ziegler, Mir- tionship to a character variety. Hence, there is no obvi- ror symmetry for moduli spaces of Higgs bundles via p-adic in- ous reason for the Betti numbers to be invariant with re- tegration, arXiv:1707.06417, 2017. gard to the choice of deg(퐸), which in the classical setting [9] Tam´asHausel, Global topology of the , Hand- would follow from nonabelian Hodge theory. However, book of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, the independence holds in direct calculations of the Betti Int. Press, Somerville, MA, 2013, pp. 29–69. MR3184173 numbers in low rank and was recently shown for GL(푛, ℂ) [10] N. J. Hitchin, The self-duality equations on a Riemann sur- and SL(푛, ℂ)-Higgs bundles by Groechenig–Wyss–Ziegler face, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59– in [8]. This suggests that some topological properties of 126, DOI 10.1112/plms/s3-55.1.59. MR887284 [11] Nigel Hitchin, Stable bundles and integrable systems, Duke Hitchin systems are independent of the hyperkähler geom- Math. J. 54 (1987), no. 1, 91–114, DOI 10.1215/S0012- etry (see references in [9,16] for more details). 7094-87-05408-1. MR885778 Finally, it should be mentioned that an alternative de- [12] Victoria Hoskins, Parallels between moduli of quiver repre- scription of the Hitchin fibration can be given through sentations and vector bundles over curves, SIGMA Symmetry cameral data, as introduced by Donagi and Gaitsgory, and Integrability Geom. Methods Appl. 14 (2018), Paper No. this perspective presents many advantages, in particular 127, 46 pp., DOI 10.3842/SIGMA.2018.127. MR3882963 when considering correspondences arising from mirror [13] Fanny Kassel, Geometric structures and representations of symmetry and Langlands duality, as those mentioned in discrete groups, Proceedings of the International Congress previous sections studied by Donagi–Pantev. of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 1115– 1151. MR3966802 ACKNOWLEDGMENTS. The author would like to [14] François Labourie, Cyclic surfaces and Hitchin components thank Steve Bradlow, Laura Fredrickson, Tamas Hausel, in rank 2, Ann. of Math. (2) 185 (2017), no. 1, 1–58, DOI Nigel Hitchin, Qiongling Li, Sara Maloni, Du Pei, James 10.4007/annals.2017.185.1.1. MR3583351 Unwin, Richard Wentworth, Anna Wienhard, the re- [15] Qiongling Li, An introduction to Higgs bundles via har- viewers of the paper, and very especially Steve Rayan monic maps, SIGMA Symmetry Integrability Geom. Meth- for useful comments on a draft. ods Appl. 15 (2019), Paper No. 035, 30 pp., DOI 10.3842/SIGMA.2019.035. MR3947036

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 633 FEATURED TITLES FROM THE [16] Steven Rayan, Aspects of the topology and combinatorics of EUROPEAN Higgs bundle moduli spaces, SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), Paper No. 129, 18 pp., MATHEMATICAL DOI 10.3842/SIGMA.2018.129. MR3884746 [17] Laura P. Schaposnik, Higgs bundles and applications, Ober- SOCIETY wolfach Reports 31 (2015), 31–33, available at arXiv: 1603.06691. [18] Laura P. Schaposnik, Advanced topics in gauge theory: Ge- Gösta Mittag-Leffler ometry and physics of Higgs bundles (2019), available at and Vito Volterra arXiv:1907.09800. 40 Years of [19] Carlos T. Simpson, Higgs bundles and local systems, Correspondence Inst. Hautes Etudes´ Sci. Publ. Math. 75 (1992), 5–95. Frédéric Jaëck, Ecole Normale MR1179076 Supérieure, Paris, France, Laurent [20] Anna Wienhard, An invitation to higher Teichmüller Mazliak, Université Pierre et theory, Proceedings of the International Congress of Marie Curie, Paris, France, Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lec- Emma Sallent Del Colombo, tures, World Sci. Publ., Hackensack, NJ, 2018, pp. 1013– Universitat de Barcelona, Spain, 1039. MR3966798 and Rossana Tazzioli, Université Lille 1, Villeneuve-d’Ascq, France, Editors Volterra and Mittag-Leffler’s exchanges illustrate how gener- al analysis, especially , gained a dramatic momentum during those years, and how Volterra became one of the major leaders of the field, opening the path for several fundamental developments over the following decades. Heritage of European Mathematics, Volume 12; 2019; 438 pages; Hardcover; ISBN: 978-3-03719-199-6; List US$98; AMS members US$78.40; Order code EMSHEM/12 Laura P. Schaposnik Microlocal Analysis of Quantum Credits Fields on Curved All figures are by the author. Spacetimes Photo of the author is by James Unwin. Christian Gérard, Université de Paris 11, Orsay, France This monograph focuses on free fields and the correspond- ing quasi-free states, and, more precisely, on Klein Gordon fields and Dirac fields. It will be useful to both mathemati- cians and mathematical physicists. ESI Lectures in Mathematics and Physics, Volume 10; 2019; 228 pages; Softcover; ISBN: 978-3-03719-094-4; List US$58; Individual member US$46.40; Order code EMSESILEC/10

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634 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Can You Pave the Plane with Identical Tiles?

Chuanming Zong

Everybody knows that identical regular triangles or squares and one type of decagon that can form nontrivial fivefold can tile the whole plane. Many people know that identi- translative tilings. In fact, parallelograms, centrally sym- cal regular hexagons can tile the plane properly as well. In metric hexagons, and these three types of polygons are fact, even the bees know and use this fact! Is there any the only convex polygons that can form fivefold transla- other convex domain that can tile the Euclidean plane? Of tive tilings. This paper reviews the dramatic progress. course, there is a long list of them! To find the list and to show the completeness of the list is a unique drama Introduction in mathematics which has lasted for more than one cen- Tiling the plane is an ancient subject in our civilization. It tury, and the completeness of the list has been mistakenly has been considered in the arts by craftsmen since antiq- announced more than once! Up to now, the list consists uity. According to Gardner [4], the ancient Greeks knew of triangles, quadrilaterals, fifteen types of pentagons, and that, among the regular polygons, only the triangle, the three types of hexagons. In 2017, Michaël Rao announced square, and the hexagon can tile the plane. Aristotle ap- a computer proof for the completeness of the list. Mean- parently knew this fact, since he made a similar claim in while, Qi Yang and Chuanming Zong made a series of un- the space: Among the five Platonic solids, only the tetrahedron expected discoveries in multiple tilings in the Euclidean and the cube can tile the space. Unfortunately, he was wrong: plane. For example, besides parallelograms and centrally Identical regular tetrahedra cannot tile the whole space! symmetric hexagons, there is no other convex domain that The first recorded scientific investigation into tilings was can form any two-, three-, or fourfold translative tiling made by Kepler. Assume that 풯 is a tiling of the Euclidean in the plane. However, there are two types of octagons plane 피2 by regular convex polygons. If the polygons are identical (congruent), the answer was already known to Chuanming Zong is a distinguished professor of mathematics at Tianjin Uni- the ancient Greeks. When different polygons are allowed, versity, P.R. China. His email address is [email protected]. the situation becomes more complicated and more inter- Communicated by Notices Associate Editor Emilie Purvine. esting. In particular, an edge-to-edge tiling 풯 by regular For permission to reprint this article, please contact: polygons is said to be of type (푛1, 푛2, … , 푛푟) if each ver- [email protected]. tex 퐯 of 풯 is surrounded by an 푛1-gon, an 푛2-gon, and DOI: https://doi.org/10.1090/noti2075 so on in a cyclic order, where edge-to-edge means that

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 635 every pair of neighbors shares an entire common edge. In 1917 Reinhardt was an assistant of Hilbert’s at Göttin- Usually, they are known as Archimedean tilings. In gen and likely discussed this problem with him. It is worth 1619, Kepler enumerated all such tilings as (3, 3, 3, 3, 3, 3), mentioning that in 1911, Bieberbach himself solved the (3, 3, 3, 3, 6), (3, 3, 3, 4, 4), (3, 3, 4, 3, 4), (3, 4, 6, 4), (3, 6, 3, 6), first part of Hilbert’s 18th problem: Is there in 푛-dimensional (3, 12, 12), (4, 4, 4, 4), (4, 6, 12), (4, 8, 8), and (6, 6, 6). Beau- Euclidean space also only a finite number of essentially different tiful illustrations of the Archimedean tilings can be found kinds of groups of motions with a fundamental region? in many references. Reinhardt’s List If 퐚1, 퐚2, … , 퐚푛 are 푛 linearly independent vectors in the 푛-dimensional Euclidean space 피푛, then the set In 1918, Reinhardt received his doctoral degree under the supervision of Bieberbach at Frankfurt am Main with a the- Λ = {∑ 푧푖퐚푖 ∶ 푧푖 ∈ ℤ} sis “On Partitioning the Plane into Polygons” (Über die Zerlegung der Ebene in Polygone). This is the first ap- is an 푛-dimensional lattice. Clearly, lattices are the most nat- proach to characterizing all the convex domains that can ural periodic discrete sets in the plane and space. There- tile the whole plane. First, he studied the tiling networks fore, many pioneering scientists like Kepler, Huygens, (the vertices, edges, and faces of the tilings) and obtained Haüy, and Seeber took lattice packings and lattice tilings an expression for the mean of the number of vertices over as the scientific foundation for crystals. In 1885, the fa- faces. As a corollary of the formula, he obtained the fol- mous crystallographer Fedorov [3] discovered that A con- lowing result. vex domain can form a lattice tiling of 피2 if and only if it is a parallelogram or a centrally symmetric hexagon; a convex body Theorem 1 (Reinhardt [14]). A convex 푚-gon can tile the 2 can form a lattice tiling in 피3 if and only if it is a parallelotope, whole plane 피 by identical copies only if a hexagonal prism, a rhombic dodecahedron, an elongated oc- 푚 ≤ 6. tahedron, or a truncated octahedron. Usually, tilings allow very general settings without re- In fact, as Reinhardt and several other authors pointed striction on the shapes of the tiles and the number of the out (see [4,10,12,14]), this theorem can be easily deduced different shapes. However, to avoid complexity and confu- by Euler’s formula sion, in this paper we deal only with the tilings by identical 푣 − 푒 + 푓 = 1, (1) convex polygon tiles. In other words, all the tiles are con- where 푣, 푒, and 푓 stand for the numbers of vertices, edges, gruent to one convex polygon. In particular, we call it a and faces of a polygonal division of a finite polygon. translative tiling if all the tiles are translates of one another Let 풯 = {푇 , 푇 , 푇 , …} be a tiling of 피2 such that all tiles and call it a lattice tiling if it is a translative tiling and all 1 2 3 푇 are congruent to a convex 푚-gon 푃 , and let 퐻 be a the translative vectors together form a lattice. 푖 푚 ℓ regular hexagon of edge length ℓ centered at the origin of In 1900, Hilbert [8] proposed a list of mathematical 피2. Assume that 퐻 contains 푓(ℓ) tiles 푇 , 푇 , … , 푇 of 풯 problems in his ICM lecture in Paris. As a generalized in- ℓ 1 2 푓(ℓ) and the boundary of 퐻 intersects 푔(ℓ) tiles 푇′, 푇′, … , 푇′ verse of Fedorov’s discovery, he wrote in the second part ℓ 1 2 푔(ℓ) of 풯, and let 푢 denote the number of vertices of the tiling of his 18th problem that a fundamental region of each group 푖 network on the boundary of 푇 . Clearly we have 푢 ≥ 푚 of motions, together with the congruent regions arising from the 푖 푖 and group, evidently fills up space completely. The question arises: 푔(ℓ) whether polyhedra also exist which do not appear as fundamen- lim = 0. (2) ℓ→∞ 푓(ℓ) tal regions of groups of motions, by means of which nevertheless by a suitable juxtaposition of congruent copies a complete filling Applying Euler’s formula to 풯 ∩ 퐻ℓ, when ℓ is sufficiently up of all space is possible. Here Hilbert did not restrict to large we get convex ones. 푓 ≤ 푓(ℓ) + 푔(ℓ) ≲ 푓(ℓ), (3)

Hilbert proposed his problem in the space; perhaps he 푓(ℓ) 1 푚 believed that there is no such domain in the plane. When 푒 ≥ ∑ 푢 ≥ ⋅ 푓(ℓ), (4) 2 푖 2 Reinhardt started his doctoral thesis at Frankfurt am Main 푖=1 in the 1910s, Bieberbach (see [14]) suggested that he de- 푓(ℓ) termine all the convex domains which can tile the whole plane 1 푚 푣 ≲ ∑ 푢 ≲ ⋅ 푓(ℓ), (5) 3 푖 3 and to verify in this way that Hilbert’s problem indeed has 푖=1 a positive answer in the plane. This is the origin of the 푚 following natural problem: 푣 − 푒 + 푓 ≲ (1 − ) 푓(ℓ), (6) 6 Bieberbach’s Problem. To determine all the two-dimen- and therefore sional convex tiles. 푚 ≤ 6, (7)

636 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 where 푔(ℓ) ≲ 푐푓(ℓ) means Type 1. 푔(ℓ) lim ≤ 푐. ℓ→∞ 푓(ℓ) v4 v3

v2 v5

v6 v1

Type 2.

v Figure 1. Two quadrilaterals make a centrally symmetric 4 v3 hexagon. v5 v2 v Apparently, two identical triangles can make a parallelo- 6 v1 gram and two identical quadrilaterals can make a centrally symmetric hexagon (see Figure 1). Thus, by Fedorov’s the- orem, identical triangles or quadrilaterals can always tile the plane nicely. However, it is easy to see that identical Type 3. regular pentagons or some particular hexagons cannot tile the plane. Then, Bieberbach’s problem can be reformu- lated as: v4 What kind of convex pentagons or hexagons can tile the v3 v5 plane? v2 Let 푃푛 denote a convex 푛-gon with vertices 퐯1, 퐯2, … , 퐯푛 퐺 퐯 in an anti-clock order, let 푖 denote the edge with ends 푖−1 v6 v1 and 퐯푖, where 퐯0 = 퐯푛, let ℓ푖 denote the length of 퐺푖, and let 훼푖 denote the inner angle of 푃푛 at 퐯푖. Reinhardt’s thesis obtained the following solution to the hexagon case of Bieberbach’s problem. Figure 2. Reinhardt’s hexagonal tiles and their local tilings. is a square of edge length ℓ in which the tiling is edge-to-edge Theorem 2 (Reinhardt [14]). A convex hexagon 푃6 can tile the whole plane 피2 by identical copies if and only if it satisfies and every vertex is surrounded by three hexagons. one of the three groups of conditions: Let 푆 be a big square of edge length ℓ centered at the (1) 훼1 + 훼2 + 훼3 = 2휋, and ℓ1 = ℓ4. origin of the plane and consider the network of 푁 = 풯 ∩ (2) 훼1 + 훼2 + 훼4 = 2휋, ℓ1 = ℓ4, and ℓ3 = ℓ5. 푆. Let 푛1 denote the number of vertices in 푁 that appear 2 (3) 훼1 = 훼3 = 훼5 = 휋, ℓ1 = ℓ2, ℓ3 = ℓ4, and ℓ5 = ℓ6. also in the relative interior of some edges, let 푛2 denote 3 the number of vertices in 푁 at which three hexagons join The “if” part of this theorem is relatively simple. It is properly at their vertices, and let 푛3 denote the number of illustrated by Figure 2. However, the “only if” part is much all other vertices of 푁. Lemma 1 can be proved by studying more complicated. Reinhardt deduced the only if part by the quotients considering six cases with respect to how many edges of the 푛2 considered hexagon are equal. His proof was very sketchy 푛1 + 푛3 and difficult to understand and check. It seems that he for 푆 and its subsquares for sufficiently large ℓ. It is in- considered only the edge-to-edge tilings. teresting to notice that there are hexagon tilings in which Fortunately, this theorem has been verified by several 푛3 ≠ 0. other authors. For example, without knowledge of Rein- For the pentagon tilings, by considering five cases with hardt’s thesis, in 1963 Bollob´asmade the following sur- respect to how many edges are equal, Reinhardt obtained prising observation, which guarantees the sufficiency of the following result. Reinhardt’s consideration. Theorem 3 (Reinhardt [14]). A convex pentagon 푃5 can tile Lemma 1 (Bollob´as[2]). If 풯 is a tiling of the plane by iden- the whole plane 피2 by identical copies if it satisfies one of the tical convex hexagons and ℓ is any given positive number, there five groups of conditions:

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 637 (1) 훼1 + 훼2 + 훼3 = 2휋. Type 1. (2) 훼1 + 훼2 + 훼4 = 2휋, and ℓ1 = ℓ4. v v 2 4 3 (3) 훼1 = 훼3 = 훼4 = 휋, ℓ1 = ℓ2, and ℓ4 = ℓ3 + ℓ5. 3 v2 1 (4) 훼1 = 훼3 = 휋, ℓ1 = ℓ2, and ℓ3 = ℓ4. 2 v5 v1 1 2 (5) 훼1 = 휋, 훼3 = 휋, ℓ1 = ℓ2, and ℓ3 = ℓ4. 3 3 Figuring out the list is nontrivial. However, as shown Type 2. in Figure 3, it is easy to check that all the pentagons listed in Theorem 3 indeed can tile the plane. Reinhardt him- v3 v self did not claim the completeness of the pentagon tile 4 v list. However, according to Gardner [4] it is quite clear that 2 Reinhardt and everyone else in the field thought that the Rein- v5 v hardt pentagon list was probably complete. 1 As observed by Reinhardt [14], all triangles, quadrilat- erals, the three types of hexagons listed in Theorem 2, and the five classes of pentagons listed in Theorem 3 are indeed Type 3. fundamental domains of some groups of motions. Hilbert and Bieberbach would have been happy to know this. v In 1928, Reinhardt discovered a (nonconvex) three- 3 v v2 dimensional polytope that can form a tiling in the space 4 but is not the fundamental domain of any group of mo- tions! This is the first counterexample to the second part v5 v1 of Hilbert’s 18th problem. Inspired by Reinhardt’s discovery, in 1935 Heesch [7] Type 4. obtained a two-dimensional nonconvex counterexample to Hilbert’s problem. In other words, there exists a non- convex polygon that can tile the whole plane; however, it v3 v4 is not the fundamental region of any group of motions. v2 Thirty years later, Heesch and Kienzle presented a rather detailed treatment of plane tilings in a book enti- tled Flächenschluß: System der Formen lückenlos aneinander- v5 v1 schliessender Flachteile, including nonconvex tiles. No new convex tile was discovered. It was claimed that their treat- ment was complete. Type 5. An End, or a New Start In 1968, fifty years after Reinhardt’s pioneering thesis, Ker- v shner surprisingly discovered three new classes of pen- v3 2 tagons that can pave the whole plane without gaps or over- lapping. v4

v5 v1 Theorem 4 (Kershner [10]). A convex pentagon 푃5 can tile the whole plane 피2 by identical copies if it satisfies one of the Figure 3. Reinhardt’s pentagonal tiles and their local tilings. three groups of conditions:

(6) 훼1 +훼2 +훼4 = 2휋, 훼1 = 2훼3, ℓ1 = ℓ2 = ℓ5, and ℓ3 = ℓ4. (7) 2훼2 + 훼3 = 2훼4 + 훼1 = 2휋, and ℓ1 = ℓ2 = ℓ3 = ℓ4. (8) 2훼1 + 훼2 = 2훼4 + 훼3 = 2휋, and ℓ1 = ℓ2 = ℓ3 = ℓ4. According to Kershner, having been intrigued by this prob- lem for some 35 years, he finally discovered a method of classify- ing the possibilities for pentagons in a more convenient way than Figure 4. Heesch’s counterexample to Hilbert’s problem. Reinhardt’s to yield an approach that was humanly possible to carry to completion. Unfortunately, Kershner’s paper con- of new pentagon tiles were indeed surprising, though ver- tains no hint of his method. Of course, the three classes ifications are simple (see Figure 5).

638 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Type 6.

v3 v2

v1

v4 v5

Type 7.

v4 v3 Figure 6. The Archimedean tiling of (4, 8, 8) type. v2

v5 v1 Theorem 5 (James [9]). A convex pentagon 푃5 can tile the whole plane 피2 by identical copies if it satisfies the following group of conditions: 휋 Type 8. (9) 훼5 = , 훼1 + 훼4 = 휋, 2훼2 − 훼4 = 2훼3 + 훼4 = 휋, and 2 ℓ1 = ℓ2 + ℓ4 = ℓ5.

v3 This result can be easily verified by argument based on Figure 7. In principle, Lemma 1 guarantees that every v4 v2 hexagon tiling is edge-to-edge. However, James’s discovery shows that this is no longer true in some pentagon tilings.

v5 v1 Theorem 5 also served to point out that Kershner had taken edge-to-edge as a hidden assumption in his consideration. Figure 5. Kershner’s pentagonal tiles and their local tilings. Type 9.

Remark 1. Kershner’s discovery was unexpected. Even more surprising was that all the pentagons of Types 6–8 are counterexamples to the second part of Hilbert’s 18th v 4 v problem! In other words, they can tile the whole plane; 3 nevertheless they are not the fundamental regions of any v2 group of motions. Hilbert, Bieberbach, Reinhardt, Heesch, v v and others would have been surprised by Kershner’s ele- 5 1 gant examples! Kershner himself did not mention this fact in his papers. Perhaps he overlooked it. This fact has been mentioned in many books and survey papers (see [6]). Inductively, 푛-dimensional counterexamples to Figure 7. James’s pentagonal tile and its local tiling. Hilbert’s problem can be constructed as cylinders over (푛 − 1)-dimensional ones. For example, if 퐷 is a domain Meanwhile, mathematical amateur Marjorie Rice made of Type 6 and 퐻 is the cylinder of height one over 퐷, then some truly astonishing discoveries that made the news. 퐻 is a counterexample to the second part of Hilbert’s 18th She was a true amateur. According to Schattschneider [15], problem in 피3. Rice had no mathematical training beyond “the bare min- imum they required...in high school over 35 years ago.” In 1975, Reinhardt and Kershner’s discoveries were in- Even so she was able to consider the problem with a sys- troduced and popularized by Martin Gardner, a famous tematic method based on the possible local structures of scientific writer, in the “Mathematical Games” column of the pentagon tilings at a given vertex. By dealing with more the Scientific American magazine. Since then, the tiling than sixty cases, she discovered four types of new pentagon problem has stimulated many amateurs who went on to tiles! make significant contributions to this problem. Soon after Gardner’s popular paper, based on the Theorem 6 (Rice [15]). A convex pentagon 푃5 can tile the 2 known Archimedean tiling (4, 8, 8) by octagons and whole plane 피 by identical copies if it satisfies one of the four squares together, as shown in Figure 6, a computer scien- groups of conditions: tist, Richard James III, discovered a class of new tiles. (10) 훼2 + 2훼5 = 2휋, 훼3 + 2훼4 = 2휋, and ℓ1 = ℓ2 = ℓ3 = ℓ4.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 639 휋 Type 10. (11) 훼1 = , 훼3 + 훼5 = 휋, 2훼2 + 훼3 = 2휋, and 2ℓ1 + ℓ3 = 2 ℓ4 = ℓ5. v3 휋 (12) 훼1 = , 훼3+훼5 = 휋, 2훼2+훼3 = 2휋, and 2ℓ1 = ℓ3+ℓ5 = v 2 2 ℓ4. v 휋 4 (13) 훼1 = 훼3 = , 2훼2 + 훼4 = 2훼5 + 훼4 = 2휋, ℓ3 = ℓ4, and 2 v5 v1 2ℓ3 = ℓ5. It is routine to verify this theorem based on Figure 8. Type 11. Nevertheless, it is rather surprising to notice that the tilings v produced by the pentagons of Type 10 are edge-to-edge, a 3 fact that was missed by both Reinhardt and Kershner. It v2 is even more surprising that all the pentagons of Types 9– v4 13 are counterexamples to Hilbert’s problem as well (see [6]). In other words, they can tile the whole plane; how- v5 v1 ever, they are not the fundamental domains of any group of motions. Type 12. Marjorie Rice died on July 2, 2017, at the age of 94. A lobby floor of the Mathematical Association of America in v3 Washington is paved with one of Rice’s pentagon tiles in v2 her honor. On July 11, 2017, Quanta Magazine published v an article in her memory. 4 v v Rice’s method was systematic, in the sense that it was 5 1 based on a geometric principle. In any case, the method was not strong enough to guarantee the completeness of Type 13. the list. In 1985, Rolf Stein reported another one.

Theorem 7 (Stein [16]). A convex pentagon 푃5 can tile the v 2 3 whole plane 피 by identical copies if it satisfies the following v2 conditions: v4 휋 (14) 훼1 = , 2훼2 + 훼3 = 2휋, 훼3 + 훼5 = 휋, and 2ℓ1 = 2ℓ3 = 2 v v ℓ4 = ℓ5. 5 1

Fifteen, and Only Fifteen Figure 8. Rice’s pentagonal tiles and their local tilings. Let 풯 denote a tiling of 피2 with congruent tiles. A symmetry of 풯 is an isometry of 피2 that maps the tiles of 풯 onto tiles of 풯, and the symmetry group 풢 of 풯 is the collection of all Type 14. such symmetries associated with isometry multiplications. v3 Two tiles, 푇1 and 푇2 of 풯, are said to be equivalent if there v2 is a symmetry 휎 ∈ 풢 such that 휎(푇1) = 푇2. If all the tiles of 풯 are equivalent to one tile 푇, the tiling 풯 is said to v4 be transitive (or isohedral) and 푇 is called a transitive tile. Then, the second part of Hilbert’s 18th problem can be v5 v1 reformulated as: Figure 9. Stein’s pentagonal tile and its local tiling. Is every polytope that can tile the whole space a transitive tile? A tiling 풯 of 피2 by identical convex pentagons is called Types 5–14 except Type 9 are two-block transitive, and the an 푛-block transitive tiling if it has a block 퐵 consisting of 푛 tiles of Type 9 are three-block transitive. (minimum) connected tiles such that 풯 is a transitive tiling From the intuitive point of view, it is reasonable to of 퐵. If a convex pentagon 푇 can form an 푛-block transitive believe that periodic structure is inevitable in pentagon tiling but not an 푚-block transitive tiling for any 푚 < 푛, tilings and the period cannot be too large. Based on this then we call it an 푛-block transitive tile. Clearly, all the tiles belief, Mann, McLoud-Mann, and Von Derau [12] devel- of Types 1–5 are one-block transitive. In other words, they oped an algorithm for enumerating all the 푛-block tran- are transitive tiles. According to [12, 15], all the tiles of sitive pentagon tiles. When they checked the three-block

640 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 case, surprisingly, they discovered a new type of pentagon mathematical problems ranges from zero to infinity, and tiles. there indeed exist problems that have no transparent logi- cal proofs. Theorem 8 (Mann, McLoud-Mann, and Von Derau [12]). In 1980, Grünbaum and Shephard [6] made the follow- A convex pentagon 푃 can tile the whole plane 피2 by identical 5 ing comment when they wrote about the tiling problems: copies if it satisfies the following conditions: 휋 3휋 7휋 휋 5휋 Current fashions in mathematics applaud abstraction for its own (15) 훼1 = , 훼2 = , 훼3 = , 훼4 = , 훼5 = , and 3 3 12 2 6 sake, regarding it as the highest intellectual activity—whether ℓ1 = 2ℓ2 = 2ℓ4 = 2ℓ5. or not it is, in any sense, useful or related to other endeavors. Mathematicians frequently regard it as demeaning to work on Type 15. problems related to “elementary geometry” in Euclidean space of two or three dimensions. In fact, we believe that many are unable, both by inclination and training, to make meaningful contributions to this more “concrete” type of mathematics; yet v4 v3 it is precisely these and similar considerations that include the

v5 results and techniques needed by workers in other disciplines. Clearly, the proof history of Bieberbach’s problem indeed v2 confirms their comment. v1 Multiple Tilings Intuitively speaking, tiling the plane is to pave the whole plane flat with identical tiles. As one can see from previous sections, only a few types of polygons are qualified for the job. However, if multiple layers are permitted, we will have Figure 10. Mann, McLoud-Mann, and Von Derau’s pentagonal tile and its local tiling. many more choices for the shape of the tile. Let 퐾 denote an 푛-dimensional convex body with inte- rior int(퐾) and boundary 휕(퐾). In particular, let 퐷 denote Remark 2. It was shown by Mann, McLoud-Mann, and Von a two-dimensional convex domain. Derau [12] that there is no other 푛-block transitive penta- Assume that ℱ = {퐾1, 퐾2, 퐾3, …} is a family of convex 푛 gon tile with 푛 ≤ 4. The completeness of the list emerges bodies in 피 and 푘 is a positive integer. We call ℱ a 푘-fold 푛 푛 again. tiling of 피 if every point 퐱 ∈ 피 belongs to at least 푘 of these convex bodies and every point 퐱 ∈ 피푛 belongs to Since Hales’s computer proof for the Kepler conjecture, at most 푘 of the int(퐾푖). In other words, a 푘-fold tiling of more and more geometers have turned to computers for 피푛 is both a 푘-fold packing and a 푘-fold covering in 피푛. In help when their mathematical problems can be reduced particular, we call a 푘-fold tiling of 피푛 a 푘-fold congruent into a large number of cases. Characterizing all the penta- tiling, a 푘-fold translative tiling, or a 푘-fold lattice tiling if all gon tiles seems to be a perfect candidate for such purpose. 퐾푖 are congruent to 퐾1, all 퐾푖 are translates of 퐾1, or all In 2017, one century after Bieberbach proposed the 퐾푖 are translates of 퐾1 and the translative vectors form a characterization problem, Michaël Rao announced a com- lattice in 피푛, respectively. In these particular cases, we call puter proof for the completeness of the known pentagon 퐾1 a 푘-fold congruent tile, a 푘-fold translative tile, or a 푘-fold tile list. Rao’s approach is based on a graph expression. lattice tile, respectively. Clearly, a 푘-fold translative tiling First he proved that if a pentagon tiles the plane, then it of the plane 피2 is a nice pavement with identical copies. can form a tiling such that every vertex type has positive In other words, it covers every point of the plane with the density. Clearly, this is a weak version of the periodic tiling. same multiplicity, excepting the boundary points of the Second, it was shown that there are only a finite number of tiles. possible vertex types in the modified pentagon tiling. In For a fixed convex body 퐾, we define 휏•(퐾) to be the fact, he reduced them to 371 types. Then, by testing the smallest integer 푘 such that 퐾 can form a 푘-fold congru- 371 cases, Rao announced the following theorem. ent tiling in 피푛, 휏(퐾) to be the smallest integer 푘 such that 푛 ∗ Theorem 9 (Rao [13]). A convex pentagon 푃5 can tile the 퐾 can form a 푘-fold translative tiling in 피 , and 휏 (퐾) to whole plane 피2 by identical copies if and only if it belongs to be the smallest integer 푘 such that 퐾 can form a 푘-fold one of the fifteen types listed in Theorems 3–8. lattice tiling in 피푛. For convenience, if 퐾 cannot form any multiple congruent tiling, translative tiling, or lattice Computer proofs are still not as acceptable as transpar- tiling, we will define 휏•(퐾) = ∞, 휏(퐾) = ∞, or 휏∗(퐾) = ∞, ent logical proofs within the mathematical community. However, we have to admit that the complexity of the

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 641 respectively. Clearly, for every convex body 퐾 we have Lemma 2 (Bolle [1]). A convex polygon 퐷 is a 푘-fold lattice Λ 푘 휏•(퐾) ≤ 휏(퐾) ≤ 휏∗(퐾). (8) tile for a lattice and some positive integer if and only if the following conditions are satisfied: By looking at the separating hyperplanes between tangent (1) It is centrally symmetric. neighbors, it is obvious that a convex body can form a mul- (2) When 퐷 is centered at the origin, the relative interior tiple tiling only if it is a polytope. 1 of each edge 퐺 of 퐷 contains a point of Λ. If 휎 is a nonsingular affine linear transformation from 2 1 피푛 피푛 ℱ = {퐾 , 퐾 , 퐾 , …} 푘 (3) If the midpoint of 퐺 is not in Λ, then 퐺 is a lattice to , then 1 2 3 forms a -fold tiling of 2 푛 ′ 피 if and only if ℱ = {휎(퐾1), 휎(퐾2), 휎(퐾3), …} forms a 푘- vector of Λ. fold tiling of 피푛. Consequently, for any 푛-dimensional Based on Bolle’s criterion, Gravin, Robins, and Shiryaev convex body 퐾 and any nonsingular affine linear transfor- [5] discovered the following example. mation 휎 we have both Example 1. Let Λ denote the two-dimensional integer 2 휏(휎(퐾)) = 휏(퐾) (9) lattice ℤ and let 푃8 denote the octagon with vertices 퐯1 = 1 3 3 1 3 1 1 3 ( , − ), 퐯2 = ( , − ), 퐯3 = ( , ), 퐯4 = ( , ), 퐯5 = −퐯1, and 2 2 2 2 2 2 2 2 ∗ ∗ 휏 (휎(퐾)) = 휏 (퐾). (10) 퐯6 = −퐯2, 퐯7 = −퐯3, and 퐯8 = −퐯4, as shown in Figure 11. 2 Unfortunately, 휏•(퐾) is not an invariant for the linear trans- Then 푃8 + Λ is a sevenfold lattice tiling of 피 . formation group. Clearly, onefold tilings are the usual tilings. In the plane, we have v5 v4 휏(퐷) = 휏∗(퐷) = 1 (11) if and only if 퐷 is a parallelogram or a centrally symmetric hexagon, and v6 v3 휏•(퐷) = 1 (12) if and only if 퐷 is a triangle, a quadrilateral, a pentagon o belonging to one of the fifteen types listed in Theorems 3– v7 P8 v2 8, or a hexagon belonging to one of the three types listed in Theorem 2. Taking a usual tiling and stacking it on top of itself 푘 times forms a 푘-fold tiling. Similarly, by stacking 푗 copies v8 v1 of a 푘-fold tiling on top of each other, we get a 푗푘-fold tiling. However, we are interested in the nontrivial multi- Figure 11. Gravin, Robins, and Shiryaev’s octagonal sevenfold ple tilings. lattice tile. Since 1936, multiple tilings have been studied by P. Furtwängler, G. Haj´os, R. M. Robinson, U. Bolle, Let 풟 denote the family of all two-dimensional convex N. Gravin, M. N. Kolountzakis, S. Robins, D. Shiryaev, and domains and let 풫2푚 denote the family of all centrally sym- many others. Nevertheless, many natural problems are metric convex 2푚-gons. Since the octagon of Example 1 is still open. In the forthcoming sections we will introduce the simplest centrally symmetric polygon (except parallel- some fascinating new results about multiple tilings in the ograms and hexagons) satisfying the criterion of Lemma 2, plane. one may conjecture that Multiple Lattice Tilings min 휏∗(퐷) ≥ 7. 퐷∈풟⧵{풫4∪풫6} In 1994, Bolle studied the two-dimensional lattice multi- ple tilings. Let 퐷 be a convex domain, let Λ be a lattice, However, based on the known results on multiple lattice ́ and assume that 퐷 + Λ is a 푘-fold lattice tiling of 피2. It is packings by V. C. Dumir, R. J. Hans-Gill, and G. Fejes Toth easy to see that 퐷 must be a polygon. Let 퐸 be an edge of (see Zong [19]), in 2017 Yang and Zong discovered the fol- lowing unexpected result. 퐷, let 퐿 be the straight line containing 퐸, let 퐻1 and 퐻2 de- note the two closed half-planes with 퐿 as their boundary, Theorem 10 (Yang and Zong [17]). If 퐷 is a two- and for a general point 퐩 ∈ int(퐸) define dimensional convex domain that is neither a parallelogram nor a centrally symmetric hexagon, then we have 푛푖(퐩) = ♯{퐠 ∶ 퐠 ∈ Λ, 퐷 + 퐠 ∈ 퐻푖, 퐩 ∈ 휕(퐷) + 퐠}. (13) 휏∗(퐷) ≥ 5, By studying 푛1(퐩) and 푛2(퐩) for general 퐩 ∈ int(퐸), Bolle proved the following criterion: where the equality holds at some particular decagons.

642 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 ∗ v6 휏 (푃12) ≥ 6, (16)

∗ 휏 (푃10) ≥ 5, (17) v7 v5 where equality in (17) holds if and only if (after a suit- v8 푃 v4 able affine linear transformation) 10 is a centrally sym- metric decagon whose edge midpoints are 퐮1 = (0, −1), o 1 3 3 1 v 퐮2 = (1, − ), 퐮3 = ( , 0), 퐮4 = ( , ), 퐮5 = (1, 1), 9 v3 2 2 2 2 퐮 = −퐮 , 퐮 = −퐮 , 퐮 = −퐮 , 퐮 = −퐮 , and 퐮 = −퐮 P 6 1 7 2 8 3 9 4 10 5 10 (as shown by Figure 12), and v10 v2 ∗ 휏 (푃8) ≥ 5, (18) v1 where the equality holds if and only if (after a suitable Figure 12. Yang and Zong’s decagonal fivefold lattice tile. affine linear transformation) 푃8 is either a centrally sym- metric octagon 푃8(훼) (see Figure 13, top) with vertices 퐯1 = 3 3 1 1 2 (−훼, − ), 퐯2 = (1−훼, − ), 퐯3 = (1+훼, − ), 퐯4 = (1−훼, ), Let Λ be the integer lattice ℤ and let 푃10 denote a 2 2 2 2 decagon whose edge midpoints are 퐮1 = (0, −1), 퐮2 = 퐯5 = −퐯1, 퐯6 = −퐯2, 퐯7 = −퐯3, and 퐯8 = −퐯4, where 1 3 3 1 1 (1, − ), 퐮3 = ( , 0), 퐮4 = ( , ), 퐮5 = (1, 1), 퐮6 = −퐮1, 0 < 훼 < , or a centrally symmetric octagon 푃8(훽) (see Fig- 2 2 2 2 4 퐮7 = −퐮2, 퐮8 = −퐮3, 퐮9 = −퐮4, and 퐮10 = −퐮5, as shown ure 13, bottom) with vertices 퐯1 = (훽, −2), 퐯2 = (1+훽, −2), in Figure 12. By Lemma 2, it can be easily verified that 퐯3 = (1 − 훽, 0), 퐯4 = (훽, 1), 퐯5 = −퐯1, 퐯6 = −퐯2, 퐯7 = −퐯3, 2 1 1 푃10 + Λ is indeed a fivefold lattice tiling of 피 . 퐯8 = −퐯4, where < 훽 < . Let Λ be the integer lattice 4 3 Even more unexpected, by studying lattice polygons, all 2 ℤ ; it can be verified that both 푃8(훼) + Λ and 푃8(훽) + Λ are the fivefold lattice tiles can be nicely characterized. There indeed fivefold lattice tilings of 피2. are two classes of octagons and one class of decagons besides the parallelograms and the centrally symmetric hexagons. v6 v5 Theorem 11 (Zong [20]). A convex domain 퐷 can form a 퐷 fivefold lattice tiling of the Euclidean plane if and onlyif is v v a parallelogram or centrally symmetric hexagon or, up to affine 7 4 linear transformation, 퐷 is a centrally symmetric octagon with 3 3 1 P8(α) o vertices 퐯1 = (−훼, − ), 퐯2 = (1 − 훼, − ), 퐯3 = (1 + 훼, − ), 2 2 2 v v3 1 8 퐯4 = (1 − 훼, ), 퐯5 = −퐯1, 퐯6 = −퐯2, 퐯7 = −퐯3, and 2 1 퐯8 = −퐯4, where 0 < 훼 < , or with vertices 퐯1 = (훽, −2), 4 v1 v2 퐯2 = (1 + 훽, −2), 퐯3 = (1 − 훽, 0), 퐯4 = (훽, 1), 퐯5 = −퐯1, 1 1 퐯6 = −퐯2, 퐯7 = −퐯3, 퐯8 = −퐯4, where < 훽 < , or 4 3 a centrally symmetric decagon whose edge midpoints are 퐮1 = 1 3 3 1 v v5 (0, −1), 퐮2 = (1, − ), 퐮3 = ( , 0), 퐮4 = ( , ), 퐮5 = (1, 1), 6 2 2 2 2 퐮6 = −퐮1, 퐮7 = −퐮2, 퐮8 = −퐮3, 퐮9 = −퐮4, and 퐮10 = −퐮5.

v4 Let 푃2푚 be a centrally symmetric convex 2푚-gon cen- 2 tered at the origin 퐨 of 피 . It is reasonable to believe that P8(β) ∗ 휏 (푃2푚) is big when 푚 is sufficiently large. In fact, by study- o v3 ing the local structure of a multiple tiling (see next section), v7 Yang and Zong [18] proved that v 푚 − 1 if 푚 is even, 8 휏∗(푃 ) ≥ { (14) 2푚 푚 − 2 if 푚 is odd. v Furthermore, by detailed geometric analysis based on v1 2 Lemma 2, Lemma 4, and Pick’s theorem, Zong [20] proved that Figure 13. Zong’s octagonal fivefold lattice tiles. ∗ 휏 (푃14) ≥ 6, (15)

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 643 Clearly, Theorem 11 follows from (14)–(18). The v6 v5 proofs of these inequalities are complicated, in particular (17) and (18). Their proofs rely on carefully designed area estimations by dealing with many cases. Nevertheless, un- u4 v like Rao’s proof for Theorem 9, computer checking is not 7 necessary here. P8 v To describe the structure of the decagon in Theorem 11 4 more explicitly we have the following theorem. v8 o u3 푊 Theorem 12 (Zong [20]). Let denote the quadrilateral v3 1 1 3 2 2 with vertices 퐰1 = (− , 1), 퐰2 = (− , ), 퐰3 = (− , ), 2 2 4 3 3 u 3 3 2 and 퐰4 = (− , ). A centrally symmetric convex decagon can 4 4 1 3 3 1 v1 v2 take 퐮1 = (0, −1), 퐮2 = (1, − ), 퐮3 = ( , 0), 퐮4 = ( , ), 2 2 2 2 퐮5 = (1, 1), 퐮6 = −퐮1, 퐮7 = −퐮2, 퐮8 = −퐮3, 퐮9 = −퐮4, and 퐮 = −퐮 as the middle points of its edges if and only if one 10 5 u2 u3 of its vertices is an interior point of 푊. u 1 u Similarly, the sixfold lattice tiles can be characterized as 4 u follows: 10 o u5 Theorem 13 (Zong [20]). A convex domain 퐷 can form a u9 P10 u sixfold lattice tiling of the Euclidean plane if and only if 퐷 is 6 a parallelogram or centrally symmetric hexagon or, up to affine u u8 7 linear transformation, 퐷 is a centrally symmetric octagon with vertices 퐯1 = (훼 − 1, 2), 퐯2 = (훼, 2), 퐯3 = (1 − 훼, 0), 퐯4 = (1 + 훼, −1), 퐯5 = −퐯1, 퐯6 = −퐯2, 퐯7 = −퐯3, and 퐯8 = −퐯4, 1 u u where 0 < 훼 < , a centrally symmetric decagon whose edge 1 2 6 1 1 3 u10 u3 midpoints are 퐮1 = (−1, ), 퐮2 = ( , 1), 퐮3 = ( , 1), 퐮4 = 2 2 2 1 (2, ), 퐮5 = (2, 0), 퐮6 = −퐮1, 퐮7 = −퐮2, 퐮8 = −퐮3, 퐮9 = 2 u9 o u4 −퐮4, and 퐮10 = −퐮5, or a centrally symmetric decagon whose 1 1 3 1 P edge midpoints are 퐮1 = (− , 1), 퐮2 = ( , 1), 퐮3 = ( , ), u 10 u 2 2 2 2 8 5 3 1 퐮4 = (2, 0), 퐮5 = ( , − ), 퐮6 = −퐮1, 퐮7 = −퐮2, 퐮8 = −퐮3, 2 2 u7 u6 퐮9 = −퐮4, and 퐮10 = −퐮5. Figure 14. Zong’s sixfold lattice tiles, one class of octagons and two classes of decagons. Theorem 14 (Zong [20]). Let 푄 denote the quadrilateral with 5 1 3 vertices 퐪1 = (0, 1), 퐪2 = (0, ), 퐪3 = (− , ), and 퐪4 = 6 4 4 Multiple Translative Tilings 1 5 (− , ). A centrally symmetric convex decagon 푃10 can take 3 6 In 2012, Gravin, Robins, and Shiryaev [5] proved that an 푛- 1 1 3 1 dimensional convex body can form a multiple translative 퐮1 = (−1, ), 퐮2 = ( , 1), 퐮3 = ( , 1), 퐮4 = (2, ), 퐮5 = 2 2 2 2 tiling of the space only if it is a centrally symmetric poly- (2, 0), 퐮 = −퐮 , 퐮 = −퐮 , 퐮 = −퐮 , 퐮 = −퐮 , and 6 1 7 2 8 3 9 4 tope with centrally symmetric facets. Therefore, to study 퐮 = −퐮 as the middle points of its edges if and only if one 10 5 multiple translative tilings in the plane, we need to deal of its vertices is an interior point of 푄. ∗ 5 only with the centrally symmetric polygons. Let 푄 denote the quadrilateral with vertices 퐪1 = (0, ), 4 Let 푃2푚 denote a centrally symmetric convex 2푚-gon 1 7 1 7 퐪2 = ( , ), 퐪3 = (0, 1), and 퐪4 = (− , ). A centrally centered at the origin, with vertices 퐯1, 퐯2, … , 퐯2푚 enumer- 6 6 6 6 ∗ 1 ated in the clock-order, and write 푉 = {퐯1, 퐯2, … , 퐯2푚}. As- symmetric convex decagon 푃 can take 퐮1 = ( , −1), 퐮2 = 10 2 푃 + 푋 휏(푃 ) 피2 3 1 3 1 1 sume that 2푚 is a 2푚 -fold translative tiling in , ( , − ), 퐮3 = (2, 0), 퐮4 = ( , ), 퐮5 = ( , 1), 퐮6 = −퐮1, 2 2 2 2 2 where 푋 = {퐱1, 퐱2, 퐱3, …} is a discrete multiset with 퐱1 = 퐨. 퐮7 = −퐮2, 퐮8 = −퐮3, 퐮9 = −퐮4, and 퐮10 = −퐮5 as the By studying the local structure of 푃2푚 + 푋 at the vertices middle points of their edges if and only if one of its vertices is 퐯 ∈ 푉 + 푋, Yang and Zong [18] discovered some fascinat- an interior point of 푄∗. ing results.

644 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Theorem 15 (Yang and Zong [18]). If 퐷 is a two- dimensional convex domain that is neither a parallelogram nor v a centrally symmetric hexagon, then we have v5 6 휏(퐷) ≥ 5, v4 where the equality holds if 퐷 is some particular centrally sym- v7 metric octagon or some particular centrally symmetric decagon. v3 o

v8 Remark 3. It is known that  P8(α) 휏(퐷) ≤ 휏∗(퐷) v2 v1 holds for every convex domain 퐷. Therefore, Theorem 15 implies Theorem 10. At this point, it is natural to ask for a characterization of all fivefold translative tiles and in particular to determine v5 v6 if these are just the known fivefold lattice tiles. Theorem 16 (Yang and Zong [18]). A convex domain 퐷 can form a fivefold translative tiling of the Euclidean plane ifand v4 v7 only if 퐷 is a parallelogram or centrally symmetric hexagon or, up to affine linear transformation, 퐷 is a centrally symmetric o 3 5훼 1 5훼 octagon with vertices 퐯1 = ( − , −2), 퐯2 = (− − , −2), v 2 4 2 4 3 v8 훼 3 훼 3 퐯3 = ( − , 0), 퐯4 = ( − , 1), 퐯5 = −퐯1, 퐯6 = −퐯2, 4 2 4 2  2 P8(β) 퐯7 = −퐯3, and 퐯8 = −퐯4, where 0 < 훼 < , or with vertices 3 v2 v1 퐯1 = (2−훽, −3), 퐯2 = (−훽, −3), 퐯3 = (−2, −1), 퐯4 = (−2, 1), 퐯5 = −퐯1, 퐯6 = −퐯2, 퐯7 = −퐯3, and 퐯8 = −퐯4, where 0 < 훽 ≤ 1, or a centrally symmetric decagon whose edge midpoints Figure 15. Yang and Zong’s octagonal fivefold translative tiles. 1 3 3 1 are 퐮1 = (0, −1), 퐮2 = (1, − ), 퐮3 = ( , 0), 퐮4 = ( , ), 2 2 2 2 퐯5 = −퐯1, 퐯6 = −퐯2, 퐯7 = −퐯3, and 퐯8 = −퐯4, where 퐮5 = (1, 1), 퐮6 = −퐮1, 퐮7 = −퐮2, 퐮8 = −퐮3, 퐮9 = −퐮4, and 0 < 훽 ≤ 1. 퐮10 = −퐮5. Clearly, Theorems 15 and 16 follow by (19)–(23) and The proofs for Theorems 15 and 16 are extremely com- (17). plicated. They consist of a series of lemmas showing that Though the statements (19)–(23) are more or less iden- tical to (14)–(18), their proofs are very different. While 푚 − 1 if 푚 is even, 휏(푃 ) ≥ { (19) the lattice case was based on lattice polygon checking, the 2푚 푚 − 2 if 푚 is odd, translative case is based on combinatorial analysis. ′ 휏(푃14) ≥ 6, (20) In fact, the two classes 푃8(훼) and 푃8 (훽) shown in Figures 13 and 15, respectively, are equivalent under suitable lin- 휏(푃12) ≥ 6, (21) ear transformations, as well as the two classes 푃8(훽) and 휏(푃10) ≥ 5, (22) ′ 푃8 (훼). Therefore, we have the following theorem. where in (22) equality holds if and only if 푃10 is a centrally symmetric decagon that can form a fivefold lattice tiling of Theorem 17 (Yang and Zong [18]). A convex domain can 피2, and form a fivefold translative tiling of the Euclidean plane ifand only if it can form a fivefold lattice tiling in 피2. 휏(푃8) ≥ 5, (23) where the equality holds if and only if (after a suitable Open Problems affine linear transformation) 푃8 is either a centrally sym- To end this paper, let us list three open problems about ′ metric octagon 푃8 (훼) (see Figure 15, top) with vertices multiple tilings that are closely related to the known re- 3 5훼 1 5훼 훼 3 퐯1 = ( − , −2), 퐯2 = (− − , −2), 퐯3 = ( − , 0), sults. 2 4 2 4 4 2 훼 3 퐯4 = ( − , 1), 퐯5 = −퐯1, 퐯6 = −퐯2, 퐯7 = −퐯3, and 퐷 4 2 Problem 1. Is there a two-dimensional convex domain 2 satisfying 휏(퐷) ≠ 휏∗(퐷)? 퐯8 = −퐯4, where 0 < 훼 < , or a centrally symmet- 3 ′ ric octagon 푃8 (훽) (see Figure 15, bottom) with vertices In 2000, Kolountzakis [11] proved that if a convex 퐯1 = (2−훽, −3), 퐯2 = (−훽, −3), 퐯3 = (−2, −1), 퐯4 = (−2, 1), polygon that is not a parallelogram can form a multiple

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 645 translative tiling of the plane, then the translative set must [6] B. Grünbaum and G. C. Shephard, Tilings with con- be a finite union of translated lattices. To improve this gruent tiles, Bull. Amer. Math. Soc. (N.S.) 3 (1980), result and to answer a question of Gravin, Robins, and no. 3, 951–973, DOI 10.1090/S0273-0979-1980-14827-2. Shiryaev [5], B. Liu and Q. Yang independently proved that MR585178 [7] H. Heesch, Aufbau der Ebene aus kongrenten Bereichen, if a convex domain 퐷 can form a multiple translative tiling Nachr. Ges. Wiss. Göttingen 1 (1935), 115–117. of the plane, then it also can form a multiple lattice tiling ∗ [8] D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc. of the plane. However, we do not know if 휏(퐷) = 휏 (퐷) 8 (1902), no. 10, 437–479, DOI 10.1090/S0002-9904- holds for every convex domain 퐷. 1902-00923-3. MR1557926 [9] R. James, New pentagonal tiling reported by Martin Gardner, Problem 2. Is there an integer 푘 ≥ 6 such that 휏(퐷) ≠ 푘 Scientific American (December 1975), 117–118. ∗ (or 휏 (퐷) ≠ 푘) holds for all the two-dimensional convex [10] R. B. Kershner, On paving the plane, Amer. Math. Monthly domains 퐷? 75 (1968), 839–844, DOI 10.2307/2314332. MR236822 [11] M. N. Kolountzakis, On the structure of multiple transla- As noticed by Yang and Zong [17], based on the two- tional tilings by polygonal regions, Discrete Comput. Geom. dimensional examples, for any 푛 ≥ 3 one can construct 23 (2000), no. 4, 537–553, DOI 10.1007/s004540010014. 푛-dimensional centrally symmetric polytopes 푃 satisfying MR1753701 [12] C. Mann, J. McLoud-Mann, and D. Von Derau, Convex 2 ≤ 휏∗(푃) ≤ 5 pentagons that admit 푖-block transitive tilings, Geom. Dedi- cata 194 (2018), 141–167, DOI 10.1007/s10711-017-0270- and 9. MR3798065 2 ≤ 휏(푃) ≤ 5. [13] M. Rao, Exhaustive search of convex pentagons which tile the plane, arXiv:1708.00274, 2017. Then, we have the following natural problem. [14] K. Reinhardt, Über die Zerlegung der Ebene in Polygone, Dissertation, Universität Frankfurt am Main, 1918. Problem 3. Assume that 푘 = 2, 3, or 4, and 푛 ≥ 3. Is [15] D. Schattschneider, Tiling the plane with congruent there an 푛-dimensional polytope 푃 satisfying 휏(푃) = 푘 (or pentagons, Math. Mag. 51 (1978), no. 1, 29–44, DOI 휏∗(푃) = 푘)? 10.2307/2689644. MR493766 [16] R. Stein, A new pentagon tiler, Math. Magazine 58 (1985), Besides these results and open problems for 휏∗(푃) and 308. 휏(푃), analogous problems for 휏•(푃) are interesting and [17] Q. Yang and C. Zong, Multiple lattice tilings in Euclidean worth studying as well. spaces, Canad. Math. Bull. 62 (2019), no. 4, 923–929, DOI 10.4153/s0008439518000103. MR4028498 ACKNOWLEDGMENTS. The author is grateful to the [18] Q. Yang and C. Zong, Characterization of the two- dimensional five-fold translative tiles, arXiv:1712.09732, referees and the handling editor for their helpful sug- 2017. gestions, which essentially improved the quality of the [19] C. Zong, Packing, covering and tiling in two-dimensional paper. This work was supported by the National Natu- spaces, Expo. Math. 32 (2014), no. 4, 297–364, DOI ral Science Foundation of China (NSFC11921001) and 10.1016/j.exmath.2013.12.002. MR3279483 the National Key Research and Development Program [20] C. Zong, Characterization of the two-dimensional of China (2018YFA0704701). five- and six-fold lattice tiles, arXiv:1712.01122, 2017, arXiv:1904.06911, 2019.

References [1] U. Bolle, On multiple tiles in 퐸2, Intuitive geome- try (Szeged, 1991), Colloq. Math. Soc. J´anos Bolyai, vol. 63, North-Holland, Amsterdam, 1994, pp. 39–43. MR1383609 [2] B. Bollob´as, Filling the plane with congruent convex hexagons without overlapping, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 6 (1963), 117–123. MR166683 [3] E. S. Fedorov, Naˇcala uˇceniya o figurah (Russian), Izdat. Akad. Nauk SSSR, Moscow, 1953. MR0062061 [4] M. Gardner, On tessellating the plane with convex polygon Chuanming Zong tiles, Scientific American (July 1975), 112–117. Credits [5] N. Gravin, S. Robins, and D. Shiryaev, Translational tilings by a polytope, with multiplicity, Combinatorica 32 All images are courtesy of Chuanming Zong. (2012), no. 6, 629–649, DOI 10.1007/s00493-012-2860-3. MR3063154

646 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Finding Solitons

Jorge Lauret

1. Introduction The concept of soliton provides a useful way to find or discover elements in a given set that are somehow distin- soliton guished. Heuristically, only three ingredients are needed to define a soliton: • A set Γ endowed with some kind of tangent space or space of directions 푇훾Γ at each 훾 ∈ Γ. 푞(훾) • An equivalence relation ≃ on Γ collecting in each equivalence class [훾] all the elements that cannot be 훾 distinguished from 훾 in relation to the question to be [훾] studied. Γ • An optimal or preferred direction at each point, 푞(훾) ∈ 푇훾Γ, viewed as a “direction of improvement” in some sense. Figure 1. Solitons are not actually improved toward the In that case, 훾 ∈ Γ is called a soliton if “direction of improvement.” 푞(훾) ∈ 푇 [훾]; (1) 훾 The existence of solutions is not guaranteed. Solitons are that is, 훾 is in a way nice enough that it cannot be improved not in general fixed points of this evolution flow (i.e., ze- toward 푞(훾) (see Figure 1). The relation ≃ is typically de- roes of 푞). However, 훾 is a soliton if and only if 훾(푡) ∈ [훾] fined by the action of a group 퐻, and if 푞 is 퐻-equivariant, for all 푡, called a self-similar solution (see Figure 1 and the then 훾 is a soliton if and only if the whole class [훾] consists opener image on the left). In other words, solitons are not of solitons. improved by the flow, and so their existence is not that wel- By assuming enough differentiability in the situation, comed if one is hoping to use the flow to find a fixed point we may consider the evolution differential equation de- in Γ. Indeed, an element may be attracted or stopped in fined by 푞 on Γ, its way to a fixed point by a soliton. 휕 On the other hand, the existence of solitons is great 훾(푡) = 푞(훾(푡)), 훾(0) = 훾. 휕푡 news for the search for canonical or distinguished ele- ments in Γ beyond the zeroes of 푞. Jorge Lauret is a professor at Universidad Nacional de C´ordoba and principal We note that if 푞(훾) = − grad(퐹)|훾 for some functional researcher of CIEM, CONICET (Argentina). His email address is lauret 퐹 ∶ Γ → ℝ that is constant on equivalence classes (or @famaf.unc.edu.ar . 퐻-invariant), then 훾 is a soliton if and only if 푞(훾) = 0; Communicated by Notices Associate Editor Chikako Mese. that is, solitons are precisely the critical points of 퐹 or the For permission to reprint this article, please contact: fixed points of the corresponding flow. Most interesting [email protected]. phenomena occur when this is not the case. DOI: https://doi.org/10.1090/noti2082

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 647 The evocative word soliton first appeared in PDE the- Remark 2.1. Arguing as above, one obtains that the even ory in the context of the Korteweg–de Vries equation to simpler possibility 푞(퐴) = [퐴, 퐴푡] gives that only normal name certain solutions resembling solitary water waves matrices are solitons. that evolve only by translation without losing their shape. Besides normal matrices, it is straightforward to check More generally, in the study of geometric flows, solitons re- that the following nilpotent matrices are also solitons: fer to geometric structures that evolve along symmetries of 0 0 the flow (i.e., self-similar solutions). The use of the word √ 0 √2 0 3 0 soliton was initiated by Hamilton in the 1980s in the con- [ 1 0 ] , [ ] , [ 0 2 0 ] , 0 √2 0 text of Ricci flow to name Ricci solitons (see [8]) and nowa- 0 0 √3 0 0 (2) days is spread over the fields of and 2 0 0 0 √6 0 푎1 0 geometric analysis. [ ] , ⋯ , [ ⋱ ⋱ ] , 0 0 √6 0 푎푘−1 0 In this article we discuss and find solitons in many dif- 0 0 0 2 0 ferent contexts, including matrices, polynomials, plane where 푎 ≔ √푖(푘 − 푖) (rather than the expected matrices curves, Lie group representations (moment maps), and the 푖 with 푎 = 1 for all 푖). Any nilpotent matrix is therefore variety of Lie algebras, as well as in the context of geomet- 푖 conjugate to a soliton by using the Jordan canonical form. ric structures (Riemannian, Hermitian, almost-Kähler, and 1 An easy computation gives that actually 푞(퐴) ≔ 퐺2) and their homogeneous versions on Lie groups. 1 − grad(E)|퐴, where E ∶ 픤픩 → ℝ is the functional E(퐵) ≔ 4 푛 2. Matrices |[퐵, 퐵푡]|2 measuring how far a matrix is from being normal. Consider Γ = 픤픩푛, the vector space of all real 푛 × 푛 matri- Note that E is not constant on [퐴] due to scaling, but it is ces. As is well known, a matrix is semisimple (i.e., diago- easy to see that 퐴 is a soliton if and only if it is a critical nalizable over ℂ) if and only if it is conjugate to a normal point of the normalized functional E(퐵) ≔ E(퐵)/|퐵|4. matrix (i.e., [퐴, 퐴푡] = 0). The subset of normal matrices An interesting characterization of normal matrices, per- is invariant under scaling and the action of the orthogo- haps less known, is that they have minimal norm among nal group O(푛) by conjugation, and there is exactly one their conjugacy classes. Also, it is not hard to show that O(푛)-orbit of normal matrices in each semisimple conju- if 퐴 = 푆 + 푁, where 푆 is semisimple, 푁 nilpotent, and gacy class. With the aim of finding distinguished matrices [푆, 푁] = 0, then 푆 ∈ GL푛 ⋅ 퐴. In particular, if a conju- other than normal matrices, we consider orthogonal con- gacy class GL푛 ⋅ 퐴 is closed, then 퐴 is necessarily semisim- jugation and scaling as the equivalence relation, that is, ple (also note that 0 ∈ GL푛 ⋅ 푁 for any nilpotent matrix [퐴] ≔ 퐻⋅퐴 = {푐ℎ퐴ℎ−1 ∶푐 ∈ ℝ∗, ℎ ∈ O(푛)}, 퐻 ≔ ℝ∗O(푛). 푁). The following nice properties of soliton matrices fol- low from well-known results in geometric invariant theory It is easy to see that the tangent space at a matrix 퐴 of its (GIT for short) and the fact that the moment map (to be conjugacy class GL ⋅ 퐴 is given by 푇 GL ⋅ 퐴 = [퐴, 픤픩 ], 푛 퐴 푛 푛 defined later) for the GL푛-action on 픤픩푛 by conjugation is 푡 where GL푛 is the group of all invertible real 푛 × 푛 matrices, precisely m(퐴) = [퐴, 퐴 ]: so the simplest preferred direction that will make normal • For any nilpotent matrix 퐴, there is exactly one matrices solitons is ∗ ℝ O(푛)-orbit of solitons in its conjugacy class GL푛 ⋅ 퐴. 푡 푞(퐴) ≔ [퐴, [퐴, 퐴 ]]. • Any soliton 퐴 is a minimum of E restricted to GL푛 ⋅ 퐴; that is, a nilpotent soliton is in a sense the matrix According to (1), a matrix 퐴 is a soliton if and only if closest to being normal in its conjugacy class. 푡 • A conjugacy class GL푛 ⋅ 퐴 is closed if and only if 퐴 is [퐴, [퐴, 퐴 ]] = 푐퐴 + [퐴, 퐵] ∈ 푇퐴[퐴], 푐 ∈ ℝ, 퐵 ∈ 픰픬(푛), semisimple, if and only if GL푛 ⋅ 퐴 contains a matrix of where 픰픬(푛) denotes the space of skew-symmetric matrices. minimal norm (or normal matrix). Since [퐴, 퐵] ⟂ 퐴, 퐴푡 (relative to the usual inner product • The negative gradient flow solution 퐴(푡) of the func- tr 푋푌 푡), we obtain that 퐴 is a soliton if and only if either 퐴 tional E starting at 퐴 stays in the conjugacy class 푡 2 푡 is normal (푐 = 0 implies −|[퐴, 퐴 ]| = ⟨[퐴, [퐴, 퐴 ]], 퐴⟩ = 0) GL푛 ⋅ 퐴, and if 퐴 = 푆 + 푁 as above, then 퐴(푡) con- or 퐴 is nilpotent (푐 ≠ 0 implies tr 퐴푘 = 0 for any 푘) and verges as 푡 → ∞ to either a normal matrix in the con- satisfies the following matrix equation: jugacy class of 푆 or, in the case 푆 = 0, to a soliton in

|[퐴,퐴푡]|2 the conjugacy class of 푁. [퐴, [퐴, 퐴푡]] = − 퐴. 2 |퐴| 3. Polynomials Γ 푃 1In order to limit the number of references to twenty, most citations have been The space is now given by 푛,푑, the vector space of all omitted, and mainly survey articles where the precise references can be found homogeneous polynomials of degree 푑 in 푛 variables with have been included. coefficients in ℝ (e.g., quadratic (푑 = 2) and binary (푛 = 2)

648 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 forms). There is a natural left GL푛-action on 푃푛,푑 given by ℎ ⋅ 푓 ≔ 푓 ∘ ℎ−1 and the inner product for which the basis of monomials

퐷 푑1 푑푛 {푥 ≔ 푥1 ⋯ 푥푛 ∶ 푑1 + ⋯ + 푑푛 = 푑} , 퐷 ≔ (푑1, … , 푑푛),

퐷 2 is orthogonal and |푥 | ≔ 푑1! ⋯ 푑푛! /푑! is O(푛)-invariant. The role of normal matrices in the previous section is 훾″(푡) played here by the renowned harmonic polynomials, i.e., Δ푓 = 0, where Δ is the Laplace operator defined by 훾 휕2 휕2 Δ ∶ 푃푛,푑 ⟶ 푃푛,푑−2, Δ ≔ 2 + ⋯ + 2 . 휕푥1 휕푥푛

Δ is O(푛)-equivariant, and so the subspace ℋ푛,푑 ⊂ 푃푛,푑 of Figure 2. Curvature of a plane curve. all harmonic polynomials is invariant under scalings and orthogonal maps. Moreover, ℋ푛,푑 is known to be O(푛)- Concerning evolution, given 푓 = 푓푗 + 푓푗+1 + ⋯ + 푓푘, irreducible (i.e., the only O(푛)-invariant subspaces are {0} 푓푗, 푓푘 ≠ 0, 푗 < 푘, relative to decomposition (3) (i.e., 푓푖 ∈ ℋ and 푛,푑). 푟2푖ℋ ), the solution to the corresponding flow It is therefore natural to consider the equivalence de- 푛,푑−2푖 ∗ 푑 2 fined by 퐻 = ℝ O(푛) ⊂ GL푛 and as a preferred direction 푓(푡) = −푟 Δ푓(푡), 푓(0) = 푓, 푑푡 at 푓 ∈ 푃푛,푑, −푡휆푗 −푡휆푘 2 is given by 푓(푡) = 푒 푓푗 + ⋯ + 푒 푓푘. This implies that 푞(푓) ≔ −푟 Δ푓 ∈ 푃푛,푑 = 푇푓푃푛,푑, 1 1 lim 푓(푡) = 푓푗, where 푡→∞ |푓(푡)| |푓푗 |

2 2 2 and so each polynomial in the open and dense subset of 푟 ≔ 푥1 + ⋯ + 푥푛 ∈ 푃푛,2. 푃푛,푑 defined by 푓0 ≠ 0 flows to some harmonic polynomial. Note that harmonic polynomials are the fixed points of On the contrary, any polynomial 푓 with 푓0 = 0 will be the corresponding flow. A polynomial 푓 is a soliton (see stopped in its way to ℋ푛,푑 by a soliton. (1)) if and only if 4. Plane Curves 푟2Δ푓 = 푐푓 + 휃(퐴)푓 ∈ 푇 (퐻 ⋅ 푓), 푐 ∈ ℝ, 퐴 ∈ 픰픬(푛), 푓 Plane curves naturally flow according to their curvature, 푑 | 푡퐴 where 휃(퐴)푓 ≔ | 푒 ⋅ 푓. Are there solitons other than and this may be considered as the gene of all geometric 푑푡 0 harmonic polynomials? flows. We first note that Let Γ be the space of all regular plane curves, 2 2 4 Γ ≔ {훾 ∶ ℝ ⟶ ℝ ∶ 훾 is differentiable}. 푃푛,푑 = ℋ푛,푑 ⊕ 푟 ℋ푛,푑−2 ⊕ 푟 ℋ푛,푑−4 ⊕ ⋯ (3) Two curves are considered equivalent if their traces coin- is a decomposition of 푃푛,푑 in irreducible O(푛)-invariant cide up to rotations, translations, and scaling. We note subspaces2 (note that 푟푘 is fixed by O(푛) for any 푘 ∈ ℕ). that an element of 푇 Γ consists of a vector field along the This is suggesting candidates for solitons. For instance, a 훾 curve 훾 or, in other words, a smooth family of vectors, one straightforward computation gives that if 푓 = 푟2푘푔 with at each point of the trace of the curve (see Figure 2). After 푔 ∈ ℋ , then 푛,푑−2푘 assuming that 훾 is parametrized by arc length (i.e., |훾′| ≡ 1), 2 푟 Δ푓 = 휆푘푓, where 휆푘 ≔ 2푘(2푑 − 2푘 + 푛 − 2), (4) the most natural preferred or optimal direction is its “cur- and thus 푓 is a soliton.3 Moreover, by writing any poly- vature,” 푞(훾) ≔ 훾″. nomial according to (3) and using that 휆0 < 휆1 < 휆2 < ⋯ and 휃(픰픬(푛))푓 ⟂ 푓 for any 푓 ∈ 푃푛,푑, it is easy to see that any Being a measure of how sensitive your constant velocity soliton is actually of this form. Thus the subset of solitons car 훾′ is to passing through the point 훾(푠), 훾″(푠) certainly is precisely the union of the O(푛)-irreducible subspaces in provides a good perception of how curved the trace of 훾 is decomposition (3). at that point (see Figure 2). The evolution equation defined by this preferred direc- 2 2 Since 푟 푃푛,푑−2 is the orthogonal complement of ℋ푛,푑 in 푃푛,푑, which is tion is called the curve shortening flow (CSF for short); the straightforward to check, and hence Δ푃푛,푑 = 푃푛,푑−2. following are just a few of its several wonderful properties: 3This is strongly related to the fact that 푑(푑 + 푛 − 2) is precisely the 푑th eigen- value of the Laplace–Beltrami operator on the sphere 푆푛−1 with eigenspace • Each of the following kinds of curves is invariant un- ℋ푛,푑|푆푛−1 . der the flow: embedded, closed, simple, and convex.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 649 y

y

1 x 1 1 x 1

Figure 5. Three expanding solitons (left) and one rotating soliton (right).

Figure 3. The unique translating soliton is the Grim Reaper found by Calabi, given by the graph of the function y y 푦 = − log(cos 푥). Its curvature produces the precise gentle breeze needed to move it up without losing its shape.

• For closed curves, the CSF is precisely the negative gra- 1 ″ dient flow of the length; that is, 푞(훾) = 훾 is the op- 1 x x timal direction to shorten a closed curve, explaining 1 1 the name of the flow. • Grayson proved that under CSF, any simple curve be- comes convex (cf. Figure 2 and the opener image on the right), and Gage–Hamilton showed that once it is convex, it converges toward a round point (i.e., asymp- Figure 6. Rotating and expanding soliton (left) and rotating totically becoming a circle), collapsing in finite time. and shrinking soliton (right). We note that according to the equivalence relation on 5. Lie Group Representations Γ considered above, a curve is a soliton if and only if it evolves under CSF without losing its shape, i.e., by only a As a massive generalization of the matrices example given 4 combination of rotations, translations, and scalings (pos- above, we may consider any linear action of a Lie group 퐺 sibly expanding or shrinking). It is therefore easy to con- on a real vector space 푉. Thus Γ = 푉, and a natural ques- vince ourselves that a circle 훾 is a soliton; indeed, 푞(훾) is tion arises: What would be a distinguished vector 푣 ∈ 푉 in the appropriate sense tangent to the subset of all circles analogous to a normal matrix? If we endow 푉 with an in- with the same center as 훾. It is not so easy, however, to fig- ner product, then natural candidates are minimal vectors; ure out what would be another example of soliton, other i.e., |푣| ≤ |ℎ ⋅ 푣| for any ℎ ∈ 퐺 (recall the characteriza- than straight lines, which are the trivial solitons with 푞 = 0. tion of normal matrices as minimal vectors in their conju- The complete classification of CSF-solitons was ob- gacy classes). Note that any closed 퐺-orbit contains a min- tained by Halldorsson in [7]. We give in Figures 3, 4, 5, imal vector. The next question, more intriguing, is, What and 6 examples of all the behaviors that appear. Note that should the other solitons be, playing the role of nilpotent in particular solitons that translate and are scaled at the soliton matrices (see (2)) in this much more general con- same time do not exist. text? Motivated by the following equation satisfied in the case of matrices,

푡 1 푑 | 푡퐵 −푡퐵 2 ⟨[퐴, 퐴 ], 퐵⟩ = | |푒 퐴푒 | , 2 푑푡 푡=0 we fix inner products on the Lie algebra 픤 of 퐺 and on 푉 and consider for each 푣 ∈ 푉 the element m(푣) ∈ 픤 implicitly defined by Figure 4. Three shrinking solitons from the infinite discrete 1 푑 | 2 ⟨m(푣), 푋⟩ = | |exp 푡푋 ⋅ 푣| = ⟨휃(푋)푣, 푣⟩ ∀푋 ∈ 픤, family obtained in the classification by Abresch–Langer. Their 2 푑푡 푡=0 evolution consists of the very same flower, just reducing its (5) size.

4A group that is also, compatibly, a differentiable manifold.

650 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 where 휃 ∶ 픤 → 픤픩(푉) is the corresponding Lie algebra rep- • A 퐺-orbit is closed if and only if it contains a minimal 푑 | vector. The closure of any 퐺-orbit contains exactly one resentation (i.e., 휃(푋)푣 ≔ | exp 푡푋 ⋅ 푣 ∈ 푇푣(퐺 ⋅ 푣)). Thus 푑푡 0 퐾-orbit of minimal vectors. m(푣) encodes the behavior of the norm of vectors inside • The subset of solitons of a given 퐺-orbit is either the orbit 퐺 ⋅ 푣 in a neighborhood of 푣. Moreover, by (5), empty or consists of exactly one 퐾-orbit (up to scal- −휃(m(푣))푣 ∈ 푇 (퐺 ⋅ 푣) is the direction of fastest norm de- 푣 ing). creasing tangent to the orbit 퐺 ⋅ 푣 at 푣, in the sense that • Every soliton 푣 is a minimum of the functional E re- 푑 | | exp (−푡 m(푣)) ⋅ 푣|2 = −2⟨휃(m(푣))푣, 푣⟩ = −2| m(푣)|2 stricted to 퐺 ⋅ 푣. Solitons are therefore the vectors 푑푡 | 0 closest to being a minimal vector in their 퐺-orbit in ≤ 2⟨m(푣), 푋⟩ = ⟨2휃(푋)푣, 푣⟩ a sense. 푑 | 2 = | | exp 푡푋 ⋅ 푣| , • The negative gradient flow solution of E starting at any 푑푡 0 푣 ∈ 푉 stays in 퐺 ⋅ 푣 and converges as 푡 → ∞ to a for any 푋 ∈ 픤 such that |푋| = | m(푣)|, where equality holds soliton 푤 ∈ 퐺 ⋅ 푣. Moreover, there is exactly one 퐾- if and only if 푋 = − m(푣) (note that any minimal vector 푣 orbit (up to scaling) of solitons 푧 ∈ 퐺 ⋅ 푣 such that satisfies m(푣) = 0). m(푧) ∈ 퐾 ⋅ m(푤), which is the limit set towards which We assume from now on that the following conditions the whole orbit 퐺 ⋅ 푣 is flowing. on the 퐺-action on 푉 hold. If In what follows, we analyze the existence of solitons on 푡 픨 ≔ {푋 ∈ 픤 ∶ 휃(푋) = −휃(푋)}, some particular examples of representations. 푡 픭 ≔ {푋 ∈ 픤 ∶ 휃(푋) = 휃(푋)}, Ternary cubics. Consider 퐺 = SL3 acting on 푉 = 푃3,3, the then 픤 = 픨 ⊕ 픭 and 퐺 = 퐾 exp 픭, where 퐾 ≔ {ℎ ∈ 퐺 ∶ vector space of all homogeneous polynomials of degree 3 푣 ↦ ℎ ⋅ 푣 is orthogonal}. It follows that 픤 is reductive on 3 variables with real coefficients. It follows that 픤 = 픰픩 퐾 = SO(3) 픨 = 픰픬(3) 픭 = sym (3) (i.e., semisimple modulo an abelian factor), 픤 = 픨 ⊕ 픭 3, , , and 0 , the space of is a Cartan decomposition (i.e., [픨, 픨] ⊂ 픨, [픨, 픭] ⊂ 픭, and traceless symmetric 3 × 3 matrices. It is easy to compute m ∶ 푃 → sym (3) [픭, 픭] ⊂ 픨), 퐾 is a maximal compact subgroup of 퐺 with Lie that the moment map 3,3 0 is given by algebra 픨, and the function 퐾 × 픭 → 퐺, (ℎ, 푋) ↦ ℎ exp 푋 1 휕푓 m(푓) = 퐼 − [⟨푥 , 푓⟩] . 2 푗 is a diffeomorphism. By (5), m(푣) ∈ 픭 for any 푣 ∈ 푉, and |푓| 휕푥푖 the function m ∶ 푉 → 픭, called in GIT the moment map5 m(푥퐷) = Diag(1−푑 , 1−푑 , 1−푑 ) (or 퐺-gradient map) for the action, is 퐾-equivariant. Thus 1 2 3 for any monomial 푥퐷 퐷 = (푑 , 푑 , 푑 ) Similarly to matrices, we consider Γ = 푉, [푣] = 퐾 ⋅ 푣, , 1 2 3 , and so any monomial is a soliton by E(푥퐷) = −1+∑ 푑2 and as the preferred direction, (6) with critical value 푖 (this holds on any 푃푛,푑). It also easily follows that E(푝) = 0 for 푝 ≔ 푥1푥2푥3; 푞(푣) ≔ − grad(E)|푣, where E ∶ 푉 ⧵ {0} → ℝ, that is, 푝 is a minimal vector and its SL3-orbit is therefore closed. On the other hand, the polynomials |m(푣)|2 E(푣) ≔ , |푣|2 1 2 2 2 5 2 3 푔 = 푥 푥3 + 푥1푥 and 푓 = 푥 푥3 + ( ) 푥 is the 퐾-invariant functional measuring how far 푣 is from 1 2 1 27 2 being a minimal vector. Therefore, solitons are precisely 1 푑 are both solitons with critical values E(푔) = and E(푓) = 2 the critical points of E (i.e., 푞(푣) = 0), and 푣(푡) = 푞(푣(푡)) 155 1 푑푡 − 3 < , respectively, which in particular implies that is the negative gradient flow of the functional E. A straight- 49 2 forward computation gives that 푓 ∉ SL3 ⋅ 푔. 4 2 Algebras. We consider the vector space grad(E)|푣 = (휃(m(푣))푣 − | m(푣)| 푣) ; |푣|2 푉 = 풜 ≔ {휇 ∶ ℝ푛 × ℝ푛 → ℝ푛 ∶ 휇 is bilinear}, hence 푣 is a soliton if and only if 휃(m(푣))푣 ∈ ℝ푣. (6) parametrizing the set of all 푛-dimensional algebras over ℝ. Note that isomorphism classes of algebras are precisely 퐺- This suggests that, as in the case of matrices, there may be orbits, where 퐺 = GL푛, relative to the standard GL푛-action solitons other than minimal vectors, probably having non- on 푉 given by ℎ ⋅ 휇 ≔ ℎ휇(ℎ−1⋅, ℎ−1⋅). The corresponding closed 퐺-orbits (see examples below). The following are representation, nice and important results from real GIT (see [3, 9]): 휃(퐴)휇 = 퐴휇 − 휇(퐴⋅, ⋅) − 휇(⋅, 퐴⋅) ∀퐴 ∈ 픤픩푛, 휇 ∈ 풜, (7) 5The suggestive name comes from the fact that if we complexify everything, then m is precisely the moment map for the Hamiltonian action of 퐾 on the complex measures how far 퐴 is from being a derivation of the al- projective space 푃(푉ℂ). gebra 휇. Thus 픨 = 픰픬(푛), 픭 = sym(푛), 퐾 = O(푛), and it

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 651 is straightforward to see that the moment map m ∶ 풜 → The study of soliton Lie algebras was strongly motivated sym(푛) is given by by their relationship with left-invariant Ricci solitons and

1 2 Einstein metrics on Lie groups (see [12]). The author is not ⟨m(휇)푋, 푋⟩ = − ∑⟨휇(푒푖, 푒푗), 푋⟩ (8) 2 aware of any study of solitons in other classes of algebras, 1 2 such as associative or Jordan algebras. + ∑⟨휇(푋, 푒푖), 푒푗⟩ 8 1 2 푛 6. Geometric Structures + ∑⟨휇(푒푖, 푋), 푒푗⟩ ∀푋 ∈ ℝ . 8 Let 푀 be a differentiable manifold. We consider the Since tr m(휇) = −|휇|2 by (5) and (7), the only minimal space Γ of all geometric structures on 푀 of a given type, vector is the trivial algebra 휇 = 0, which is also the only e.g., Riemannian metrics, almost-Hermitian structures, 퐺2- closed GL푛-orbit (note that actually 0 ∈ GL푛 ⋅ 휇 for any structures, etc. As usual, Γ is identified with a subset of the 푟,푠 휇 ∈ 풜). However, there are closed SL푛-orbits and SL푛- vector space 풯 푀 of all tensor fields of some type (푟, 푠), minimal vectors. or tuples of tensors, and the equivalence relation is scal- According to (6), an algebra product 휇 ∈ 풜 is a soliton ing and pulling back by diffeomorphisms. Thus the equiv- if and only if the following nice compatibility condition alence class of 훾 ∈ Γ is determined by the natural action between 휇 and the fixed inner product ⟨⋅, ⋅⟩ holds: of the group 퐻 ≔ Diff(푀) × ℝ∗ on tensor fields: m(휇) = 푐퐼 + 퐷, 푐 ∈ ℝ, 퐷 ∈ Der(휇). (9) [훾] ≔ 퐻 ⋅ 훾 = {푐ℎ∗훾 ∶ 푐 ∈ ℝ∗, ℎ ∈ Diff(푀)}. Which algebras are isomorphic to a soliton? How special The preferred direction 푟,푠 are they? 훾 ↦ 푞(훾) ∈ 푇훾Γ ⊂ 풯 푀 Lie algebras. We now list only a few of several known is typically given by a curvature tensor associated to some results on solitons in the case of Lie algebras (see [12]). affine connection associated with 훾, or the gradient of Note that the set of all 푛-dimensional Lie algebras is a natural geometric functional, or the Hodge–Laplacian parametrized by the GL -invariant algebraic subset 푛 on differential forms, etc. Thus 푞 is in most cases dif- ℒ ⊂ 풜 (10) feomorphism equivariant, i.e., 푞(ℎ∗훾) = ℎ∗푞(훾) for any ℎ ∈ Diff(푀), which implies that if 훾 is a soliton, then any of all algebras that are in addition skew-symmetric and sat- ∗ 6 ℎ 훾 is also a soliton. isfy the Jacobi condition, called the variety of Lie algebras. Γ is many times open in a vector subspace 풯 ⊂ 풯푟,푠푀, • SL푛 ⋅ 휇 is closed if and only if 휇 is semisimple. More- in which case one has that 푇훾Γ = 풯, and so any tensor over, 휇 is an SL푛-minimal vector if and only if the field in 풯 can be the “direction of improvement” 푞(훾) at Killing form 퐵휇 is either a negative multiple of ⟨⋅, ⋅⟩ and the structure 훾 ∈ Γ. 휇 is compact semisimple, or 퐵휇 has exactly two oppo- Once the space Γ and the preferred direction 푞 have site eigenvalues (relative to ⟨⋅, ⋅⟩) and the eigenspace been specified, it follows from (1) that 훾 ∈ Γ is a soliton if decomposition is a Cartan decomposition. and only if • There is a soliton in the isomorphism class of each of 푞(훾) ∈ 푇 (퐻 ⋅ 훾), 퐻 = Diff(푀) × ℝ∗, the fifty nilpotent Lie algebras of dimension ≤ 6 (see 훾 [20]). In dimension 7, there are infinitely many nilpo- which is equivalent to tent Lie algebras that are not isomorphic to a soliton, 푞(훾) = 푐훾 + ℒ푋 훾, 푐 ∈ ℝ, 푋 ∈ 픛(푀), (11) and a complete classification was obtained in [6]. • The only known general obstruction in the nilpotent where ℒ푋 denotes the Lie derivative with respect to the case is that any soliton 휇 has to admit an ℕ-gradation. vector field 푋 of 푀. Indeed, recall that if 푋 is defined by a Everything seems to indicate that a full structural char- one-parameter family 푓(푡) ∈ Diff(푀) with 푓(0) = id, then 푑 | ∗ acterization of nilpotent solitons may be hopeless. | 푓(푡) 훾 = ℒ푋 훾. 푑푡 0 • A Lie algebra 휇 is a soliton if and only if its nilradical 픫 is a soliton and the orthogonal complement 픯 of 픫 is Example 6.1 (Ricci solitons). Consider Γ = ℳ, the space 푡 of all Riemannian metrics on 푀. Thus ℳ is open in 풯 = a reductive Lie algebra such that ad휇 푋|픫 ∈ Der(픫) for 풮2푀 ⊂ 풯2,0푀, the vector space of all symmetric 2-tensors any 푋 ∈ 픯. In that case, ad휇 푋|픫 is a normal operator for any 푋 in the center of 픯 and the subspaces 픨 ≔ {푋 ∶ on 푀. A natural preferred direction is 푞(푔) ≔ −2 Ric푔, 푡 푡 where Ric is the Ricci tensor of the metric 푔 ∈ ℳ, giv- ad휇 푋|픫 = − ad휇 푋|픫} and 픭 ≔ {푋 ∶ ad휇 푋|픫 = ad휇 푋|픫} 푔 give rise to a Cartan decomposition [픯, 픯] = 픨 ⊕ 픭 of ing rise to the well-known Ricci solitons (see [5, Chap- the semisimple Lie subalgebra [픯, 픯]. ter 1]). Note that the corresponding evolution equation 휕 is precisely the famous Ricci flow 푔(푡) = −2 Ric푔(푡) in- 휕푡 6 휇(휇(푒푖, 푒푗), 푒푘) + 휇(휇(푒푗, 푒푘), 푒푖) + 휇(휇(푒푘, 푒푖), 푒푗) = 0 for all 푖, 푗, 푘. troduced in the 1980s by Hamilton and used as a primary

652 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 tool by Perelman to prove the Poincar´eand geometriza- respectively given by tion conjectures. (−푇훼, ∞) , (−∞, ∞), (−∞, 푇훼) , In the case we want to consider a space Γ of geomet- 1 where 푇 ≔ > 0, ric structures satisfying some extra properties, e.g., an 훼 (1 − 훼)|푐| integrability-like condition such as Hermitian, almost- often called immortal, eternal, and ancient solutions, respec- Kähler, or closedness/coclosedness for 퐺 -structures, we 2 tively. For instance, 훼 = 0 if 푞 is the Ricci tensor or form have to reduce accordingly the group 퐻 ⊂ Diff(푀) × ℝ∗ of any connection associated to a metric or to an almost- determining the equivalence between structures. The pos- 1 Hermitian structure, and 훼 = for most of the known sibilities for preferred directions also decrease, and the vec- 3 tor field 푋 in the definition of soliton (11)must be tangent flows for 퐺2-structures. In what follows, we give an overview of different kinds to 퐻 as 푞(훾) ∈ 푇훾(퐻 ⋅ 훾). Note that 푞 is assumed to be only 퐻-equivariant in this situation rather than diffeomor- of solitons in complex, symplectic, and 퐺2 geometries. phism equivariant. On the other hand, Γ is no longer open Chern–Ricci solitons. For a given complex manifold in 풯. (푀, 퐽), consider the space Γ of all Hermitian metrics on 푀 (or 퐽-invariant, i.e., 푔(퐽⋅, 퐽⋅) = 푔). Thus 풯 ⊂ 풯2,0푀 is the Example 6.2. If a complex manifold (푀, 퐽) is fixed and vector space of holomorphic (or 퐽-invariant) symmetric 2- Hermitian metrics or any other kind of geometric struc- tensors, and the group providing the equivalence relation tures on (푀, 퐽) are to be considered, then 퐻 = Aut(푀, 퐽) × is the subgroup 퐻 ⊂ Diff(푀) of bi-holomorphic diffeomor- ∗ ℝ , where Aut(푀, 퐽) is the group of bi-holomorphic diffeo- phisms (i.e., ℎ∗퐽 = 퐽). The Chern connection ∇퐶, being morphisms, 푞 is assumed to be only Aut(푀, 퐽)-equivariant, the only Hermitian connection (i.e., ∇퐶푔 = 0, ∇퐶퐽 = 0) and 푋 has to be a holomorphic field. Analogously, in the with an anti-퐽-invariant torsion, provides us with the nat- symplectic case, 퐻 = Aut(푀, 휔), the group of symplecto- 퐶 퐶 ural preferred direction 푞(푔) ≔ −2 Ric푔 ∈ 풯, where Ric푔 morphisms of a fixed (푀, 휔). is the corresponding Chern–Ricci tensor of the Hermitian metric 푔. Note that 푔 is a soliton, called the Chern–Ricci Concerning the associated evolution equation, 퐶 soliton, if and only if Ric푔 = 푐푔 + ℒ푋 푔, for some 푐 ∈ ℝ 휕 and holomorphic vector field 푋 on 푀. The correspond- 훾(푡) = 푞(훾(푡)), 훾(0) = 훾, (12) 휕푡 ing Chern–Ricci flow was introduced by Gill and has been studied by Tosatti–Weinkove among others. Examples of one easily obtains that 훾 is a soliton if and only if homogeneous Chern–Ricci solitons were given in [16]. ∗ 훾(푡) = 푐(푡)푓(푡) 훾, 푐(푡) ∈ ℝ, 푓(푡) ∈ Diff(푀); (13) Remark 6.3. Another possible preferred direction for Her- mitian metrics is to just take the 퐽-invariant part of the that is, 훾(푡) is a self-similar solution. It is worth pointing Ricci tensor, given by out at this point that the natural preferred direction 푞 cho- 1,1 1 sen may or may not produce a flow, as the existence of 푞(푔) ≔ − Ric푔 = − (Ric푔 + Ric푔(퐽⋅, 퐽⋅)) . 2 solutions to the PDE (12) is not always guaranteed. So possibly, a study of solitons can be worked out without This choice does not give rise to any geometric flow on the any reference to a flow. An example of this situation is set of all Hermitian metrics. However, as recently shown Ricci solitons in pseudo-Riemannian geometry (see [4]). by Lafuente–Pujia–Vezzoni, it coincides with the Hermit- On the other hand, even though the flow is not defined on ian curvature flow (HCF for short) introduced by Streets– the whole space Γ, there may be special subclasses Γ′ ⊂ Γ Tian among left-invariant Hermitian metrics on complex on which solutions to (12) do exist, e.g., homogeneous unimodular Lie groups. structures (see below). Pluriclosed solitons. Consider now on (푀, 퐽) the space Γ Assuming that the scaling behavior of the preferred di- of all Hermitian metrics on 푀 that satisfy the pluriclosed 훼 ∗ rection 푞 is given by 푞(푐훾) = 푐 훾 for any 푐 ∈ ℝ , 훾 ∈ Γ, for condition 휕휕휔 = 0, where 휔 = 푔(퐽⋅, ⋅) ∈ Ω2푀 (also some fixed 훼 < 1, it is easy to check that the scaling in (13) called SKT metrics). A natural preferred direction here is given by 1,1 is given by 푞(푔) ≔ − (Ric퐵) , where Ric퐵 denotes the 1 푔 푔 푐(푡) = ((1 − 훼)푐푡 + 1) 1−훼 , Bismut–Ricci tensor of 푔 associated to the Bismut connec- 7 퐵 tion. Pluriclosed solitons (i.e., Ric푔 = 푐푔 + ℒ푋 푔) and the where 푐 is the constant appearing in the soliton equation corresponding pluriclosed flow, which coincides with HCF (11). The soliton 훾 is therefore called expanding, steady, or shrinking depending on whether 푐 > 0, 푐 = 0, or 푐 < 0. 7The only Hermitian connection whose torsion satisfies that (푋, 푌, 푍) ↦ The maximal time intervals of the self-similar solutions are 푔(푇 퐵(푋, 푌), 푍) is a 3-form.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 653 on SKT metrics, have been studied by several authors (see Laplacian solitons. A 퐺2-structure on a 7-dimensional dif- [1]). ferentiable manifold 푀 is a differential 3-form 휑 that is Anticomplexified Ricci solitons. Let (푀, 휔) be a symplec- positive (or definite), in the sense that 휑 (uniquely) de- tic manifold and let Γ denote the space of all compatible termines a Riemannian metric 푔 on 푀 together with an 2 metrics (i.e., 퐽 = −퐼 if 휔 = 푔(퐽⋅, ⋅)). For each 푔 ∈ Γ, orientation. The space Γ of all 퐺2-structures on 푀 is an 3 3,0 the pair (휔, 푔) is called an almost-Kähler structure (in other open subset of 풯 = Ω 푀 ⊂ 풯 , and the equivalence is ∗ words, an almost-Hermitian structure (휔, 푔, 퐽) such that determined by 퐻 = Diff(푀) × ℝ . A very natural preferred 푑휔 = 0). It follows that 풯 is the vector space of anti- direction is 푞(휑) = Δ휑휑, where Δ휑 = ∗푑∗푑−푑∗푑∗ denotes 퐽-invariant symmetric 2-tensors (i.e., 푞(퐽⋅, 퐽⋅) = −푞) and the Hodge Laplace operator on forms and ∗ the Hodge star 퐻 ⊂ Diff(푀) is the subgroup of symplectomorphisms (i.e., operator attached to 푔 and the orientation. Indeed, if 푀 is ∗ ℎ 휔 = 휔). As a preferred direction, the simplest option is compact and Δ휑휑 = 0, then 푔 is Ricci flat and has holo- 2,0+0,2 1 nomy group contained in the exceptional compact simple 푞(푔) ≔ Ric푔 = (Ric푔 − Ric푔(퐽⋅, 퐽⋅)) , 2 Lie group 퐺2. 휕 The corresponding Laplacian flow 휑(푡) = Δ휑(푡)휑(푡) was the anti-퐽-invariant part of the Ricci tensor Ric푔. This flow 휕푡 was introduced by Le–Wang, and examples of homoge- introduced back in 1992 by Bryant as a tool to try to de- neous solitons were given by Fern´andez–Culma(see [13]). form closed 퐺2-structures toward holonomy 퐺2 and has Symplectic curvature flow solitons. We now consider the recently been deeply studied by Lotay–Wei (see [18]). We larger and trickier space Γ of all almost-Hermitian struc- refer to [14] for an account of the existence and structure tures (i.e., a 3-tuple (휔, 푔, 퐽) such that 푔 = 휔(퐽⋅, ⋅)) on a of Laplacian solitons. Interestingly, the shrinking Lapla- manifold 푀. It is easy to see that in this case cian solitons found on certain solvable Lie groups are the only Laplacian flow solutions with a finite-time singularity 2 2 1,1 1,1 풯 = {(휔, 푔) ∈ Ω 푀 × 풮 푀 ∶ 푔 = 휔 (⋅, 퐽⋅)} , (14) known so far. and we take the full 퐻 = Diff(푀)×ℝ>0, so the equivalence 7. Geometric Structures on Lie Groups class of an almost-Hermitian structure 훾 = (휔, 푔) is given The role of (locally) homogeneous manifolds in Ricci flow by [(휔, 푔)] = {(푐ℎ∗휔, 푐ℎ∗푔) ∶ 푐 ∈ ℝ∗, ℎ ∈ Diff(푀)}. 퐶 theory has been very important (see [2]). More recently, As above, we consider the Chern–Ricci tensor Ric(휔,푔) Lie groups have also played an even stronger role in the and the corresponding Chern–Ricci form 푝(휔, 푔) ≔ 퐶 study of geometric flows in complex, symplectic, and ex- Ric(휔,푔)(퐽⋅, ⋅), which is a very natural preferred direction at ceptional holonomy geometries, due mainly to the lack of 휔 to choose (indeed, 푑푝 = 0 if 푑휔 = 0, so 푝 is tangent explicit examples (see [13]). to the set of almost-Kähler structures). According to (14), We continue in this section the study of solitons in dif- the 퐽-invariant part of the preferred direction at 푔 must be ferential geometry initiated above. However, our fixed 1,1 given by 푝(휔, 푔) (⋅, 퐽⋅); hence it only remains to set the manifold is here a Lie group8 퐺, and we assume that Γ con- anti-퐽-invariant part, for which we can just choose the sim- ∗ sists of left-invariant geometric structures, i.e., 퐿푎훾 = 훾 for plest one considered above. In this way, we arrive at the any 푎 ∈ 퐺, where 퐿푎 is the diffeomorphism of 퐺 defined following natural preferred direction: by 퐿푎(푏) = 푎푏 for all 훾 ∈ Γ. Thus each 훾 is determined by 1,1 2,0+0,2 its value at the identity 푒 ∈ 퐺 and so is identified with the 푞(휔, 푔) ≔ (푝(휔, 푔), 푝(휔, 푔) (⋅, 퐽⋅) + Ric푔 ) ∈ 풯. tensor 훾푒 (or a tuple of tensors) on the Lie algebra 픤 = 푇푒퐺 Thus (휔, 푔) is a soliton if and only if there exist 푐 ∈ ℝ and of 퐺. In this way, 푋 ∈ 픛(푀) such that Γ ⊂ 푇푟,푠픤,

푝(휔, 푔) = 푐휔 + ℒ푋 휔, the finite-dimensional vector space of all left-invariant ten- { 1,1 2,0+0,2 푝 (⋅, 퐽⋅) + Ric = 푐푔 + ℒ푋 푔. sor fields of type (푟, 푠) on 퐺. Note that Γ is usually con- tained in a single GL(픤)-orbit, which is many times open in The flow is called symplectic curvature flow (SCF for short) some suitable vector subspace 푇 ⊂ 푇푟,푠픤 (e.g., inner prod- and was introduced by Streets–Tian; see [17] for a study of ucts, nondegenerate 2-forms, almost-Hermitian structures, homogeneous SCF-solitons. The fact that the three struc- SU(3)-structures, positive 3-forms, etc.). tures 휔, 푔, and 퐽 are actually evolving is particularly chal- Accordingly, we consider the equivalence between left- lenging. invariant structures to be defined by scalings and the partic- Remark 6.4. The classification of complex surfaces is ama- ular diffeomorphisms of 퐺 that are also group morphisms, jor problem motivating the study of all the above flows i.e., by the group among different subclasses of structures. They all coincide 퐻 ≔ Aut(퐺) × ℝ∗, with the Kähler–Ricci flow among Kähler stuctures (i.e., ∇퐽 = 0 for the Levi–Civita connection ∇ of 푔). 8A differentiable manifold that is also, compatibly, a group.

654 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 which is identified with Aut(픤) × ℝ∗, where Aut(픤) is the to 퐺/Λ. The solution (퐺/Λ, 훾(푡)) is very peculiar though: automorphism group of 픤, as 퐺 is assumed to be simply it is “locally self-similar” in the sense that 훾(푡) is locally connected from now on. equivalent to 훾 up to scaling for all 푡. Any preferred direction 푞 from the general case that is The moving-bracket approach. The following viewpoint diffeomorphism equivariant produces a left-invariant one. is suggested by the fact that all the geometric information Indeed, one obtains a preferred direction determined by a on a Lie group endowed with a left-invariant geometric function structure, say (퐺, 훾), is encoded in just the tensor 훾 ∈ 푇푟,푠픤 훾 ↦ 푞(훾) ∈ 푇 Γ ⊂ 푇푟,푠픤. 훾 and the Lie bracket 휇 of 픤. We consider the variety of Lie al- We therefore have that 훾 ∈ Γ is a soliton, called a semi- gebras ℒ ⊂ Λ2픤∗ ⊗ 픤 as in (10) (i.e., the algebraic subset of algebraic soliton in the literature, if and only if all Lie brackets on the vector space 픤) and fix a suitable ten- 훾 픤 휇 ∈ ℒ (퐺 , 훾) 푞(훾) = 푐훾 + ℒ푋퐷 훾, 푐 ∈ ℝ, 퐷 ∈ Der(픤), (15) sor on . Each is therefore identified with 휇 , the simply connected Lie group 퐺휇 with Lie algebra (픤, 휇) where 푋퐷 ∈ 픛(퐺) is defined at each point 푎 ∈ 퐺 by 푑 | endowed with the left-invariant geometric structure on 퐺휇 푋퐷(푎) = | 푓(푡)(푎) and 푓(푡) ∈ Aut(퐺) is determined 푑푡 푡=0 defined by the fixed 훾: 푑푓(푡)| = 푒푡퐷 ∈ Aut(픤) 푋 (푒) = 0 by 푒 (since 퐷 , these fields are 휇 ⟷ (퐺 , 훾). (18) never left invariant). Note that (퐺, 훾) is also a soliton from 휇 the general point of view considered in (11). It is easy to The natural GL(픤)-actions on tensors provide the following check that, algebraically, the Lie derivative is simply given key equivalence between geometric structures: ∗ by (퐺ℎ⋅휇, 훾) −→ (퐺휇, ℎ 훾) ∀ℎ ∈ GL(픤), (19) ≃ ℒ푋퐷 훾 = −휃(퐷)훾, (16) where 휃 denotes the usual 픤픩(픤)-representation on tensors. given by the Lie group isomorphism 퐺ℎ⋅휇 → 퐺휇 with de- −1 As in the general case, additional conditions on the deriva- rivative ℎ . Since Γ ⊂ GL(픤) ⋅ 훾, it follows from (18) and tion 퐷 may apply if Γ satisfies extra properties (see Example (19) that the isomorphism class GL(픤) ⋅ 휇 contains all geo- 6.2). metric structures of the same type of 훾 (up to equivalence) on the Lie group 퐺휇 for each 휇 ∈ ℒ. Thus one has inside ℒ, Remark 7.1. These vector fields 푋퐷 on Lie groups attached all together, all Lie groups of a given dimension endowed to a derivation 퐷 may be viewed as a generalization of lin- with left-invariant geometric structures of a given type. 푛 ear vector fields on ℝ (i.e., 푋푣 = 퐴푣, 퐴 ∈ 픤픩푛) and have been strongly used in control theory since a pioneering ar- Remark 7.3. The usual convergence of a sequence of brack- ticle by Ayala–Tirao. ets produces convergence of the corresponding geomet- ric structures in well-known senses such as pointed (or It is important to point out that in the Lie group case Cheeger–Gromov) and smooth up to pull-back by diffeo- considered in this section, the corresponding geometric morphisms, under suitable conditions (see [13]). In par- flow 푑 ticular, a degeneration (i.e., 휆 ∈ GL(픤) ⋅ 휇 ⧵ GL(픤) ⋅ 휇) gives 훾(푡) = 푞(훾(푡)), 훾(0) = 훾, (17) 푑푡 rise to the convergence of a sequence of geometric struc- is actually an ODE rather than a PDE. In particular, short tures on a given Lie group toward a structure on a different time existence and uniqueness of solutions are always guar- Lie group, which may be nonhomeomorphic. anteed. It is also known that |푞(훾(푡))| must blow up at any The moving-bracket approach has actually been used finite-time singularity (see [13]). Note that 훾 is a semialge- for decades in homogeneous geometry (see [13, Section 5] braic soliton if and only if the solution 훾(푡) to (17) is given and [15]). In most applications, concepts and results from by 훾(푡) = 푐(푡)푓(푡)∗훾 for some 푐(푡) ∈ ℝ and 푓(푡) ∈ Aut(퐺). GIT, including moment maps and their convexity proper- Due perhaps to its neat definition as a combination ties, closed orbits, stability, categorical quotients, and Kir- of geometric and algebraic aspects of (퐺, 훾) (cf. (15) and wan stratification, have been exploited in one way oran- (16)), the concept of semialgebraic soliton has a long and other. fruitful history in the Ricci flow case (see [10–12]) and has A particularly fruitful interplay occurs in the Riemann- also been a quite useful tool to address the existence prob- ian case, which relies on the fact that if 휇 is nilpotent, then lem of soliton structures for all the geometric flows in com- the Ricci operator of (퐺 , ⟨⋅, ⋅⟩) is precisely the moment plex, symplectic, and 퐺 geometries given above. 휇 2 map m(휇) defined in (8) (up to scaling). This implies that, Remark 7.2. Given a semialgebraic soliton (퐺, 훾), if 퐺 has remarkably, soliton nilpotent Lie algebras (see (9)) and a cocompact discrete subgroup Λ, then the solution 훾(푡) semialgebraic Ricci solitons (see (15) and (16)) on nilpo- also solves (17) on the compact manifold 퐺/Λ. However, tent Lie groups (called nilsolitons) are the same thing. In in general, the locally homogeneous manifold (퐺/Λ, 훾) is particular, the uniqueness up to isometry and scaling of no longer a soliton, since the field 푋퐷 does not descend nilsolitons on a given nilpotent Lie group follow from the

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 655 uniqueness of critical points of E(휇) = | m(휇)|2 on a given As expected, algebraic solitons are distinguished from nilpotent GL(픤)-orbit up to the action of O(픤) and scaling. many other points of view. Some of the results supporting The bracket flow. Provided by equivalence (19), a main this follow: tool to study the geometric flow (17) is a dynamical system • Consider the Ricci pinching functional defined on the variety of Lie algebras ℒ called the bracket 2 flow, which is equivalent in a precise sense to the geometric scal푔 퐹(푔) ≔ 2 , flow (17). It is defined by | Ric푔 | 푑 휇(푡) = 휃(푄 )휇(푡), 휇(0) = 휇, (20) 푑푡 휇(푡) measuring in a sense how far a homogeneous metric 푔 is from being Einstein (indeed, 퐹(푔) ≤ dim 푀, and where 푄휇 ∈ 픤픩(픤) is a suitable (unique) operator such that equality holds if and only if (푀, 푔) is Einstein). As 9 푑 휃(푄휇)훾 = 푞(퐺휇, 훾). Since 휇(푡) ∈ 푇휇(푡) (GL(픤) ⋅ 휇(푡)), the shown by Lauret–Will, algebraic Ricci solitons are pre- 푑푡 solution 휇(푡) ∈ GL(픤) ⋅ 휇 for all 푡, and so each 휇(푡) rep- cisely the global maxima for 퐹 restricted to the set of resents a structure on 퐺휇. However, 휇(푡) may converge to all left-invariant metrics on any unimodular Lie group, a Lie bracket 휆 ∈ GL(픤) ⋅ 휇, i.e., toward a structure on a as well as on any solvable Lie group with codimension different Lie group 퐺 (cf. Remark 7.3). For instance, this one nilradical. 휆 • occurs already in dimension 3 for the Ricci flow. Böhm–Lafuente proved that the dimension of the The bracket flow is useful to better visualize the possi- isometry group of an algebraic Ricci soliton on a solv- 푆 ble pointed limits of solutions under diverse rescalings, as able Lie group (called solvsolitons) is maximal among 푆 well as to address regularity issues. Immortal, ancient, and all left-invariant metrics on . A stronger symmetry self-similar solutions naturally arise from the qualitative maximality condition was shown to hold in the case 푆 analysis of the bracket flow (see [1, 2,13, 17]). when is in addition unimodular by Jablonski: the isometry group of a solvsoliton contains all possible Algebraic solitons. In light of the equivalence between isometry groups of left-invariant metrics on 푆 up to the flows (17) and (20), a natural question arises: Howdo conjugation by a diffeomorphism. solitons evolve according to the bracket flow? It is natural • A closed 퐺 -structure 휑 is called extremally Ricci-pinched to expect an evolution of a very special kind. 2 (ERP for short) if If 휇 is a fixed point (up to scaling) of the bracket flow 1 1 (20), i.e., 휇(푡) = 푐(푡)휇 for 푐(푡) ∈ ℝ, then 휃(푄휇)휇 = −푐휇 for 푑휏 = |휏|2휑 + ∗ (휏 ∧ 휏), some 푐 ∈ ℝ by evaluating (20) at 푡 = 0, and one obtains 6 6 from (7) that where 휏 = −∗푑∗휑 is the torsion 2-form of 휑. They are characterized in the compact case as the structures at 푄휇 = 푐퐼 + 퐷, 푐 ∈ ℝ, 퐷 ∈ Der(휇). (21) which equality holds in the following Ricci curvature In that case, (퐺휇, 훾) is called an algebraic soliton. Note that estimate for closed 퐺2-structures discovered by Bryant: these are semialgebraic solitons, since by (16), 푞(퐺휇, 훾) = 2 2 푐훾 − ℒ푋퐷 훾 for some 푐 ∈ ℝ. They are distinguished though; ∫ scal ∗1 ≤ 3 ∫ | Ric | ∗ 1. (22) indeed, it is proved in [15] that in terms of the operator 푄휇, 푀 푀 (퐺 , 훾) is a semialgebraic soliton if and only if 푄 = 푐퐼 + 휇 휇 It was proved by Lauret–Nicolini that any left- 푝(퐷) for some 푐 ∈ ℝ and 퐷 ∈ Der(휇), where 푝 ∶ 픤픩(픤) → 픮 invariant ERP 퐺 -structure on a Lie group is necessarily is the projection with respect to the decomposition10 2 a steady algebraic Laplacian soliton and its attached 픤픩(픤) = 픨 ⊕ 픮, 픨 ≔ {퐴 ∈ 픤픩(픤) ∶ 휃(퐴)훾 = 0}. metric is an expanding algebraic Ricci soliton. Thus algebraic solitons are the solitons for which 푄 = 푐퐼+ Concerning bracket flow evolution of a semialgebraic 휇 휇(푡)/|휇(푡)| 푝(퐷) holds for a special derivation 퐷 such that 푝(퐷) = 퐷 soliton that is not algebraic, we know that is ei- (see (21)). ther periodic or not periodic and the following chaotic be- 푡 푡 → ±∞ Any Ricci soliton on a Lie group is isometric to an alge- havior occurs: for each 0 there exists a sequence 푘 휇(푡 )/|휇(푡 )| 휇(푡 )/|휇(푡 )| braic soliton (see [10]). On the other hand, examples of such that 푘 푘 converges to 0 0 (see [15]). Laplacian and pluriclosed semialgebraic solitons that are The existence of a soliton of this last kind is an open prob- not isometric to any algebraic soliton were found in [19] lem. and [1], respectively. Remark 7.4. More generally, the whole picture developed in this section essentially works for 퐺-invariant geometric 9The existence of such an operator relies on the fact that GL(픤) ⋅ 훾 is open in 푇. structures on a homogeneous space 퐺/퐾, though a more 1 10For instance, 푝(퐷) = (퐷 + 퐷푡) if 훾 is a metric or a closed 퐺 -structure. 2 2 technical exposition would be necessary (see [13, 15]).

656 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 8. Concluding Remarks [7] H. P. Halldorsson, Self-similar solutions to the curve short- As discussed in the Introduction, solitons play the role of ening flow, Trans. Amer. Math. Soc. 364 (2012), no. 10, 5285–5309, DOI 10.1090/S0002-9947-2012-05632-7. “best” elements in a given set in the case when the most MR2931330 natural ones are not available. A main aim of this article [8] R. S. Hamilton, The Ricci flow on surfaces, Mathematics and was to show how fruitful this has been in the study of geo- general relativity (Santa Cruz, CA, 1986), Contemp. Math., metric structures on manifolds, with particular strength on vol. 71, Amer. Math. Soc., Providence, RI, 1988, pp. 237– Lie groups. 262, DOI 10.1090/conm/071/954419. MR954419 On each solvable Lie group, there is at most one solv- [9] P. Heinzner, G. W. Schwarz, and H. Stötzel, Strati- soliton up to isometry and scaling. This allows us to en- fications with respect to actions of real reductive groups, dow several Lie groups that do not admit Einstein metrics Compos. Math. 144 (2008), no. 1, 163–185, DOI (e.g., nilpotent or unimodular solvable Lie groups) with a 10.1112/S0010437X07003259. MR2388560 [10] M. Jablonski, Homogeneous Ricci solitons are algebraic, canonical Riemannian metric. Analogously, Chern–Ricci, Geom. Topol. 18 (2014), no. 4, 2477–2486, DOI pluriclosed, and HCF (resp., SCF) algebraic solitons pro- 10.2140/gt.2014.18.2477. MR3268781 vide distinguished Hermitian (resp., almost-Kähler) struc- [11] R. Lafuente and J. Lauret, Structure of homogeneous Ricci tures for Lie groups on which Kähler metrics do not exist. solitons and the Alekseevskii conjecture, J. Differential Geom. Laplacian algebraic solitons play the same role in the ho- 98 (2014), no. 2, 315–347. MR3263520 mogeneous case, where holonomy 퐺2 is out of reach since [12] J. Lauret, Einstein solvmanifolds and nilsolitons, New de- Ricci flat implies flat. velopments in Lie theory and geometry, Contemp. Math., The moving-bracket approach allows the rich interplay vol. 491, Amer. Math. Soc., Providence, RI, 2009, pp. 1–35, between soliton geometric structures on Lie groups and DOI 10.1090/conm/491/09607. MR2537049 soliton Lie algebras, paving the way to many beautiful ap- [13] J. Lauret, Geometric flows and their solitons on homogeneous spaces, Rend. Semin. Mat. Univ. Politec. Torino 74 (2016), plications of GIT to differential geometry. no. 1, 55–93. MR3772582 [14] J. Lauret, Distinguished G2-structures on solvmanifolds, ACKNOWLEDGMENTS. The author is very grateful to Fields Institute Commun., Springer, in press. Ramiro Lafuente, Emilio Lauret, and Cynthia Will for [15] J. Lauret, The search for solitons on homogeneous spaces, invaluable conversations during the writing of this pa- preprint, 2019 (arXiv). per. Current and past support from CONICET, FON- [16] J. Lauret and E. A. Rodr´ıguezValencia, On the Chern-Ricci CyT (ANPCyT), and SECyT (UNC) is acknowledged. flow and its solitons for Lie groups, Math. Nachr. 288 (2015), no. 13, 1512–1526, DOI 10.1002/mana.201300333. MR3395824 References [17] J. Lauret and C. Will, On the symplectic curvature flow [1] R. M. Arroyo and R. A. Lafuente, The long-time behavior for locally homogeneous manifolds, J. Symplectic Geom. 15 of the homogeneous pluriclosed flow, Proc. Lond. Math. Soc. (2017), no. 1, 1–49, DOI 10.4310/JSG.2017.v15.n1.a1. (3) 119 (2019), no. 1, 266–289, DOI 10.1112/plms.12228. MR3652072 MR3957836 [18] J. Lotay, Geometric flows of 퐺2 structures, Fields Institute [2] C. Böhm and R. A. Lafuente, Immortal homogeneous Ricci Commun., Springer, in press (arXiv). flows, Invent. Math. 212 (2018), no. 2, 461–529, DOI [19] M. Nicolini, Laplacian solitons on nilpotent Lie groups, Bull. 10.1007/s00222-017-0771-z. MR3787832 Belg. Math. Soc. Simon Stevin 25 (2018), no. 2, 183–196. [3] C. Böhm and R. Lafuente, Real geometric invariant theory, MR3819121 preprint, 2017 (arXiv). [20] C. Will, The space of solvsolitons in low dimensions, Ann. [4] M. Brozos-V´azquez, G. Calvaruso, E. Garc´ıa-R´ıo, and Global Anal. Geom. 40 (2011), no. 3, 291–309, DOI S. Gavino-Fern´andez, Three-dimensional Lorentzian homoge- 10.1007/s10455-011-9258-0. MR2831460 neous Ricci solitons, Israel J. Math. 188 (2012), 385–403, DOI 10.1007/s11856-011-0124-3. MR2897737 Credits [5] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isen- berg, T. Ivey, D. Knopf, P. Lu, F. Luo, and L. Ni, The Opener images are courtesy of Getty. Ricci flow: techniques and applications. Part I: Geometric as- Figures 1–3 are by the author. pects, Mathematical Surveys and Monographs, vol. 135, Figures 4–6 are courtesy of Höskuldur American Mathematical Society, Providence, RI, 2007. Halld´orsson. MR2302600 Author photo is courtesy of Archives [6] E. A. Fern´andez-Culma, Classification of 7-dimensional Ein- of the Mathematisches Forschungsin- stein nilradicals, Transform. Groups 17 (2012), no. 3, 639– stitut Oberwolfach. 656, DOI 10.1007/s00031-012-9186-5. MR2956161 Jorge Lauret

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 657 AMS Membership for Early- Career Mathematicians The AMS supports early-career mathematicians with opportunities for career development, access to information and job listings, connections with prospective employers, and valuable member bene ts, including:

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Photo courtesy of AMS Chapter of Brown University Join OR Renew your AMS membership today by visiting: www.ams.org/membership EARLY CAREER

The Early Career Section offers information and suggestions for graduate students, job seekers, early career academics of all types, and those who mentor them. Angela Gibney serves as the editor of this section. Next month’s theme will be variations in academic math jobs.

I interpret this breakdown literally and typically set POST- DO CS aside two full days each week for undistracted research time, two days for teaching (prep, lectures, office hours, UDENTS meeting with students, grading, reading courses, student ST TE learning seminars), and one day for service. In practice the A U teaching and service days blur together, and I also set aside D A N a day for (mostly) nonacademic work (errands, shopping, R E

W

G food prep, coordinating travel, processing email (usually

F in batch mode), and planning out my week). A

C My research time is sacred. One would never skip teach- U

L T ing a lecture or a committee meeting for reasons related to

Y research, and I encourage the converse. Work off campus S G if necessary, or at least with a closed door, headphones on, R O

O V immune to distraction. T

E

R N

E N Create (and protect) research time. I used to “joke”

M

M with contemporaneous early career faculty that our work- E

N

T

S load doubled each year. The joke quickly aged.

E

L I

A

R

B O O T R

A I recommend reading some book like Getting Things

Y Done by David Allen or Deep Work by Cal Newport, or the R

T

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U archives of an academic blog with a heavy advice subtheme D

N I (e.g., by Terry Tao1 or Matt Might2). Here are a few ways that I implement their ideas. Become organized. A main message of books like Get- ting Things Done is: if you can do something in 1–2 minutes, Research just do it; otherwise, schedule when you are planning to do it and put it out of sight and (especially) out of mind How to Balance Research until then. I do this via a calendar (synced with my phone, tablet, with Everything Else etc.). If I have a recommendation letter to write (or slides to make, PhD applications to read, etc.), I schedule a block We Have to Do of time (not just a reminder) to work on the task. When I get a request to (for example) upload an already David Zureick-Brown written recommendation to yet another UC school, I do it immediately (and organize so that it takes literally under My institution’s tenure and promotion guidelines describe a minute). my position as “40% research, 40% teaching, 20% service.” Email. Process in batch mode. Usually I’ll set aside a few Teaching includes three courses each year, but also advising hours over the weekend to handle as many menial tasks as PhD or honors thesis students, and service includes com- I can all at once. Throughout the week, I’ll usually set aside mittees, organizing conferences, refereeing, recommenda- half an hour for email in the mornings and another fifteen tion letters, etc. minutes at the end of the day, and otherwise I mostly don’t check email (especially not on my phone, which might David Zureick-Brown is an associate professor of mathematics at Emory reveal some task that I can’t attend to right away). University. His email address is [email protected]. For permission to reprint this article, please contact: reprint-permission 1https://terrytao.wordpress.com/career-advice/ @ams.org. 2http://matt.might.net/articles/productivity-tips-hints DOI: https://dx.doi.org/10.1090/noti2080 -hacks-tricks-for-grad-students-academics/

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 659 Early Career

Focus. I turn off most mobile notifications and set my work with little guidance, and reaching out to experienced phone to “do not disturb.” I stay off social media when colleagues can be productive and therapeutic. working, and (worth emphasizing again) I don’t check email as a distraction (new work will manifest, and if I can’t dedicate 10–20 minutes to process, then this lingers and distracts). Any emails related to a joint project are printed to pdf and put in a folder related to that project as they arrive so that I don’t need to check email when working. Automate. I coordinate our weekly Algebra and Number Theory research seminar. For each outside speaker, I send 5–6 emails (announcing the seminar, requesting that the staff make and post flyers, processing reimbursement, etc.). David Zureick-Brown The emails are basically the same every week, with a few variables; I have a bash script that writes the emails for me. Credits Saying no. Initially I accepted every reasonable referee Author photo is courtesy of Sarah Zureick-Brown. request. Eventually, several senior colleagues explained that they turn down many requests. Good citizens seem to referee roughly three papers for each paper they submit, How to Read and I try to stick to that (even if the paper looks interesting and I’m an appropriate referee). a Research Paper It’s difficult to say no, and psychologically, I needed “permission” to start declining requests. It was helpful Matt Baker that my department’s service expectations were clear and direct. Similarly, I found writing for Mathematical Reviews Before attempting to read a research paper, I recommend stressful and difficult but organizing conferences enjoyable first deciding why you want to read it, what you hope to and natural; these days I do more of the latter. get out of the paper, and how much time you’re willing to Efficiency. I could fill another article with habits and commit. Then place the paper into one of the following routines that create and protect time. three categories: I bike to work: it takes about 25 minutes, compared to • Speed Read: A paper whose introduction you plan 35–40 to drive and park. Our department has a shower on to read in order to get an overview of the results my floor. Cycling doubles as cardio and boosts my mood and then possibly skim further. and mental health. • Substantial Skim: A paper that you plan to skim Each semester I poll students to find a time that accom- all the way through, perhaps reading certain parts modates everyone who might possibly attend office hours. in detail. I reserve a classroom for office hours. Students can show up • Deep Dive: A paper that you wish to thoroughly even if they don’t have focused questions, and I can leave study and understand. immediately when finished. (In any case I can’t fit more It’s useful to have different categories because there’s so than 3–4 people in my office.) much interesting math research produced every day, and I scan my lecture notes after each class; I include the date, it’s impossible to keep up with everything. I go through the lecture number, and course number in the filename, and arXiv preprint listings in three different categories almost keep a terse outline of what I covered each lecture. Teaching religiously every single morning. I get email notifications a course a second (or ninth…) time is a breeze, and this for all new postings and revisions in these categories, and I type of organization creates time and space to focus on make an effort most days to “speed read” at least one new improving the course. And if I’m stuck somewhere (e.g., paper while drinking my morning coffee. I also bookmark delayed flight), I can access the notes via Dropbox and use papers to come back to later (although to be honest, I that time for review and preparation. end up not having time to come back to many of these). In fact, I scan everything; my office is across from the Skimming through the arXiv abstracts in the daily digest printer/scanner, and Dropbox has a useful mobile app (if and then speed reading at least one paper a day keeps me I have a meal receipt that needs to be reimbursed, I scan it feeling in touch with what’s happening in the fields I’m immediately). If I give a chalkboard talk, I scan the notes most interested in. (and put the scan somewhere easy to find again). Discuss! I’ve had countless conversations about these Matt Baker is a professor of mathematics at Georgia Institute of Technology. topics (at tea, conference dinners, in hallways) and have His email address is [email protected]. benefited greatly. Our profession tends to pile on extra DOI: https://dx.doi.org/10.1090/noti2077

660 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Early Career

With a Speed Read, I don’t spend a lot of time trying my current research or I’ve agreed to referee it. But that’s to come to terms with complex definitions or understand an artifact of choices I’ve made and priorities I’ve set in precisely why the hypotheses in various theorems are what my own life, and of course if you really want to build up a they are; I just want to get an overall impression of the paper formidable knowledge base, you should take Deep Dives and learn something new, even if it’s mostly superficial as often as possible. If, like me, you’re constantly feeling knowledge. This kind of reading will take me about 15 busy and overwhelmed, another way to “read” a paper is minutes for an average paper. to assign it to a student to explain to you! Or put together If the paper seems really interesting and I’m wanting a study group and parcel out the task to various students to read and learn more, I’ll upgrade it to the “Substantial and colleagues. I did this recently with the paper “Lorent- Skim” category and come back to it later in more detail. zian Polynomials” by Petter Brändén and June Huh over I’ve found that over time, the collection of quick impres- the course of a whole semester. (For an example of how sions I obtain through my Speed Reads actually provides to organize such a seminar, see https://sites.google a reasonably deep foundation for understanding various .com/view/gtlorentzian.) bits of the mathematical landscape at large. One other thing I think is important is to read widely For a smaller number of papers (roughly one a week, and try to push the boundaries of your understanding. Es- in my case, though the variance is high), I will do a more pecially with Speed Reads and Substantial Skims, don’t just Substantial Skim. This involves reading through all of the read a bunch of papers about the same topic all the time. definitions and statements, and at least some of the proofs, By reading about different topics within a short time frame, in the whole paper and making an attempt to actually your brain will automatically start to explore connections understand what’s going on. My goal here is to embed and begin thinking “outside the box.” the paper firmly enough into my mind that I’ll be able to This is my own personal approach to reading math pa- incorporate some of the ideas into my thinking later on. pers, and I don’t claim that the same techniques will work When I read a paper in this way, I will sometimes jot down for everyone. But I do think that every working mathemati- key definitions or statements (either in a physical notebook cian needs to develop a system for reading research papers or Evernote file), and I might go through certain arguments in order to attain, over time, both the breadth and depth carefully in order to understand particular points in detail. of knowledge required to keep up with the relentless but However, I don’t make an attempt to check the paper for thrilling march of progress in modern mathematics. correctness or try to fully understand all of the technical points. This is actually my favorite category of reading, be- cause I learn a tremendous amount without having to put in too many hours of work. Depending on the complexity of the paper, this kind of reading might take me anywhere between one and three hours per 10 pages. Finally, there’s the Deep Dive. These days, I probably do this for only about one paper every month or two (though when I was younger and less overwhelmed with respon- sibilities, it was quite a bit more frequent). Here, I spend as much time as it takes to understand the definitions, Matt Baker the conditions of the theorems, and the logic behind the Credits arguments. I will sometimes write out detailed notes in a Author photo is courtesy of the author. notebook, and as often as possible I’ll stop reading and see if I can figure out the next step myself. If I’m really interested in understanding the paper and retaining the knowledge, I’ll try to explain the results to someone else; there’s no better way than teaching to internalize complex informa- tion! I’ll also look up background facts in the references and, as long as it doesn’t lead to an exponential recursion, try to understand that material as well. And I’ll keep a list of typos or other errors I find and send it to the author as appropriate. This kind of reading can take anywhere from 15 minutes to a couple of hours per page, depending on the subject matter and how familiar I am with the under- lying concepts. To be perfectly honest, these days I find the time for a Deep Dive only when a paper is either directly relevant to

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 661 Early Career Changing Focus Postdoc Numero Uno: Midwest Postdoc! First job! Two years. Upon arrival, in the early fall of so many years ago, I submitted my first NSF grant Mark Andrea de Cataldo research proposal. In writing it, I realized, to my horror, that I was not interested in the problems I came up with Alla memoria di mamma e papà, con amore. and was proposing. The subsequent decline letter from the foundation only confirmed my suspicion that many others I write down here a few thoughts about changing the were in total agreement with me. Meantime, I had stumbled – direction of one’s research, mostly by referring to my per- upon some new beautiful math, estimates for solutions to ∂ sonal experience. It is not my intention to have the reader and applications to . I started a learning interpret my writing as a description of a move towards seminar with faculty and graduate students. I asked myself “better” math (whatever that means). Very plainly, I have a question, answered it, wrote two papers and my second tried to share selected memories of what happened when I NSF proposal, which was funded just in time for my second met new math that I found to be interesting and beautiful. postdoc to begin. Math that I could not resist. It would be silly to try to pose Postdoc Numero Due: Germany that what follows contains any kind of universal truth. I Postdoc! Second job! One year. Upon my arrival in Ger- hope the young reader glances at these recollections and many, a math physicist said, “Ah! You are an algebraic ge- draws some hopefully useful conclusions. ometer. Why don’t you explain to us, in a series of talks (not In My Youth one, not two…a series of them, as many as you need), the action of the Heisenberg algebra on the cohomology of the In my youth, I had lots of fun playing rugby as a left-wing Hilbert scheme of points on a surface?” I said, “Uh?” The for the University of Milano. In this role, change of pace truth is, I did not want to do it: I did not know that field; it – and direction are very important, and they occur as if in a would have been too much work; I wanted to focus on ∂; I dizzying ride, the result of the split-second decisions you could not care less. I needed to get out of this predicament. make running with the ball. The speed at which they occur But the physicist was…my office mate. I tried avoiding him. notwithstanding, these changes are planned. On the other I failed. He was gently persistent. Hey, he was a fun guy hand, I think that, more often than not, the changing of with a great disposition—towards math, and towards life. research direction does not occur because of a choice made I started reading up. Wow! Love at fifteenth sight. It took at one point in time. Oftentimes, it is the result of a series a lot of work to start realizing that I liked very much what of (mostly) fortunate events. This new evolution is not a I was not fully understanding. By the end of the year, my second postdoc was over; I was working a little bit on the sudden occurrence but the result of a mental disposition – towards the beautiful math that comes to us by reading, ∂ project, and more on Hilbert schemes. thinking, working out problems, writing, and, most impor- Postdoc Numero Tre: East Coast tantly, talking to people. Postdoc! Third job! One year. Fancy US institution. Fellow- Grad School: Midwest ship from the AMS (thank you). Stellar mentor. But (there As a graduate student, I had been working on a problem I is always a “but”) I felt something was a bit off: I had been awarded the NSF grant, the fellowship, and the postdoc be- did not like. Ah! What a unique situation I found myself in. – cause of ∂, but I could not stop binging on Hilbert schemes. At that time, higher-dimensional algebraic geometry was in Was I even on the right track? Upon arrival, a newly minted a boom (still is), and, darn it, I wanted to be part of it. So I postdoc, a fellow algebraic geometer, explained in very, set myself to work on it, drop my problem, change advisor, very lengthy detail his achievements and his goals. He and do great new things. Bust! It came to nothing. The field asked me about my work, and ten seconds into my pitch, itself was very exciting, but I was unprepared to enter it he dismissed me quickly: “That’s not algebraic geometry.” for what I now think was the wrong reason. I backtracked. What should you do when something like this happens? However, I did not go back to the old line of work. With I smiled. During that year, I ended up devoting a lot of the help of my (very patient) advisor, I found myself a new my attention to finding a tenure-track job, giving over two problem, involving the special properties of codimension dozen talks, and straddling the two research topics. I know two submanifolds of complex hyperquadrics, solved it for now that nothing was off. Math, as a whole, must invest in the most part, and got a few papers out of it. And a degree. its own future: encourage young people to try new things.

Mark Andrea de Cataldo is a professor of mathematics at SUNY Stony Tenure-Track: Long Island Brook. His email address is [email protected]. Tenure-track! Move to Long Island. Benefits, tenure-clock, DOI: https://dx.doi.org/10.1090/noti2079 dean, sub-dean, uber-dean, mini-dean, provost, president,

662 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Early Career chair, departmental politics, grant proposals, teaching, set- assign to myself (now that I'm the mentor!) to make sure ting up a webpage (1999! Still have the same one; looks like that my students and I have a fun and productive experi- it’s from 1993). More importantly, what to do mathwise? ence. – My good old friend ∂ was no longer on my horizon. Hilbert But first, a reminder: Whom you mentor matters. Stu- schemes? Yes, but it was not the Hilbert schemes anymore, dents form their mathematician identity through their really. Somehow, the topology of complex algebraic vari- experiences with us, the mathematicians in their lives. Our eties, with its difficulties, was occupying my thoughts. It outside recognition of them as “the kind of people who do had started dwelling in my mind without much fanfare as math” is a component of this development, especially for a result of thinking about Hilbert schemes. It was there, students who don’t see themselves represented in math- and it did not budge. What was going to happen? I was on ematics often or at all. For an entry into the literature on tenure-track, needed to write papers, get grants, and this this topic, see the recent paper [RCJ19] and its thorough math was so new to me (and so weirdly exciting). What if bibliography. The upshot? Inviting students from under- it did not pan out? What about the grant renewal? Tenure represented groups in mathematics to work on our research (tick-tock)? In the end, there was really no choice. It was projects is one way of committing to diversity and inclusion too interesting, difficult, and beautiful. The light was too in our profession. blinding. I went towards it, and that was it. Luckily, as it turned out, it was not a truck. Exercises I recommend completing these exercises several months Today in advance of your anticipated research project with un- Undergrad years, compulsory military service, grad school, dergraduates. For example, if you are thinking of working postdocs, tenure-track, promotions… It’s all a vivid, awe- with students over a summer, consider working on them some blur. There have been more changes in direction since between the fall and spring semesters. then, but, at least mathwise, none more pronounced than Exercise 1. Answer two questions: What do you hope the ones of my younger years. When I look back (when do to get from the collaboration? What do you hope your I do that? There’s no time!), it seems to me that the real students get from it? changes of direction in my research were not recognized Your answers here (and below) don’t need to be deep, by me as such when they started happening: I was simply philosophical reflections; keep it workable. For example, drawn to work on something that fascinated me, and to when I was a pretenure REU mentor, my goal for me was to do that, I needed to learn and discover new math. And it produce a paper with my team. My goal for my team was to was fun, pure and simple. At some point, suddenly, I found give them each ownership over specific parts of the project. myself already moving in a new direction. It seems that It’s a good time to state that every piece of advice comes there was no moment in time when I made a conscious with exceptions! While I usually ask students to work on a decision to change. Quite simply, it just happened. And I project I designed, I’ve had gung-ho students pursue their am very, very happy it did. own problems, and they’ve come up with surprising (to me) approaches and results. In those cases, I revised my goals for myself and made sure I felt like I had enough Mentoring Undergraduate background to jump in and help. Exercise 2. Write up a project proposal, including the Research: Advanced problem(s) and a potential pathway to a solution. The first time I did this, I didn’t have a choice. I had Planning Tools and Tips agreed to be a visiting REU mentor, and I had to write this as part of the grant proposal. I think it’s the most useful Courtney R. Gibbons exercise on the list. From this, you will figure out what kind of background your future research students will need and My first experience with mentoring undergraduate re- what supplemental material they’ll need to learn. You will searchers was as the undergraduate mentee, and the critical start to see how to divide up the project into parts. importance of that summer features prominently in my You can guide students along a path and stay a few mathematical coming-of-age story. That summer was a suc- steps ahead of them. You can even assign students some cess because my mentor had a pretty clear idea of what we preliminary homework if they seem interested in working would do: the problem, the schedule, the follow-up. With on research with you! this in mind, I’m sharing five homework exercises that I Exercise 3. Decide on the length of the collaboration and structure your time. Courtney R. Gibbons is an associate professor of mathematics at Hamilton 1. Break the project into phases: ramp-up (literature re- College. Her email address is [email protected]. view, “classes” to cover background material, and time DOI: https://dx.doi.org/10.1090/noti2078 for students to build up stamina), active research, and

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 663 Early Career

ramp-down (writing up results in various formats). Fill References in some of the details for each phase. What topics will [RCJ19] Sarah Rodriguez, Kelly Cunningham, and Alex Jor- you cover? What software will your students learn to dan, STEM identity development for Latinas: The role of self- use? What will students be responsible for presenting? and outside recognition, Journal of Hispanic Higher Educa- tion 18 (July 2019), 254–72. 2. Draft a sample weekly and daily schedule. Big blocks of unstructured time are great for research if you already know how to support your research process. Those doing research for the first time benefit from having finite chunks of time planned for specific purposes. Your weekly schedule can include a lunch outing, times for group meetings, or whatever you like to break the time up. For first-time researchers, I schedule 2–3 hours twice a week to sit side by side and work on math so that I can help them develop research practices for what to do when they’re stuck. If there are other faculty working with stu- Courtney R. Gibbons dents along a similar timeline, I coordinate with them to have joint “show-and-tell” meetings. Exercise 4. Write a paragraph dedicated to your future Credits research student to share your expectations for them and Author photo is by Victor Lou. yourself. How much time do you expect them to work on the project each day/week, and how will you keep them accountable? I have found that remote mentoring is far less effective than mentoring in person. For me, this means when I decide to work with undergraduates, I’m committing to being in the same room (at the beginning) and in the same building (later on) for most of the collaboration. Exercise 5. Describe how you will commit to your stu- dents’ professional development after the time is officially up. Will you send them info about conferences where they can present? Will you write down some preliminary notes for letters of recommendation? I like to keep a file with notes about what my students accomplished during our project and any other tidbits I think might be useful for future formal or informal recom- mendations. It can be as simple as a text file. Parting Advice Working with undergraduates was a highlight of my early career. Not every project turned into a bullet point in my research portfolio for tenure and promotion, but the energy that each young researcher brought to a collaboration gave me a booster shot of enthusiasm for all of my projects. Without other commutative algebraists around to talk to mathscinet.ams.org in my department, I found it was talking to my research students that did the most to keep me motivated and pro- ductive. I’ll finish with the advice I received that I keep in mind when I start a new project with undergraduates: be ambitious for your students; they can learn a lot and do ex- © Marijn Heule cellent work. They may just be your best local collaborators.

664 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 MEMORIAL TRIBUTE

Ola Bratteli and His Diagrams Tone Bratteli, Trond Digernes, George A. Elliott, David E. Evans, Palle E. T. Jorgensen, Aki Kishimoto, Magnus B. Landstad, Derek W. Robinson, and Erling Størmer

Trondheim (now Introduction NTNU) 1980–91, and since 1991 at the Magnus B. Landstad University of Oslo. Ola’s father, and George A. Elliott Trygve Bratteli, was a Norwegian politi- Ola Bratteli is known first and foremost for what are now cian from the Labour called Bratteli diagrams, a kind of infinite, bifurcating, Party and prime graded graph. He showed how these diagrams (cousins of minister of Norway Coxeter–Dynkin diagrams) can be used to study algebras in 1971–72 and that are infinite increasing unions of direct sums of matrix 1973–76. During algebras. They turned out to be very useful tools, giving a the Nazi invasion of large class of examples, and later led to a K-theoretical clas- Norway, he was ar- Figure 1. Ola Bratteli (1946–2015). sification both of the algebras just mentioned and, more rested in 1942, and recently, of an enormously larger class (all “well-behaved” was a Nacht und Nebel prisoner in various German con- simple amenable C*-algebras). According to MathSciNet, centration camps from 1943 to 1945 but miraculously Ola Bratteli has 113 publications with 21 coauthors and he survived. Ola’s mother, Randi Bratteli, was a respected jour- received various awards. He was a member of the Norwe- nalist and author of several books. Ola was born October gian Academy of Science and Letters and a member of the 24, 1946, and died February 8, 2015. He is survived by his AMS for forty-three years. wife, Rungnapa (Wasana), and their son, Kitidet. As to his biography, Ola Bratteli graduated with dis- For a detailed biography, see the MacTutor History of tinction from the University of Oslo in 1971 and took his Mathematics Archive: www-history.mcs.st-and.ac.uk doctorate there in May 1974. He was a research fellow at /Biographies/Bratteli.html. New York University 1971–73, had various postdoc posi- tions 1973–77, was an associate professor at the Univer- sity of Oslo 1978–79, a full professor at the University of

Magnus B. Landstad is a professor emeritus of mathematics at the Norwe- gian University of Science and Technology. His email address is magnus [email protected]. George A. Elliott is a professor of mathematics and Canada Research Chair at the University of Toronto. His email address is elliott@math .toronto.edu. For permission to reprint this article, please contact: reprint-permission Magnus B. Landstad George A. Elliott @ams.org. DOI: https://dx.doi.org/10.1090/noti2085

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just the reflection of the step from the first row to the sec- Bratteli Diagrams ond. Both these one-step Bratteli diagrams are obtained from a single Coxeter–Dynkin diagram by pleating it in George A. Elliott the two different possible ways. Bratteli diagrams with an order structure were intro- One of Bratteli’s most important discoveries, I think, was duced by Vershik to describe a measurable transformation, what is now called a Bratteli diagram, which he found as a and were adapted by Herman, Putnam, and Skau to de- way of codifying the data of an approximately finite-dimen- scribe a minimal transformation of the Cantor set. Using sional (AF) C*-algebra (the completion of an increasing se- this description, Giordano, Putnam, and Skau classified the quence of finite-dimensional C*-algebras, i.e., finite direct orbit structures of such transformations, the invariant being sums of full matrix algebras), in a far-reaching extension of (in the generic case) the ordered K-group of the associated the thesis of Glimm. (Glimm considered the case of simple C*-algebra, also classified by this ordered group. finite-dimensional C*-algebras, with unital embeddings. Below is a Bratteli diagram representing the GICAR alge- The nonunital case was later studied by Dixmier.) bra (see above). Note the resemblance to Pascal’s triangle. In one sense, Bratteli diagrams had perhaps already been invented, as, for one thing, the idea is so simple—a vertically arranged sequence of horizontal rows of points, with numbered lines connecting the points of each row to the points of the row below, these numbers recording the multiplicities of the partial embeddings of the simple direct summands at one stage of the sequence of algebras into those at the next stage. In the unital case, with unital embeddings, the orders of the simple direct summands at each stage, which can be written as numbers accompanying the corresponding points in the diagram, are determined in a simple way by The corresponding inductive chain system (depicted the multiplicities, assuming that the first stage is just the vertically) is complex numbers (which clearly is no loss of generality). Pascal’s triangle is a Bratteli diagram—with the multi- plicities equal to one, and the numbers appearing in the rows being of course the successive degrees of binomial coefficients. The unital AF C*-algebra this diagram encodes arises in physics as the gauge-invariant subalgebra of the so-called CAR algebra, the C*-algebra of the canonical anticommutation relations (also AF). (This well-known subalgebra is referred to as the GICAR algebra.) Bratteli was not content just to look at the diagrams— he isolated the equivalence relation between them that is determined by isomorphism of the associated C*-algebras. This was prophetic, as he in fact was noticing that the diagrams formed a category, in which his equivalence is just isomorphism. It was later noticed that this category is equivalent to the category of ordered groups arising from the algebras in question via the K-functor. This led eventu- ally to a K-theoretical classification of an enormous—still evolving!—class of amenable C*-algebras, analogous to where the (injective) connecting homomorphisms are the classification by Connes and Haagerup and others of given by amenable von Neumann algebras. Bratteli diagrams arose in a fundamental way in Jones’s theory of subfactors. Given a subfactor of Jones index less than four, the increasing sequence of relative commutants in the Jones tower are finite-dimensional and so give rise to an AF algebra. Its Bratteli diagram is periodic with period two, with the step from the second row to the third being

666 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 MEMORIAL TRIBUTE where blank means one or more zeros; a, b, c, d are complex numbers; and B, C are matrices. Ola was always a fountain, or mountain, of good sense. Once, when he was visiting Toronto, we went to the gym to go swimming. I was a member, but it was a little murky what guest privileges I had. I thought I had them as a fac- ulty member and proceeded to explain that. This met with resistance, which gradually became more protracted. In the meantime, Ola slipped past the desk, picked up a towel, and half an hour later had finished his swim—perhaps even had a sauna too. At that point I gave up and we left. Ola—A Child of Peace, a Man of the Outdoors, and a Family Man Tone Bratteli

On United Nations Day, October 24, 1946, a child of peace was born in a crowded maternity department in Oslo. He was one of many in the baby boom that followed World War II. This event was no matter of course. My father came home in 1945 after years in extermination camps in Germany. He Figure 2. The Bratteli family, 1952. survived by a hair’s breadth. Soon after his return he met my mother. Her father had also come home from concentra- some half-empty wine glasses had caught his eye. After tion camps. The two found each other quickly, despite my tasting that juice, the shy, self-conscious boy was briefly father’s shyness. My mother’s sociable nature made up for transformed into a party animal. that. And in October 1946 he could lift his son up in the But Ola was soon ready for Bolteløkka School and a air in joy. He had not been sure whether he would be able meeting with the subject that became his passion—math- to have children after the appalling treatment in the camps. ematics. My mother was rather taken aback the day she A year and a half later, May 8, 1948, I was born. And discovered that Ola had wallpapered his room with equa- after another three years, May 20, 1951, our little sister, tions. He did not play football or hang out with friends. Marianne, arrived. Ola’s sometimes insistent little sister. Instead he solved equations and went for very long walks My father was on his way into politics, with the result in the forests and the mountains. He also did my maths that when Ola was five years old we moved to a so-called homework for me, because to me that subject was a strug- official residence in which our parents could also carry out social duties. On the day we moved, Ola and I scrambled gle. Ola achieved the best grade in mathematics; mine was around searching for our old home. The little we had with the worst. us was swallowed up by the huge rooms. Many leading As a young man, Ola spent many hours skiing, and the politicians from other countries came to this apartment. trips could easily reach 50 to 60 kilometres. He also tried We children hid away in our rooms and were not especially to find detours to make them even longer. When he came eager to introduce ourselves. Ola was a quiet boy, but one home, a slice of bread or two was not enough. He ate the evening he came in to join the houseguests and started whole loaf. tugging on the men’s ties. This was a big surprise. What Early on, Ola showed an interest in music and visual had happened to Ola? He had been in the kitchen, where arts as well. He made an attempt to teach himself to play the cello. It was not a success. But he spent a lot of time Tone Bratteli, now retired, worked for the Norwegian newspaper Arbeider- bladet and the Norwegian Agency for Development Cooperation (NORAD) listening to music and he went to exhibitions. Our sister, and was deputy minister at the Ministry of Environment and political advisor Marianne, is an artist, so my brother and sister had some- for the Minister of Consumer and Administrative Affairs and the Minister thing in common there. Later in life, when I travelled to of Social Affairs. Her email address is [email protected]. several continents in my job as a journalist and in other

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 667 MEMORIAL TRIBUTE connections, Ola always knew which gallery I should head Rungnapa and Ola travelled widely all over the world. for if there was only time for one. The composition of a There were holidays, but she also accompanied him to picture and the solution to a mathematical puzzle must conferences and on visits to universities. After a while, they have something in common. settled down in Norway. They enjoyed good years together. Ola graduated from the University of Oslo in 1971. He In the last years of his life, Ola’s health deteriorated. Rung- was awarded the best possible grade and was what we call napa was enormously supportive and did all she could to “reported to the King.” That meant that the entire govern- make it possible for him to live at home as long as possible. ment was informed about his academic accomplishments. He was a lucky man. It was a big day for everyone in the family, but perhaps most Now Rungnapa and Kitidet have moved back to Phitsa- of all for my father. He was the prime minister who came nulok in Thailand. Ola and Rungnapa built a house in her from such a poor background that he never completed home city several years ago and had no doubt planned to his education himself, but was self-taught. Now he was to spend the winters there eventually. It did not turn out that inform the king and government about his son’s academic way; Ola died so early. But outside the house there is a small triumph. As usual, Ola was unassuming and self-conscious, temple for Ola. So in a way he is there too. but he was no doubt satisfied. It was difficult to be completely anonymous and work undisturbed with his own research in Oslo. Being the child of a prime minister has many sides. You have to take a vari- ety of comments and media coverage in your stride. When Ola went to New York in 1971, it was a kind of escape. From his apartment in Greenwich Village, it was easy to get to theatres and exhibitions, and Ola soaked up everything he could get to. One day when it snowed in New York—that does happen, after all—he skied up and down Fifth Avenue. Finally, skiing weather in the Big Apple.... Tone Bratteli Ola returned home in 1973 and took his doctorate in 1974. This was during Dad’s second term as prime minister and also led to coverage in the media. Reminiscences of Ola In his personal life, Ola was simply a very kind, generous, caring and family-loving man. With a twinkle in his eye, it was easy for him to establish rapport with children. Trond Digernes During the years he lived abroad, he kept in touch with I first met Ola around 1970 when we were both Master’s our mother by phone and wanted to know how we were students at the University of Oslo. At that time Ola was a doing. slender young man with a passion for the outdoors, espe- Before Ola moved back to Norway, he was fortunate cially mountain hiking. enough to meet Rungnapa (Wasana). He was to share During the year 1970–71 there were three of us who almost half his life with her. Kitidet—his son—was his spent much time together: Ola, John Erik Fornæss, and pride and joy. myself. We played bridge, took a skiing vacation in the Norwegian mountains, and went mountain hiking in the summer. At the end of summer 1971 our roads parted. We all went to the US for PhD studies, but to different institutions: Ola to the Courant Institute, John Erik to the University of Washington, and I to UCLA. Other adven- tures with Ola in the early seventies included a multiday hike in the Sierra Nevadas in the summer of 1972, and a trip by jeep through the roadless interior of Iceland in the summer of 1973. The latter involved getting stuck in rivers and sleeping out in the open. After Iceland, Ola returned to New York, whereas I was headed for a year’s stay at CPT/ CNRS, Marseille.

Trond Digernes is a professor emeritus of mathematics at the Norwe- gian University of Science and Technology. His email address is trond Figure 3. Wedding in Trondheim, 1986. [email protected].

668 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 MEMORIAL TRIBUTE Ola Bratteli, Friend and Mathematician Erling Størmer

I met Ola for the first time when he was about to start on his Master’s thesis. Then he was a dark-haired lad with a beard, radiating health and fitness, who often went for very long skiing trips. It soon struck me that he was a highly effective person who could pick up new theory extremely quickly. By then he had taught himself a great deal about the field of operator algebras, which he wanted to work on. In the late 1960s the physicists became interested, and operator algebras became a popular field. So Ola’s timing was excellent when he passed the Master’s examination in 1971 with top grades. When it became clear to me how Figure 4. Ola and Trond in Trondheim, 1983. good the thesis was, I said to Ola that we had made a big At CPT/CNRS, Marseille, the year 1973–74 was orga- mistake; this thesis should have been used for a doctoral nized as a special year dedicated to operator algebras and degree. And it was precisely the results here ([1]) that made . It attracted several high-powered Ola well known as a mathematician from an early stage. researchers, among them Alain Connes and Masamichi In 1959, James Glimm studied operator algebras that Takesaki, and Derek Robinson and Daniel Kastler were al- were achieved by taking an infinite union of an increasing ready there. Ola joined the Marseille group in January 1974, family of n n matrices. This became a famous piece of × and this was also when he started his long-time collabora- work, and Ola quickly discovered that he could generalize tion with Derek Robinson. During the decade 1980–90 I Glimm’s work by studying infinitely increasing unions of joined Ola and Derek on several occasions for discussions direct sums of matrix algebras. He then ended up with an at ANU, Canberra, and Derek also visited Trondheim. This infinitely large diagram that described all the inclusions, resulted in a few joint publications, sometimes also with which thus also described how the operator algebra was other coauthors. constructed. His main finding was that this diagram fully I was involved in only a fraction of Ola’s mathematical described the operator algebra, enabling a classification of work, but since we spent much time together, we had many all such operator algebras, now known as AF algebras. This interesting discussions. Given Ola’s incisive mind and deep result proved far more important than Ola and I had an- understanding of everything he was involved in, it was al- ticipated, and the figures are now called Bratteli diagrams. ways a rewarding experience to exchange ideas with him. After graduating, Ola studied for two years at New York He is dearly missed, both as friend and colleague. University, where Glimm was. There was here an excellent environment in mathematical physics but also for cultural life and food. His appearance changed in some ways; much of his hair disappeared, his beard was gone, and he put on so much weight that when I met him later, I did not recognize him, so I introduced myself to him. Ola returned home and took his doctorate in Oslo in 1974. After that, he went to Marseilles, where there was a very active and high-quality environment led by physicists who developed the theory of quantum physics in operator algebras. There he met Derek Robinson, and they started Trond Digernes a cooperation that lasted for the rest of Ola’s career. Rob- inson moved to Australia, and on one of Ola’s journeys to Australia he stopped in Thailand. Here he was fortunate

Erling Størmer is a professor emeritus of mathematics at the University of Oslo. His email address is [email protected].

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 669 MEMORIAL TRIBUTE

Figure 5. Ola’s PhD defense, May 1974. From left: the dean, Ola, Gert K. Pedersen. Figure 6. Derek and Ola, 1988. enough to meet Wasana, whom he married. Together, they had a good life. Life with Ola Ola had many more coauthors and was altogether a very popular person to work with. Everyone liked Ola. When there were several of us together, he was not a man Derek W. Robinson of many words, but in private he would talk. Ola radiated a good spirit, and it was easy to become fond of him. He I met Ola at the beginning of 1974 in Marseille. He was introduced to me by Trond Digernes. It was to be a pivotal was a person with a warm heart and a subtle humor that moment in each of our lives, although we had no premo- will stay with us for the rest of our lives. nition of this at the time. There were many unpredictable Ola Bratteli passed away at the age of sixty-eight after consequences: a few years later Trond was to meet his several years of steadily declining health. It was an ex- wife, Hallie, at an open air opera in Sydney; Ola became tremely sad experience to see how he had become weaker the owner of a mushroom farm in northern Thailand; and each time I saw him in recent years. His strength began Ola and I were to write a book that is still bought, read, to fail at a fairly early stage. His last research articles were and regularly cited thirty-five years later. At that time I was published in 2008, and after that he had little energy to do professor of physics at the University of Marseille, where, more. So his brilliant career as a mathematician came to an under the influence of the late Daniel Kastler, there was end far too early. Ola will be remembered and missed for a a strong visitors program in mathematical physics. This long time, both as a mathematician and as the wonderful explained Ola’s and Trond’s presence. person he was. My collaboration with Ola began the day we met. I explained to him some of the ideas I had about quantum dynamics and derivations on C*-algebras, and shortly after we wrote our first paper on these topics. By 1976 we had coauthored three other papers. When we met we had different interests, different backgrounds, and quite different personalities. Ola was quiet, well organized, and introspective, characteristics I did not share. Ola rapidly assimilated the Mediterranean lifestyle—the sun, the sand, and the seafood, especially the seafood. It was a time of calm cooperation amid the chaos of French academic life. Erling Størmer In 1975 I began to think about writing a book on opera- tor algebras and their applications in physics. I discussed the idea in spring 1976 with Ola, thinking that with our Derek W. Robinson is a professor emeritus of mathematics at the Australian National University. His email address is [email protected].

670 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 MEMORIAL TRIBUTE disparate backgrounds and successful working relationship we would be able to do justice to the subject and its recent developments. I was pleased he did not dismiss the idea immediately. Initially the book was intended as a relatively short-term project: a monograph of 300–400 pages with the early chapters on mathematical background and the later chap- ters on applications to quantum statistical mechanics. We quickly realized that we would exceed the estimated length, so the short-term project turned out to be a long-term project, and the book changed from one volume to two. We began each chapter with a tentative sketch of the intended sections. After discussing the general presentation of the material in each section, we began to draft alternate sections. We then exchanged drafts and edited each other’s work. This process would be repeated until we were each satisfied with the outcome. We often had different notions of the relative significance of the material and the emphasis to be given to various statements and results. At times my Figure 8. Sjusjøen, Lillehammer, Easter 2004. first draft would be completely changed by Ola and vice versa. Somehow the process always reached equilibrium 1000-page, two-volume book took about three years. That after a reasonably short time, with one exception, the sec- was not the end, however; we returned to it again twice, tion on modular theory. This took seven exchanges before preparing the second edition, but that is a different story. we were both satisfied. This procedure had various advan- After the book was finished, our collaboration contin- tages. It naturally introduced a uniformity of style. It also ued, with Ola visiting Australia almost every year. He was gave a fairly foolproof method of avoiding error, although lucky to survive his first visit. Whilst touring the almost we were not completely successful in that respect. deserted roads of a national park, he reverted, European The first volume of the book was completed by Septem- style, to driving on the right side, the wrong side for Austra- ber 1977, which left three months to complete the second lia. This led to a head-on collision that wrecked both cars, volume of the book before I left France to take up a position fortunately without any personal injuries. in Australia. In that time we managed to write about 40 Ola realized that by timing his Australian visits correctly percent of Volume 2, aided by discussions with Akitaka he could ensure that it was almost always summer. It also Kishimoto, who had just arrived in Marseille as a postdoc. had the advantage that we could spend a maximum of time I returned to Marseille in June 1978, and work began again. working at the family beach house. Ola would then take I then returned to Sydney and Ola moved to Trondheim, his daily swim before heading to the local oyster shop. He but we returned to Marseille in June 1979 to tackle the final bought freshly harvested oysters by the bag to be opened, work. We completed the book by working nonstop for three seasoned with a lemon from our garden, and savoured as and a half weeks. The second volume finally appeared in the sun went down. 1981, so the total operation of writing and publishing the “Those were the days my friend, I thought they’d never end.”

Derek W. Robinson

Figure 7. Lysebu, Oslo, 2017. From left: Aki Kishimoto, Palle Jorgensen, Derek Robinson, Tone Bratteli, George Elliott, Reiko Kishimoto, and David Evans.

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than me in everything. So when we had to descend a steep slope made of gravel at one point, I was surprised to find myself rather enjoying skidding down while Ola tried to walk down steadily, as if reflecting his meticulous style of doing mathematics. Ola’s paper ([1]) on AF algebras became an inspirational source for me. This class might look rather special, but now we might say if a C*-algebra is not obviously not AF, then it would be AF. As a touchstone of this credo Ola and I ex- amined the fixed point algebra Fθ of a C*-algebra Cθ(u,v) generated by two unitaries u,v with uv=e2πiθvu with θ irra- tional, under the period two automorphism σ: σ(u)=u–1, –1 σ(v)=v . (C0(u,v) with θ=0 still can be defined and com- prises the continuous functions on the torus, while F0 is on the sphere or rather a pillow with four corner points.) We managed to show Fθ, a noncommutative pillow, is AF (when θ is irrational). Another notable result in [1] was that any two irreducible representations of a simple AF algebra are bridged by an automorphism, which we established in a different setting, published as “Homogeneity of the pure state space of the Cuntz algebra” in J. Funct. Anal. Figure 9. Ola preparing bouillabaisse in Trondheim. 171 (2000). This gave me hope of successfully attacking a more general case, and I did it with some help from others. A Tribute to Ola Bratteli Ola had many works; I cannot touch on all of them. Among them the books [3,4] are a good reference for those Aki Kishimoto in the field of mathematical physics, me included. Our last work “Approximately inner derivations,” Math. Scand. 103 Before I first met him in September of 1977, I must have (2008) with Derek is an attempt to shed light on a topic read his early paper ([1]) of 1972 and his more recent se- dealt with there, which yet haunts me to this day. ries of papers on unbounded derivations, mostly written with D. W. Robinson, because his image had been firmly established in my mind as a formidable mathematician with whom I could hardly be compared. And he was, and I think I was quite lucky to come to know him in the earliest possible days, which brought me chances to collaborate with him for three decades. Ola and Derek were writing the book in Marseille, a kind of book I love and could get familiar with, when I visited in 1977. Derek soon left for Sydney, but Ola and I spent Aki Kishimoto almost a year together there. Though he was just one year my senior, he was a kind of mentor in mathematics and everything else during the stay. He also arranged for me to attend two conferences, sort of encouraging me against Mathematics Collaborations my timidity. In all these activities when transportation was required, Ola was in charge, which eliminated a practical on Three Continents worry for me. In one outing I remember we hiked on rocky coastal Palle E. T. Jorgensen paths, where I learned to mumble “Bonjour” to strangers we encountered. (Later I found this practice ubiquitous, Ola Bratteli had a profound influence on modern analy- and Ola would say “Konnichiwa” awkwardly when we sis, especially themes connected with operator algebras, hiked in Japan.) We were not sports types, but he was better classification, noncommutative harmonic analysis, and

Aki Kishimoto is a professor emeritus of mathematics at Hokkaido University. Palle E. T. Jorgensen is a professor of mathematics at the University of Iowa. His email address is [email protected]. His email address is [email protected].

672 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 MEMORIAL TRIBUTE . While Ola’s first paper was solo, Sadly, Ola’s health declined towards the end, but I am almost all that followed were joint. happy to have had the benefit of many intense research In January of 2017, a weeklong conference was organized experiences from the early part of our careers. in Oslo, with the aim of presenting some of the many collaborative advances in mathematics and its applications involving Ola. My own collaboration with Ola started by chance, dates back to the mid-1970s, and lasted for four decades. Our early work was in noncommutative geometry, and our later research moved in a diverse number of directions. My own collaborations involved my visiting Ola in Oslo. In addition, we both made research visits to Derek Robinson, Dai Evans, and George Elliott. In all, I have thirty joint research publications (including two AMS Memoirs) Palle E. T. Jorgensen with Ola, and a book. Early joint research includes the themes Lie algebras of operators, smooth Lie group actions on noncommutative Ola and Orbifolds tori, and a study of decomposition of unbounded deriva- tions into invariant and approximately inner parts. These topics are part of a systematic analysis of unbounded *-der- David E. Evans ivations as infinitesimal generators in operator algebras, with direct connections to quantum statistical mechanics. I first briefly met Ola in the new year of 1977 when I was Other applications include noncommutative geometry, a postdoc in Oslo. Later I got to know him very well, not as envisioned by Alain Connes. Our later joint research only through our work and joint papers (fifteen altogether) moved more in the direction of representations and certain but also through holidays taken together, particularly ski- applications. ing ones in Rondane and Sjusjøen. He spent six months My most recent, and substantial, joint work with Ola with me in Warwick in 1982, which was the start of our collaboration. Our first conference together was at Arco was wavelets. That part includes a book Wavelets through a Felice, Naples, in March 1978. Ola and Aki drove there from Looking Glass: The World of the Spectrum, which presents the Marseille in his Citroën deux chevaux. This was a meeting subject from a representation theoretic viewpoint. organized by Vittorio Gorini on Mathematical Problems in A common theme in my joint research with Ola is my Quantum Theory of Irreversible Processes that brought to- insistence on the central role to be played by representa- gether our mutual interests in derivations and generators of tions and decomposition theory. Some of our early work one parameter semigroups of positive maps, on which we dealt with representations of Lie groups, of C*-algebras, would later collaborate. At a meeting in Chennai organized and of multiscale systems. by Sunder, Ola took me on an expedition through the local A quite different representation theoretic theme was markets to find saffron, which was later put to good use in the theory of numerical AF invariants, representations and his bouillabaisse, the finest I have ever had, with fresh fish centralizers of certain states on the Cuntz algebras, and a re- from the harbor back in Norway. lated but different study of combinatorial notions we called In his PhD thesis ([1]), Ola classified AF C*-algebras in iterated function systems and permutation representations terms of what are now known as Bratteli diagrams. This of Cuntz algebras. They play a key role in our understand- work was not only pivotal in Elliott’s classification of AF ing of such multilevel systems as wavelet multiresolution algebras through K-theory; it is ubiquitous in operator alge- scales, in addition to multiband filters in signal processing. bras, dynamical systems, and in Jones’s theory of subfactors. Later joint work between Ola, me, K.-H. Kim, and F. My own work with Ola started with dynamical sys- Roush was inspired by Bratteli diagrams. It is known that tems of one-parameter semigroups of positive maps in the nonstationary case defies classification (order-iso- 1982. Later visits to Swansea led to the collaboration with morphism is undecidable), but we discovered that the George Elliott and Akitaka Kishimoto on noncommutative stationary case could be decided by explicit classification spheres, the irrational rotation algebra, and the K-theoretic numbers and associated finite algorithms. In this work, obstructions to classifying such amenable C*-algebras and for the stationary dimension groups, we obtained explicit their dynamical systems. This was particularly motivated computation of numerical isomorphism invariants. We by the remarkable construction by Blackadar (Annals of proved decidability of the isomorphism problem for stationary AF algebras and the associated ordered simple David E. Evans is a research professor of mathematics at Cardiff University. dimension groups. His email address is [email protected].

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Figure 11. Non-commutative orbifold from folding the Kronecker flow on a torus.

Figure 10. Kronecker flow on a torus. The flip on the irrational The work described above laid the foundations for or Kronecker flow on the torus yields a nonsingular flow on subsequent work over the last twenty-five years on the the sphere which is zero-dimensional in a strong sense. The classification of amenable C*-algebras by K-theoretic data, corresponding C*-algebra is approximately finite-dimensional. before which the classification was out of sight and did not appear feasible. Our work on orbifolds also directly Math, 1990) of a -symmetry on the Fermion algebra with Z2 led to the study of orbifold subfactors (Evans and Kawa- non-AF fixed point algebra, and the subsequent work of higashi, Comm. Math. Phys., 1994), as reported in the first Kumjian in showing that a -symmetry of a Bunce–Ded- Z2 Danish-Norwegian Workshop on Operator Algebras at dens algebra and the corresponding crossed product C(T) × Røros in 1991. dyadic rotations Z2 yielded an AF algebra. The issue of × Ola had a long connection and affection for Thailand, existence of such symmetries had been a well-known open making regular visits; his wife for more than thirty years, problem. Wasana, was from Phitsanulok. In 2016, Paulo Bertozzini This led us to consider the -symmetry on the rotation Z2 initiated at Thammasat University in Bangkok the Ola algebra with the flip : (u)=u–1, (v)=v–1 on the genera- σ σ σ Bratteli Mathematical Physics and Mathematics in Thailand tors. On the classical two torus, 2, this yields a singular T Colloquium, where I was honored to give the first talk. orbifold 2/ which can be thought of as a tetrahedron, T Z2 Ola had a generous spirit and integrity. We will miss his topologically a sphere, but with four singular vertices. presence and friendship. This led us to refer to the fixed point algebras and crossed products of rotation algebras as noncommutative spheres, but as Alain Connes pointed out, they are better described as noncommutative toroidal orbifolds, as they do not have the K-theory of a sphere. Taking matrix-valued functions constrained at the sin- gular points to have dimension drops and their inductive limits led us on a path towards studying group actions on approximately finite-dimensional AF algebras with non-AF fixed point algebras and crossed product algebras. This way we also showed the existence of non-AF C*-al- David E. Evans gebras that when tensored with a certain UHF algebra (i.e., an infinite tensor product of matrix algebras) become UHF as well, that the fixed point algebra of an irrational rotation algebra by the flip is AF, and that the irrational rotation ACKNOWLEDGMENTS. Editing has been done by algebras are inductive limits of sums of matrix algebras Magnus B. Landstad, translations from Norwegian by over the continuous functions on a circle. Margaret Forbes, Bratteli diagrams by Petter Nyland, and John Erik Fornæss assisted with the submission. A preliminary (and longer version) of this article has been posted on https://arxiv.org/abs/1903.01524.

674 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 MEMORIAL TRIBUTE

References Credits [1] Ola Bratteli, Inductive limits of finite dimensional C*-al- Figures 1 and 2 are courtesy of the Bratteli family. gebras, Trans. Amer. Math. Soc. 171 (1972), 195–234. Figure 3 is courtesy of Hallie Digernes. MR0312282 Figures 4 and 8–11 are courtesy of David Evans. [2] Ola Bratteli, Derivations, dissipations and group actions on Figure 5 is courtesy of Ivar Aaserud/Aktuell/NTB scanpix. C*-algebras, Lecture Notes in Mathematics, 1229, Spring- Figure 6 is courtesy of Marion Robinson. er-Verlag, Berlin, 1986, vi+277 pp., ISBN: 3-540-17199-1. Figure 7 is courtesy of Erik Bedos. MR0871870 Photo of Tone Bratteli is courtesy of Tone Bratteli. [3] Ola Bratteli and Derek W. Robinson, Operator algebras and Photo of Trond Digernes is by Arne Asphjell. quantum statistical mechanics. 1. C*- and W*-algebras, sym- metry groups, decomposition of states, Second edition, Texts Photo of George A. Elliott is by Mikael Rørdam. and Monographs in Physics, Springer-Verlag, New York, Photo of David E. Evans is by Twm Evans. 1987, xiv+505 pp., ISBN: 0-387-17093-6. MR0887100 Photo of Palle E. T. Jorgensen is by Myung-Sin Song. [4] Ola Bratteli and Derek W. Robinson, Operator algebras and Photo of Aki Kishimoto is by Reiko Kishimoto. quantum statistical mechanics. 2. Equilibrium states. Models Photo of Magnus B. Landstad is by Kai T. Dragland. in quantum statistical mechanics, Second edition, Texts and Photo of Derek W. Robinson is by Marion Robinson. Monographs in Physics, Springer-Verlag, Berlin, 1997, Photo of Erling Størmer is by Geir Holm, University of Oslo. xiv+519 pp., ISBN: 3-540-61443-5. MR1441540 [5] Ola Bratteli and Palle Jorgensen, Wavelets through a Looking Glass: The World of the Spectrum, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2002, xxii+398 pp., ISBN: 0-8176-4280-3. MR1913212

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www.ams.org/math-imagery Memories of Goro Shimura Don Blasius, Toni Bluher, Haruzo Hida, Kamal Khuri-Makdisi, Kenneth Ribet, Alice Silverberg, and Hiroyuki Yoshida

Goro Shimura, a mathematician who greatly influenced number theory in the second half of the twentieth cen- tury, was born in Japan on February 23, 1930. Over a career spanning six decades, he repeatedly made transfor- mational discoveries that stimulated new lines of inves- tigation and played a central role in the development of the field. Shimura earned his degrees at the University of Tokyo and held appointments at the University of Tokyo and Osaka University. He was a professor at Princeton Uni- versity from 1964 until he retired in 1999. He authored numerous influential books and papers and was awarded a Guggenheim Fellowship in 1970, the Cole Prize in Num- ber Theory in 1977, the Asahi Prize in 1991, and the Steele Prize for Lifetime Achievement in 1996. Goro Shimura passed away in Princeton on May 3, 2019, at the age of eighty-nine. Don Blasius Goro Shimura advised my 1981 Princeton thesis and he was, through his research and guidance, the central figure of my intellectual life in graduate school and for many years afterwards. His ideas about how to do mathematics have influenced me throughout my career. Figure 1. Goro, Chikako, Haru, and Tomoko returning to the Arriving at Princeton in fall 1977, I had no plan to study US from Haneda Airport in 1971. number theory, had never read a book about it, and had never heard of Shimura, Iwasawa, Dwork, Langlands, or offering of an introductory course in algebraic number even Weil. Most graduate courses had incomprehensible theory stood out as a beacon of light. Algebra already course descriptions. In this context Professor Shimura’s had great appeal for me, and I was hooked right away by this perfect course. Later I took his courses on families of Don Blasius is Distinguished Professor of Mathematics at UCLA. His email abelian varieties, special values of 퐿-functions, period rela- address is [email protected]. tions, the arithmetic theory of automorphic forms, Eisen- Communicated by Notices Associate Editor William McCallum. stein series, theta functions, etc., all topics essential to his For permission to reprint this article, please contact: current research and, as it turned out, my initial research. [email protected]. He really taught for his students, not just to have a context DOI: https://doi.org/10.1090/noti2076 to explore a topic of interest to himself. Each course started

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 677 with minimal assumptions of background and went far end as his destination after the fact. In words, he told me into its subject with both clarity and an economy of means. he starts with a general idea but no fixed goal or conjecture. He always had his lecture fully written out in a notebook, Many people who did not know him have some imag- which would be on the table in front of him. He would lec- ination of Shimura as exclusively serious or severe. This ture without it except for complex formulas. For them, he is simply not true, and I mention two among many mo- would pick it up to refresh his memory, always explaining ments of wit that we both enjoyed. After I started a conver- clearly as he went. Shimura was always perfectly prepared sation with a remark about Hecke operators, he said, “The and engaging. first thing you say to me is always interesting.” And after By early in my second year, it was clear to me that I I complained to him that I found his paper on confluent wanted Shimura to supervise my dissertation. When I hypergeometric functions hard, he said, “You need to go queried him about it, he asked me to read his famous see an analyst!” 1971 text Introduction to the Arithmetic Theory of Automor- About other mathematicians, he freely acknowledged phic Functions. This book is a masterpiece of exposition, his debts to Eichler, Hecke, and Weil, who was a friend starting from scratch and giving, mostly with proofs, the for over four decades. He also had the highest respect for key themes of his research up to that point. For me it was Siegel. Once he told me, “His proofs are correct and you the perfect preparation for reading the articles on which can just use his theorems.” He did not think that about it was based. It was a transformative undertaking. As an many mathematicians. One day we discussed Weil’s 1967 anecdote, I mention that when I told him I had finished paper on the converse theorem. After a while he said, Chapter 3, the one on Hecke algebras and the relation with with intensity, “Weil is a genius.” I never heard him say 퐿-functions, he asked me how long I had spent on it. I told that about anyone else, even when discussing major works. him I had spent about a month, and he said that was “very About ideas, although he knew a great deal, he was some- fast” (meaning too fast). He was correct! Later that year, thing of a minimalist in his preferred way of writing. For when I asked to be his student formally, he agreed but im- example, when I referred to automorphic forms as sections posed the condition that I promise not to “fire” him. He of vector bundles, he queried me sharply as to whether explained that a student with whom he had been work- this language added anything. I had to admit that, in the ing for some time had just done that, and he was visibly context, except for curb appeal to some, it did not. As a hurt when he spoke about it. I told him that there was no consequence, I avoided such usage in my dissertation. chance of that. I was asked to make a brief summary concerning Shimura did not tend, at least with me, to engage in Shimura’s theory of canonical models and its antecedents. lengthy, detailed discussion of mathematics. Usually, af- In 1953, at the start of his career, Shimura created ([1]) ter a brief discussion of math, he would shift the con- the first theory of reduction mod 푝 of varieties in arbitrary versation to something else, ranging from departmental dimensions. In a December 1953 letter to Shimura, Weil gossip to Chinese stories. He did not give me a research called it “a very important step forward” and emphasized problem right away. Instead, he asked me to read articles. its promise for the further development of complex multi- The first was [3], mentioned below, and the second was plication. He also wrote that it was “just what is needed [to [17]. Sometime while reading the second paper, he gave study] modular functions of several variables.” The first me the simple-sounding research problem concerning its direction became the famous collaboration of Shimura main theme of relations between the periods of abelian and Yutaka Taniyama, which was well underway by 1955. varieties: “Find more precise results. Find natural fields of They defined and studied abelian (group) varieties ofCM definition.” This was a great problem because it connected type and proved the Shimura–Taniyama reciprocity law, with so many topics, including values of 퐿-functions and which describes explicitly the action of a Galois group on the extension of the canonical models formalism from au- the points of finite order of the variety. The main un- tomorphic functions to automorphic forms, both new ar- derlying result here is a celebrated formula for the prime eas. ideal decomposition of the endomorphism that reduces to Before giving me this problem, Shimura had supported the Frobenius morphism attached to a given prime of the my desire to come up with my own. Several times I made field of definition. As a key application of this formula, suggestions to him, but he found compelling issues with they computed the Hasse–Weil zeta function at almost all them, such as being too hard or already done. When I places, thereby verifying Hasse’s conjecture for such func- finally found some results on his problem, he washappy tions. Their research was summarized in the well-known and exclaimed “Isn’t it nice to have some success?!” Later, 1961 monograph Complex Multiplication of Abelian Vari- I asked him about how he worked. He drew a really messy eties and Its Applications to Number Theory, which Shimura self-intersecting path on the blackboard and declared the wrote after Taniyama’s death. In fact, Shimura wrote

678 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 research articles about complex multiplication and abelian varieties over his career, even publishing in 1998 an expos- itory monograph Abelian Varieties with Complex Multiplica- tion and Modular Functions. This text includes more recent fundamental topics of his research, such as reciprocity laws for values of modular functions at CM points and the the- ory of period relations for abelian varieties of CM type. From the late 1950s until the late 1960s Shimura made a continuing study, mostly published in the Annals of Math- ematics, of the fields of definition for certain varieties de- fined by arithmetic quotients of bounded symmetric do- mains. For me, the start was the 1963 article “On ana- Figure 2. Chikako and Goro in 1993 in Nagano prefecture, lytic families of polarized abelian varieties and automor- where they had a summer house. phic functions.” In its first part, this highly readable paper showed that arithmetic quotients of three of the four clas- symmetric domains I, II,” both published in the Annals. sical domains arise, via normalizing period matrices, as In them, Shimura gave an adelic version of [8]. This view- analytic parameter spaces of the varieties of a given type. point enabled him to define an action of the finite adeles In 1966 he followed up with [7], which introduced the (a way of introducing Hecke operators) of the associated well-known notion of PEL type. He constructed a moduli on the system of models. He conjectured space for each type as a model of an arithmetic quotient, that such a theory would exist for any reductive group giv- thus providing a large supply of varieties whose points had ing rise to a product of classical domains. Indeed he wrote, definite meaning, and remarking of their fields of defini- “The completion of this task does not seem so difficult.” Shimura’s students Shih and Miyake each proved cases tion 푘Ω: “In many cases we have verified that 푘Ω is an abelian extension of 퐾′.” Here 퐾′ is a number field, fre- of Shimura’s conjecture, and Deligne made major progress, quently called the reflex field, which is central to thesub- as well as reformulating the theory in a general axiomatic ject and which first arose in the Shimura–Taniyama theory. way. In 1983 Borovoi and Milne constructed canonical In 1964 in [4], he studied the varieties (quotients of prod- models for all reductive groups of Hermitian symmetric ucts of upper half-planes) attached to quaternion algebras type. They did this by proving a conjecture of Langlands over a totally real field of arbitrary ramification behavior at that extended the reciprocity law at the fixed points to ar- infinite places. This paper introduced the cases where the bitrary automorphisms of Gal(퐐/퐐) instead of Gal(퐐/퐾′). fields of definition are abelian extensions of totally real This conjecture was based itself on Langlands’s remarkable fields. The algebraic varieties themselves had already been extension, later proven by Deligne, of the reciprocity law studied in [5] as moduli spaces associated to quaternion of Shimura and Taniyama for abelian varieties of CM type. algebras over CM fields, in which case the canonical fields Shimura himself did not really return to the theory after of definition are abelian extensions of the reflex field,a 1970, except, as mentioned, for extending it to automor- CM field. Thus in [4] it was a question of a further descent phic forms in some special cases. Instead, the 1970s were (Shimura used the term “bottom field”). In 1967, in [9] he for him a period of great and diverse achievements in new considered further the cases of [5] where the dimension is fields such as the theory of critical values of 퐿-functions. one and computed, via a congruence relation analogous to Yoshida and Khuri-Makdisi mention these in their contri- Eichler’s, the Hasse–Weil zeta function at almost all places, butions, so I will stop here. thereby proving Hasse’s conjecture for the curves. These I miss Shimura deeply. Before beginning to write, I are the famous “Shimura curves.” This article also intro- looked again at many of his articles that were so impor- duces the notion of a canonical model as one uniquely tant to me. I fell again under the spell of my teacher and I characterized by an explicit description, obtained by virtue found a problem to work on. of the uniformization, of the Galois action over the reflex field on the images of CM points (see Main Theorem 1of Toni Bluher the article). In 1967 as well, a further paper [8] extended One of the reasons that Professor Shimura had so many the canonical model theory of [9] to arithmetic quotients graduate students was his extraordinary commitment to of higher-dimensional Siegel spaces. All these papers were written in the language of ideals. Finally, in 1970 this long Toni Bluher is a senior research mathematician at the National Security Agency. development culminated with two articles, now famous: Her email address is [email protected]. “On canonical models of arithmetic quotients of bounded

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 679 teaching. He gave masterful lectures that were very pop- Somehow I had met Professor Doi in Kyoto slightly ear- ular among the graduate students. We circulated among lier, and because of a friend of mine (an ardent mathe- ourselves mimeographed copies of notes from prior years. matics addict) I had started reading mathematics books at Most influential to me was his course on Siegel modu- college level and above. This day, following a suggestion lar forms, given during my first year of graduate school of Doi, I planned to attend a lecture by Professor Shimura in 1984. Based on my experience in his course, I asked at Tokyo University of Education. I took a bullet train in to be his graduate student and told him that I particu- the morning and reached Tokyo about an hour before the larly enjoyed the material on theta functions. He remem- lecture. In the lecture, he talked about CM abelian vari- bered that comment and designed a thesis topic for me eties and their fields of moduli. I understood the content that included theta functions of half-integral weight. He well, as I had already read his red book and the English ver- prepared a series of readings that introduced all the con- sion of the book he coauthored with Taniyama. Because of cepts that I would need for my thesis topic, including my late start, I was obsessively reading, as quickly as pos- some material on bounded symmetric domains, several sible, many mathematics books. I did well, giving myself of Shimura’s articles, and Andre Weil’s Sur certain groups good background knowledge of analysis, algebraic and an- d’operateurs unitaires (Acta Math 111, 1964, 143–211). Ev- alytic number theory, and algebraic geometry, including erything fit together perfectly, and the thesis was progress- the viewpoints of both Weil and Grothendieck. Prior to ing well. I got married in my second year of graduate this point in my life, I had read a great deal but mostly to school and had a son in September of my fourth year. Pro- have fun. Reading books, including Chinese classics that fessor Shimura was supportive and nominated me for a Shimura loved, had been my way of life. Sloan Scholarship, which relieved me of teaching duties After the lecture, senior PhD students were invited into and made it possible to focus on writing my thesis. Be- a smaller room to pose questions to the speaker. Out of ing Shimura’s student was akin to hiking Mount Everest curiosity, I walked into the room also. As Shimura had a with a skilled guide—he cleared the path so that we could rare charisma, at first nobody dared to ask him questions. reach the summit. I remember him saying that he would This seemed impolite to me, so I started asking some well- have preferred an approach that would give us more time posed questions about CM abelian varieties. This went to read and gain perspective, but he adopted his style be- well, and he answered me, looking directly into my eyes, cause at that time it was expected that graduate students treating me correctly as a novice, and emphasizing the im- should finish in four years, something that is hard todo portance of studying periods of CM automorphic forms. in such a technical field. Then there was another awkward silence, so I decided to I have fond memories of when Professor Shimura in- ask a somewhat imprecise question about a minimal field vited me, my husband, and other graduate students to his of definition of a given CM abelian variety, as it seemedto house on a few occasions and also a dinner party where be related to some results on the field of moduli that Ihad we met Andre Weil. At one of those occasions, I learned seen in a preprint of his that Doi had received ([15]). Once about his sense of humor. He said that puns are not part I had described the question (saying that Doi allowed me of Japanese culture, and he could not see how they were to see his new results), he got excited and gave me a terse funny. “So what is funny to you?” I asked. He told the fol- reply, saying that you could ask questions about facts, but lowing joke. Some mosquitos were annoying the guests, trying to get some “guess” or some “way” towards a new re- so the host said he would take care of it. He put out a sult from somebody else is not morally sound. You should bowl of sake and many small pieces of tissue paper. “How think about them on your own and should find a way out. will this help get rid of the mosquitos?” the guests asked. His last words were, “Do your own mathematics.” The host replied that the mosquitos would drink the sake I am from a family belonging to a traditional commer- and then fall asleep. “Ah, very clever,” said the guests, “and cial class of people in Osaka-Kyoto. My family had been what is the tissue for?” “It’s so that the mosquitos will have successful in banking from the late shogunate era through a place to rest their heads after they drink the sake!” the Meiji Restoration. In Japan, we had a “rice exchange,” which was analogous to a European stock exchange, as Haruzo Hida early as the late seventeenth century. There were “rice stocks” (kome-tegata) that were traded at the exchange by In the mid-1970s, I was a senior undergraduate at Ky- banking officials, and the price of rice, including that col- oto University and had just started learning mathematics. lected as tax by each feudal province, was determined na- tionally at the exchange. Banking (called Ryogae, literally Haruzo Hida is Distinguished Professor of Mathematics at UCLA. His email “money-exchange”) was a prosperous business for three address is [email protected]. or four centuries. Even though Japan was governed by a

680 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 combination of the shogun’s samurai (principled and well that CM period relations were a main topic of Shimura’s literate in Chinese classics) and the emperor’s aristocrats research at the time. I knew this conjecture, but I did not (often indulging in subtle poems with hidden meanings), explicitly write it in the letter or in the published paper. In the economy at the time was essentially an unofficial cap- any case, I got a job at Hokkaido University with the help italism. This is one of the reasons Japan was able to mod- of Doi, who moved there after his trip to Germany. For ernize itself so quickly in the colonial period of other Asian my next project, at Hokkaido, I classified CM factors of Ja- nations. For people in banking, a deep understanding of cobians of Shimura curves, extending Shimura’s work for the governing samurai class is fundamental for their busi- the case of modular curves. I thus, with Shimura’s support, ness. So most children of either gender in such a family was offered a one-year visit to the Institute for Advanced had a sort of private tutor/nurse/nanny to train them how Study. to divine the undercurrent thinking of people. (I mean, I arrived at Princeton in August 1979, and right away training so that your front personality can make friendly I called Shimura on the phone. He invited me and my contact while at the same time your rear personality can wife to a dinner at his home. At the dinner, he asked us a delve into the counterpart’s intents, often by posing ques- funny question: Why do Osaka people use the plural form tions that appear innocuous.) I had learned this type of oko-tachi in referring to an only child? The part “tachi” is slightly schizophrenic approach to people. Thus, when a plural indication, although the Japanese language does Shimura made his reply to my question, I was calm, but in- not have a systematic plural form. My answer was that for side I was quite amused by his excitement (as I was looking a family in commerce, having several children is more de- for a way to cope well with him). I decided to avoid hence- sirable than having one, and hence the talker is apparently forth indulging myself too much in unfounded thoughts showing friendship by way of courtesy. He was not at all with him and instead to focus on asking him well-posed, convinced, giving me a couple of counterexamples from maybe conjectural, questions and to present him with new the usage found in Kyoto aristocracy. This was typical in ideas, not necessarily in mathematics. I made a firm note his conversation. He would come up with totally unex- in my mind that this should be my way to cope with his pected questions, and if one’s answer was off the mark, he principled personality. used it as a seed-topic of often poignant stories he loved to Immediately after his lecture, I stopped my indulgence talk about. My answer could be wrong but not bad either of reading mathematics broadly and focused on books (at least not provable in either way). After this conversa- whose content I felt I really needed. This freed my time, tion, he started calling me Haruzo-san and told me to call and I started writing a research article that became my first him Goro-san, which I never did in our conversation. (I paper, published in 1978. In March 1976, there was a called him, to his dismay, always Sensei.) I was told at the Takagi anniversary conference at RIMS, and at it I had a dinner to come to see him in Fine Hall at tea time, that is, short conversation with Shimura without much content every Thursday at 3 pm. (though he remembered me well). I felt shame that I could I kept busy every week to concoct something new to tell not produce something entertaining to him by this time, Sensei on Thursday. If I had not done that, I would have although I had the seed of an idea of creating complex had a hard time listening to his short stories. To cope with multiplication on the complex torus spanned by CM theta them I needed all my skill of conversation. Perhaps he series in the middle degree Jacobian of the Hilbert mod- was doing this intentionally to pull out the most from me. ular variety. I finished this project a year after the confer- I was fairly successful in the first year, and I wrote three ence. Doi had left for the Max Planck Institute for Math- papers, later published in Inventiones. By these, I think, I ematics just after the conference, and in April 1976, I en- earned an extension to stay at the IAS for a second year. tered the graduate school of Kyoto University. I was next This year was difficult, although I had a conjectural idea to meet Doi again in Sapporo only two years later, so I was about 푝-adic deformation of modular forms and the use of alone. Fortunately, Hiroyuki Yoshida returned to Kyoto at the Hecke algebra to make something like a GL(2)-version this time after his PhD study with Shimura in Princeton, of Iwasawa theory. They were fuzzy thoughts at the time and he had a good understanding of Hilbert modular vari- and only once or twice useful in our conversation. Repe- eties. In fact, while working on this problem, I talked only tition of topics, without much progress, was not a useful to Yoshida. When I was ready, I wrote about my results strategy for conversation with Sensei. But if you are an to Shimura at Princeton and, surprisingly, this attracted entertainer hired by somebody, you need to produce an him. Indeed, the work suggested that higher-dimensional attraction every time you perform! Thus, I had to skip the periods of Hilbert modular CM theta series are somehow meeting several times. Fortunately, I eventually came up related to periods of CM elliptic curves, at least if the base with a use of the partial Fourier transform to compute 푞- totally real field has odd degree. Here I should mention expansions of orthogonal and unitary Eisenstein series, as

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 681 well as some series that Shimura had invented (I call them Shimura series). Then I went to the tea. The first words he threw at me were “I thought you are dead.” I replied to him, “Like the characters ‘Yosaburo and Otomi’ of a Kabuki play (Japanese opera), somebody’s survival could not be known even to Buddha.” This amused him. I recov- ered, and I was to find a place twenty-five years later for this computation, in the case of Siegel’s theta series, in my arti- cle in the Coates volume of Documenta Math. dealing with the anti-cyclotomic main conjecture. Those were demanding but happy days for me at IAS. Shimura was able to squeeze out of me every bit of math- ematics I potentially had. I still have a good stock of us- able results from the notes at the time. He did not teach me much mathematics, but he guided me how to pull out Figure 3. Chikako and Goro with Kamal Khuri-Makdisi in something useful from my own mind, not from books or Beirut in May 2000. articles somebody else wrote. I am grateful for his un- usual effort for my development. Farewell to his existence, how to overcome the blockage. which was so richly difficult, rewarding, and fun for me. During my time at Princeton, Shimura’s graduate Sayou-nara! (literally, “if things have gone that way, we courses alternated between lectures at the introductory or part”). more advanced level and seminars where students had to present material out of his 1971 book Introduction to Kamal Khuri-Makdisi the Arithmetic Theory. . . or from terse notes of his on 퐿- 퐿 Personal memories. Goro Shimura was my thesis advisor functions of modular forms and Artin -functions. I have in the early 1990s, and the relationship developed into kept these notes preciously over the years. At the end of friendship over the subsequent years. While I was his grad- the semester, particularly with a seminar, he would invite uate student, he gave me full support and mentorship, pa- the students over to his house for a “dinner in lieu of final tiently guiding me through first reading a number of his exam,” an occasion to be more informal than in class. articles in preparation for my thesis, then the actual the- Shimura had a real interest in art, with an impressive sis work. Our weekly meetings would usually last one to collection of both prints and porcelain, the latter not only one and a half hours, during which he was unstintingly from East Asia but also the Middle East; his nonmathemat- generous with his attention and advice. When, as a result ical writings include a book on Imari porcelain. He and of my youthful inexperience, I made a naive mathemati- his wife Chikako traveled several times to the Middle East, cal speculation, or was mistaken about a certain point, he visiting Turkey a few times (K. Ilhan Ikeda and I did our would diplomatically correct me with the phrase, “That PhDs with him at the same time), Iran for a conference, is completely correct, but....” He also regularly exhorted and Lebanon on two occasions, when it was a real plea- me during my thesis and subsequent career choices to be sure to be able to host Goro and Chikako. He managed “practical,” especially in terms of finding research subjects to combine the mathematical aspect of his trips with visits to work on that were both attainable and interesting. He to museums and archaeological sites, plus the inevitable was so aware of my progress that when I stalled for a while antique shops. on my thesis, he was able to diagnose the problem with- Some aspects of his mathematical legacy. Shimura’s out my having told him precisely what I had been stuck on. mathematical contributions are so fundamental and wide- He simply presented me one day with a few pages of notes ranging that no one person can write about them all. I will where he explained what I had most probably overlooked go over many important topics too quickly and will skip (an issue where Maass-type Hilbert modular forms could others altogether. I hope that this discussion can at least be either odd or even at each archimedean place, which led do justice to some fraction of his work. to different constructions). His advice was of course right Shimura knew thoroughly the earlier work on modular on the mark, and he included in his notes suggestions on forms by Hecke, Siegel, Maass, Petersson, Fricke, and We- ber, among others. He had also carefully studied Lie (and Kamal Khuri-Makdisi is a professor of mathematics at the American University algebraic) groups from Chevalley’s book, as well as alge- of Beirut. His email address is [email protected]. braic geometry in the language of Weil’s Foundations. By the late 1970s, though, his articles tended to contain less

682 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 algebraic geometry and more analysis. He never worked going well beyond questions about the behavior of Eisen- explicitly in the language of automorphic representations stein series, as part of his large program on arithmeticity, but was very happy to move between the adelic language which was a large focus of his work from the mid-1970s and explicit computations with real or Hermitian symmet- through the late 1990s. ric spaces and arithmetic groups. In general, his articles Before I mention Shimura’s work on arithmeticity, how- are complete, thorough, and very demanding to read in ever, I will briefly mention his significant production of terms of the intricacy of the calculation. However, all the books and articles from the mid-1990s until 2012, dur- material is there, with very few errors (which usually get ing his retirement (“only from teaching,” he once told corrected in errata at the end of a subsequent paper). So a me). In a 1997 monograph, Euler Products and Eisenstein patient and determined reader can make it to the end but Series, he broke new ground in the explicit construction must just be prepared for a slow rate of progress per page. of 퐿-functions using essentially the “doubling method” of Readers in number theory will be familiar with Gelbart, Piatetski-Shapiro, and Shalika, obtaining all Eu- Shimura’s foundational contributions to the arithmetic of ler factors and gamma factors. Also, in two monographs, modular curves and abelian varieties, such as the decom- Arithmetic and Analytic Theories of Quadratic Forms and Clif- position of the Jacobian of a modular curve. The section by ford Groups (2004) and Arithmetic of Quadratic Forms (2010), Blasius in this memorial article summarizes his introduc- Shimura obtained new results in the theory of quadratic tion of what are now known as Shimura varieties and their forms and new explicit forms of the celebrated Siegel mass canonical models. I will only mention Shimura’s impor- formula. He further summed up and refined in mono- tant insight that in the non-PEL case one can use the CM- graph form many strands of his earlier work that had pre- points to pin down the rationality and produce a canonical viously appeared in articles: his work with Taniyama from model over the “correct” number field. the 1950s on complex multiplication in Abelian Varieties Another famous contribution by Shimura is in the area with Complex Multiplication and Modular Functions (1998); of modular forms of half-integral weight, beginning with elementary and less elementary topics in modular forms, [11]. Shimura studied many aspects, not just over SL(2, 퐐) including the Shimura correspondence and the simplest as in the first paper above, but also over symplectic groups cases of arithmeticity in the books Elementary Dirichlet Se- over totally real fields, so in the Siegel–Hilbert case. Over ries and Modular Forms (2007) and Modular Forms: Basics SL(2, 퐐), as is well known, Shimura showed that to a and Beyond (2012); and a more comprehensive treatment Hecke eigenform 푓 of half-integral weight 푘 = 푚 + 1/2 of his program of arithmeticity in Arithmeticity in the Theory there corresponds a Hecke eigenform 푔 of (even) integral of Automorphic Forms (2000). weight 2푚, so on 푃GL(2, 퐐), with matching Hecke eigen- Shimura’s program on arithmeticity, a large focus of his values. Shimura’s original proof of this went via con- work from the mid-1970s onwards, can be viewed as a very structing the 퐿-functions of twists of 푔 by Dirichlet char- large and elaborate outgrowth of the two seminal articles acters and then invoking Weil’s converse theorem. Later, [12], [14]. I shall single out two themes: from the first after work of Shintani and Niwa and with further hind- article, the arithmeticity of the values of nearly holomor- sight, this “Shimura correspondence” was recognized as phic modular forms at CM points, and from the second, an early example of a theta-correspondence, here between the arithmeticity of special values of 퐿-functions of modu- the double cover of SL(2) and 푂(2, 1), which is essentially lar forms, and the relation of these special values to more the same as 푃GL(2). Shimura revisited his correspon- fundamental periods attached to the forms. dence from this point of view in [21] and subsequent arti- The first theme above generalizes Shimura’s reciprocity cles for the Hilbert modular case, and describes the theta- law for holomorphic modular functions at CM points to correspondence viewpoint over 퐐 in a readable account certain nonholomorphic functions, where the relation to for students in his last book, Modular Forms: Basics and algebraic geometry is less direct. In the classical context, a Beyond, published in 2012. As another result using half- nearly holomorphic form is a function 푓 ∶ ℋ → 퐂 on the integral weight on SL(2), Shimura was the first to prove usual upper half-plane which transforms as expected un- the remarkable result [13] that the symmetric square 퐿- der a congruence subgroup of SL(2, 퐙). Instead of requir- function of a classical modular form has an analytic con- 푁 −푛 ing 푓 to be holomorphic, we require 푓 = ∑푛=0 푓푛(푧)푦 , 퐂 tinuation to . Prior to that, one had only a meromorphic where 푦 = Im(푧) and the 푓푛 are holomorphic. (Actually, continuation with possible poles at all the zeros of the Rie- for arithmeticity reasons, it is better to use (휋푦)−푛.) A typi- 퐿 −1 mann zeta function or a Dirichlet -function. This proof cal example is the Eisenstein series 퐸2 = (8휋푦) − (1/24) + used a careful analysis of Eisenstein series of half-integral 푛 ∑푛≥1 휎1(푛)푞 . One can also obtain nearly holomorphic weight. Shimura of course studied many other aspects of forms by applying certain differential operators to holo- half-integral weight on larger groups, including but also morphic forms. Shimura introduced an ingenious way to

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 683 evaluate such an 푓 at a CM-point 푧0 = (푎 + 푏√−퐷)/푐, by comparing 푓 and 푓|훼 for an element 훼 that stabilizes 푧0, and combining this with taking various derivatives. The generalization of this to larger groups is of course more involved. The second theme, in the setting of the arithmeticity of the special values of standard 퐿-functions of classical modular forms, can be studied in terms of the Eichler– Shimura cohomology groups or (as formulated by Manin) in terms of modular symbols. The new approach in [14] is quite different and allows for a generalization to many other groups and 퐿-functions. In the classi- cal setting, let 푓 be a newform (in Shimura’s terminol- ogy, a primitive form). Instead of considering a single (twisted) special 퐿-value 퐿(푘, 휒, 푓) for a Dirichlet character 휒, Shimura considers products of two such special values, Figure 4. Goro Shimura and Kenneth Ribet in the summer of 1973. which he obtains via an integral of Rankin–Selberg type as 퐿(푘1, 휒1, 푓)퐿(푘2, 휒2, 푓) = ⟨푓, 퐺⟩ for an explicit modular form 퐺. Here 퐺 is a product of two Eisenstein series and Kenneth Ribet can be expanded as 퐺 = 푐퐸퐸 + ∑푖 푐푖푔푖, where 퐸 itself is an I first saw Goro Shimura at the Antwerp conference Mod- Eisenstein series, and the 푔푖 are cuspidal Hecke eigenforms ular Functions of One Variable in 1972. I was a graduate (not necessarily newforms; one can have 푔푖(푧) = ℎ푖(푁푖푧) student at the time and was shy around senior mathemati- for a newform ℎ푖). Since 퐺, 퐸, and the ℎ푖 have algebraic cians. Some of the faculty members at that conference Fourier coefficients, the 푐푖 and 푐퐸 are themselves algebraic, encouraged informal contact with graduate students, but and then one obtains (after some more work) an algebraic Shimura’s body language did not convey any encourage- expression for 퐿(푘1, 휒1, 푓)퐿(푘2, 휒2, 푓) in terms of those 푐푖 ment. I was further intimidated by a comment that was where ℎ푖 = 푓. This is the heart of the idea in the classi- made by one of the Princeton graduate students: “Many cal case, and it generalizes somewhat directly to Hilbert mathematicians have knowledge. Shimura has wisdom.” modular forms. However, for the generalization to larger Twelve months later, I had finished my dissertation groups and other 퐿-functions, one requires two significant work and was about to begin a lecturer position at Prince- inputs: first, a good understanding of the analytic (not ton. Shimura welcomed me when I arrived in the Prince- just meromorphic) continuation of Eisenstein series on ton math department and gave me tips for dealing with larger groups, with a precise proof of arithmeticity of their students. In particular, he recommended that I respond to Fourier expansions, and, second, once again a thorough students’ questions by writing down answers in their note- understanding for higher groups of the differential oper- books so that they would have easy access to my comments ators that already appeared for evaluation at CM-points. even after their memories of my remarks had faded. (The differential operators are needed even in the classical Shimura then asked me a mathematical question that case but are more tractable there.) Tackling both of these turned out to be extremely fruitful. Specifically, let 푁 be questions in more general settings involved a large body a prime number and let 퐽 be the Jacobian of the modu- of work by Shimura (and his students, in their theses) over lar curve 푋0(푁) that classifies degree 푁 isogenies between some twenty-five years, and the computations are quite in- elliptic curves. Then 퐽 is an abelian variety over the field tricate. Besides the books mentioned above, Euler Products 퐐 of rational numbers. Moreover, 퐽 has endomorphisms and Eisenstein Series (1997) and Arithmeticity in the Theory 푇푛 (푛 ≥ 1) that are geometric versions of the Hecke oper- of Automorphic Forms (2000) and their references, I will sin- ators on modular forms that had been studied by Hecke gle out the articles [19], [20] as a memorable illustration and Shimura. These endomorphisms generate a subring 퐓 of the careful study that Shimura was able to undertake of of the ring of endomorphisms of 퐽 over 퐐. Shimura asked the analytic continuation and Fourier expansions of Eisen- me whether 퐓 ⊗ 퐐 is the full endomorphism algebra of 퐽. stein series in integral and half-integral weight. The diffi- culty resides largely in the number theory, but there are also genuine analytic challenges in terms of special (con- Kenneth Ribet is a professor of mathematics at the University of California, fluent hypergeometric) functions on the symmetric spaces Berkeley. His email address is [email protected]. of these higher-rank groups.

684 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Alice Silverberg The way I became a PhD student of Goro Shimura was a bit unusual. Sometime in my first year, I asked Nick Katz tobe my thesis advisor. He told me to talk to him after I passed my General Exam in the spring. After the exam, Katz went away for the summer. At a conference early that summer, I ran into John Coates, who asked me whom I planned to work with. When I said Katz, he exclaimed, “But Alice, you can’t possibly work with Nick Katz! He won’t be at Princeton. He’s accepted a job at Berkeley.” I had no way to contact Katz to check this (and to find out that it wasn’t correct). So I decided that I had bet- ter have a back-up plan. Former students of Professor Shimura had told me that the first thing he tells prospec- tive students is to read the red book, Introduction to the Figure 5. Goro, Chikako, and Haru at the Ukiyo-e Museum in Matsumoto, Japan. Arithmetic Theory of Automorphic Functions. That summer, I started to read the book and do the exercises. I found first that 퐓 ⊗ 퐐 is the algebra of all endomor- In the fall, I needed to talk to the director of graduate phisms of 퐽 that are defined over 퐐, but this was not news studies about an administrative matter. That happened to to Shimura. He asked me whether or not there were en- be Katz. One day after tea I followed him to the elevator domorphisms of 퐽 over the algebraic closure of 퐐 beyond and asked to talk with him. As we waited for the elevator the ones that are already defined over 퐐. Using results he turned to me and said, “So are you my student or aren’t that had been obtained by Deligne and Rapoport a year or you?” Based on his tone of voice, I reflexively responded, two before, I proved that all endomorphisms of 퐽 are de- “No, I’m not.” Then he asked me who my advisor was. fined over 퐐. Technically, I used the theorem of Deligne– Without thinking I blurted out “Shimura.” Then, horrified Rapoport to the effect that 퐽 has semistable reduction at at the thought that he might ask Shimura and find out it the prime 푁 (and thus at all primes, because 퐽 has good wasn’t true, I hurriedly added, “But he doesn’t know it yet!” reduction outside 푁); my result was really about abelian Luckily, Shimura agreed to be my advisor. varieties with semistable reduction. A former student told me that his experience was that Shimura was delighted by my result and asked me to he would bring a notebook to his meetings with Shimura. explain my proof to him in detail. In response, he wrote Shimura would write a problem in the notebook and ask down a polynomial identity that simplified the main com- the student to solve it for their next meeting. If the student putation that I had presented to him. Shimura then sug- didn’t solve it, Shimura wrote the solution in the student’s gested that we write a joint article with the result. Perhaps notebook. My experience was very different; I worked very selfishly, I told him that I was reluctant to write a jointpa- independently. per because I had never yet published any mathematical Shimura suggested a nice thesis problem, and I went article. Shimura accepted my answer and encouraged me away and worked on it. He then took the problem away to publish the result on my own. I did so—and credited from me. I heard through the grapevine that Shimura had Shimura for posing the original problem and for the sim- given the problem to a former student to do for his thesis, plification that he made to my argument. I was grateful but the student hadn’t solved the problem then and now to him for this act of generosity, but now worry that I was wanted it back. wrong to decline his offer. Next, perhaps to make up for the time I had spent on the By the way, my theorem shows that (End 퐽)/퐓 is a tor- first problem, Shimura gave me a choice of three problems. sion abelian group. Here, End 퐽 is the full ring of endo- When I chose the one furthest from his interests at the time, morphisms of 퐽, and 퐓 again is the subring of those en- he was pleased with my choice. domorphisms that come from Hecke operators. In 1977, Shimura gave me a first step to solve and told meto Barry Mazur proved that the quotient (End 퐽)/퐓 is torsion come back in about two weeks to report on my progress. free in his “Eisenstein ideal” article. Our results together Alice Silverberg is Distinguished Professor of Mathematics and Computer Sci- imply that the quotient is trivial, i.e., that 퐓 is the full ring ence at the University of California, Irvine. Her email address is asilverb of endomorphisms of 퐽. @uci.edu.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 685 By the end of those two weeks I managed to understand the question, but I hadn’t made progress towards a solution. I didn’t feel that was good enough, so I didn’t arrange to see him and kept working. After another two weeks I had an- swered the question, but I didn’t feel I could show up after four weeks having only accomplished what I should have done in two. So I kept working and made more progress, but it never seemed like enough, given the time I had spent on it. After a few months, I realized that I needed to let my advisor know I was still alive, so I met with Shimura and told him what I had accomplished. He could have been angry that I hadn’t kept him informed. Instead, and luckily for me, he was pleased with both the work I had accomplished and my independence. If I were restricted to one word to describe Shimura Figure 6. Group photograph taken at the NSF-CBMS as a thesis advisor, I would say that he was “responsi- conference at Texas Christian University in honor of Shimura ble.” That’s higher praise than it sounds; conscientious- in May 1996. ness seemed like a rare and unusual trait among thesis ad- visors when I was a Princeton graduate student. precise understanding, the legend of Shimura–Taniyama– As my interests moved away from automorphic forms Weil on complex multiplication and Shimura’s work on and into cryptography, I found that I continued to use Shimura curves. Hijikata selected some suitable literature Goro’s work, especially the theory of complex multiplica- for a seminar, and I was able to proceed rather quickly and tion, which is useful for modern-day cryptography. I also learned that Shimura was developing the theory of higher- used Shimura reciprocity to help construct an algorithm re- dimensional arithmetic quotients of bounded symmetric lated to point counting on elliptic curves over finite fields. domains. Hiroshi Saito also attended Hijikata’s seminar. Goro Shimura had very high standards. I do best when Next summer, Hijikata, through his friend Professor Ya- the standards for me are high, so I am very grateful to Goro sutaka Ihara, introduced me to Shimura. In the spring of for having high standards for me, for telling me that a 1971, I was admitted to graduate school in mathematics mathematician must be an optimist, and for believing in at with a scholarship, to start in the me as a mathematician. While he didn’t often communi- autumn. That spring, Professors Koji Doi and Hidehisa cate that he thought highly of me, he did it enough (to Naganuma came back to Kyoto from IAS in Princeton. I both me and mathematicians who made decisions about first met Professor Goro Shimura on July 14, 1971,on me) to have a positive effect on my life and career. I will the Ishibashi campus of Osaka University. Shimura was cherish the memories of our mathematical father-daughter invited by Professor Taira Honda to give a lecture in Os- relationship. aka. My first impression was that he resembled the famous philosopher Kitaro Nishida of the Kyoto school. Hiroyuki Yoshida Talks with Shimura in person. I arrived at Princeton on September 13, 1971, and met Shimura the next day in The impact of student protests in Paris in May 1968 spread his office, and he kindly took me to his home. Shimura to the world, and Kyoto University in Japan was swallowed had just published his now standard textbook Introduction up in big waves from the beginning of 1969. No courses to the Arithmetic Theory of Automorphic Functions. I asked were offered to students for one year. In the autumn of him, “What is your present interest?” He replied that he 1969, I visited, with classmates, Professor Hiroaki Hijikata, was studying modular forms of half integral weight. He who was then a young associate professor, in his office hinted that he had discovered a relation between modu- to ask him to give us a seminar on number theory. I lar forms of half integral weight and modular forms of in- was a senior mathematics major at Kyoto University. Hi- tegral weight. A preprint became available only the next jikata asked me what I was interested in. I responded, spring, and now this relation between modular forms of “Complex multiplication.” I knew at that time, without half integral weight (2푘 + 1)/2 and modular forms of inte- gral weight 2푘 is called the Shimura correspondence ([11]). A Hiroyuki Yoshida is emeritus professor at Kyoto University. His email address technical core of the proof is an ingenious application of is [email protected]. the Rankin–Selberg convolution and Weil’s converse theo- rem.

686 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 We also talked about complex multiplication of abelian is algebraic for 1 ≤ 푚 ≤ 푘 − 1, (−1)푚 = ±1, 푚 ∈ 퐙. (A co- varieties and construction of class fields. Though an homological approach was discovered by Shimura fifteen abelian variety 퐴 has (sufficiently many) complex multi- years earlier ([2]).) plications by an algebraic number field 퐾, the class field is Moreover, 푢+(푓)푢−(푓) = 푖1−푘휋⟨푓, 푓⟩, where ⟨푓, 푓⟩ is the obtained over the reflex field 퐾′, which is different from 퐾 normalized Petersson norm of 푓. Shimura stressed the im- in general. Shimura explained this curious phenomenon, portance of 퐷(푠, 푓, 푔) and not only 퐿(푠, 푓). Suppose that 푔 is first discovered by Hecke in a simple case, as follows. Sup- also a Hecke eigen cusp form. Then 휁(2푠+2−푘−푙)퐷(푠, 푓, 푔) pose that 퐴 is defined over a subfield 푘 of 퐂. Then the field has an Euler product of degree 4 and may be denoted as generated by the division points of 퐴 is determined as a 퐿(푠, 푓 ⊗ 푔). Then (2휋)푙−1−2푚퐿(푚, 푓 ⊗ 푔)/푢+(푓)푢−(푓) is al- subfield of 퐂. But 퐾 is not determined as a subfield of 퐂; gebraic for 푙 ≤ 푚 < 푘, 푚 ∈ 퐙 ([14], [16]). It is very inter- only its isomorphism class has a definitive meaning. In esting that the form of higher weight gives the dominant contrast to this, the reflex field is determined as a subfield contribution. Shimura discovered an ingenious but now of 퐂 from 퐾 and the CM-type of 퐴. I felt a revelation and standard way to deduce these results from (1). got deeply interested in this “thought experiment.” Shimura was a great master of using the Rankin–Selberg In the summer of 1972, Shimura took me to the convolution to squeeze arithmetical information from it. Antwerp conference on modular functions of one variable. In the half integral weight case mentioned above, 푓 is a One afternoon, we went out for sightseeing in the city. I form of half integral weight, 푔 is a classical theta func- was impressed that Shimura very efficiently found a route tion, and 퐸 is an Eisenstein series of half integral weight to visit museums and places of interest with a handy map. in (1). In the later years, Shimura generalized the method This experience turned out to be very useful for my travels to higher-dimensional cases. in later years. He went back to his hotel after buying some In the summer of 1985, Shimura suddenly called me. paper of good quality for his calculations. (In the meantime, I was communicating with Shimura by I received a PhD in 1973 under Shimura’s guidance and occasional letters.) He told me he had come to Kyoto with stayed in Princeton as a postdoc until August of 1975. In his wife but with no business. His wife was visiting her 1975, when I was preparing to leave Princeton for Kyoto, friend. So we went to Kurama temple for sightseeing. In Shimura was studying critical values of 퐿-functions associ- the bus from his hotel to the local train station, I talked ated with modular forms. This time the Rankin–Selberg about the then-popular movie Amadeus. I talked about the method was employed again. The basic formula is sad destiny of Salieri’s music. Then he said, “I am Mozart” immediately. Shimura also declared at this time that the −푠 (4휋) Γ(푠)퐷(푠, 푓, 푔) modularity conjecture for elliptic curves over 퐐 was his. Shimura explained to me some interesting work of his re- = ∫ 푓(푧)푔(푧)퐸 (푧, 푠 + 1 − 푘)푦푠−1푑푥푑푦. (1) 푘−푙 cent PhD students. The name of Don Blasius was among Γ0(푁)\퐻 them. I think Shimura wished to give me some stimula- Here 푠 is a complex variable, 퐻 is the complex upper half- tion. We parted this time after he promised to visit Kyoto plane, and 푧 ∈ 퐻 is written as 푧 = 푥 + 푖푦, 푦 > 0, 푥 ∈ 퐑; ∞ ∞ University in the summer two years later. 푓(푧) = ∑ 푎 푞푛 푔(푧) = ∑ 푏 푞푛 푞 = 푒2휋푖푧 푛=1 푛 and 푛=0 푛 , with , Shimura had a summer villa in Tateshina, Nagano pre- are holomorphic modular forms with respect to Γ0(푁) of fecture, Japan. After 2011, Shimura invited me three weights 푘 and 푙, respectively, with 푘 > 푙. For simplicity, we times to visit his villa. The villa was acquired around assume that the characters of 푓 and 푔 are trivial (Hauptty- the time when Shimura moved to Princeton from Osaka. pus). For 0 ≤ 휆 ∈ 퐙, 퐸휆(푧, 푠) is an Eisenstein series defined An old, rather small building was standing in a large site. by Tateshina is very nice to stay in during the summertime. −휆 −2푠 퐸휆(푧, 푠) = ∑ (푐푧 + 푑) |푐푧 + 푑| , Nearby are villas of celebrities. Shimura spent summers 훾∈Γ∞\Γ0(푁) here with his family every couple of years during his pro- where fessorship in Princeton. His study was small, less than 5m2. 푎 푏 훾 = ( ) It is amazing that monumental papers were written in this 푐 푑 place. As I was accompanying a young mathematician, I and asked Shimura what would you do in mathematics if you 1 푚 | were young. It was around 2013. He responded that he Γ = {± ( ) | 푚 ∈ 퐙} . ∞ 0 1 | would study Siegel. In fact, volume V of his collected pa- Our first objective is to study special values of 퐿(푠, 푓) = pers contains several important papers on quadratic forms, ∞ −푠 and the influence of Siegel and Eichler was manifest. ∑푛=1 푎푛푛 . Suppose that 푓 is a Hecke eigen cusp form. Then there exist periods 푢±(푓) such that 퐿(푚, 푓)/휋푚푢±(푓)

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 687 Shimura sometimes told me that he loved “tricks” in the chairman. He replied, “Don’t prove anybody’s conjec- mathematics. The trace formula and the Rankin–Selberg ture,” and everybody laughed. This episode is recorded in convolution are examples to him. A long winding road of How to Teach Mathematics, p. 33. In this book, Shimura reasoning leads to a simple theorem, illuminated by con- explains this advice in some depth. crete (sometimes numerical) examples that can be seen I stayed at IAS in Princeton for 1990–91. In the spring by everybody. This is a tradition among number theorists of 1991, Shimura showed me reports of senior students from ancient times. For a layman, this may look like magic. evaluating Shimura’s course. Most students evaluated the I mention [2], [6], [11], [14], and [25]. course highly; a few said that they never attended such On a few occasions, we talked about big problems splendid lectures during their student time in Princeton. such as the Hodge conjecture and the Riemann hypothesis. Shimura told me that they were not mathematics majors; Shimura was negative about attacking a big problem with- he lectured on number theory with an emphasis on histor- out having good ideas. But he had the opinion that for ical perspectives. He was very proud of the students’ eval- the Riemann hypothesis, differential operators will play a uations and saw some of them in his Princeton home for crucial role. An interested reader may consult [23], [24]. decades. In Shimura’s class. Shimura’s students know that the Shimura published two autobiographies; one in Eng- master has excellent skills of exposition and is also an en- lish (The Map of My Life), one in Japanese. The contents are tertainer. basically the same, of course, but they are independently In a lecture around 1972 at Princeton University, he written. In the book, he wrote that when he was young explained the theory of canonical models, now called (1952–56), he taught at the University of Tokyo and also Shimura varieties. To study the model further, he said, “To in a preparatory school because the salary was so low. Hi- desingularize or not to desingularize: that is the question.” jikata told me that he was then in preparatory school and Everybody, knowing Hamlet, enjoyed the performance and impressed by Shimura’s lecture; that experience led him to laughed. (This incident is recorded in his book written in study mathematics. Japanese How to Teach Mathematics, p. 20. This book is Perhaps I should comment briefly on the historical the last item of the bibliography of volume V of his col- facts concerning the Shimura–Taniyama conjecture, i.e., lected papers.) Also in another lecture around the same the modularity conjecture of elliptic curves over 퐐. The time, he talked about exotic ℓ-adic representations con- issue is analyzed in detail in his autobiographies. The structed using the theory of canonical models ([10], and English version gives a more detailed account. He under- §8 of “On canonical models of arithmetic quotients of stood well that the issue was quite social. I quote one bounded symmetric domains. I”). He said that the eigen- paragraph from Shimura’s book: “The reader may ask why values of the Frobenius automorphism have the property there were so many people who called the conjecture in of “Riemann–Ramanujan–Weil type.” Everybody enjoyed various strange ways. I cannot answer that question except it and laughed. to say that many of them had no moral sense and most In a lecture around 1989 at Kyoto University, he talked were incapable of having their own opinions.” about the critical values of Dirichlet series and periods As a man of culture. We had several interests in com- of automorphic forms ([22]). Automorphic forms are of mon, so it was not difficult for me to start a conversa- Hilbert modular type, and 퐿-functions are of standard type tion with Shimura. One interest is Japanese chess (Shogi). or of Rankin–Selberg type. But he considers all configu- The chess board is 9 × 9, and we can use captured pieces rations of weights including the half integral weight case. again. The other rules are basically the same as the western Shimura’s exposition was clear, but the situation is compli- chess game. When I visited him in his Princeton home, he cated and divided into several cases. For the half integral showed me a Japanese chess problem (Tsume-Shogi) com- case, the period to be considered is the minus period of posed by him. When I showed the correct answer of 71 the integral weight form that corresponds to the half inte- moves written on a paper in my next visit, Shimura was gral weight form by the Shimura correspondence, while it very pleased. I still keep this problem and another problem is the product of plus and minus periods when the form is of 169 moves. Shimura said he composed chess problems integral weight. He explained this by saying, “The period when he was very young, spending considerable time. becomes the half because the weight is half.” Everybody As my family served as Buddhist priests for centuries, I laughed but this time with some feeling of relief. (To catch had some training to read Buddhist scriptures in Chinese the point quickly, the reader is advised to see the introduc- translation from my childhood. Shimura liked to read Chi- tion of [18].) nese classics and published two books. He also had some In Paris in 2000, after finishing a talk at a conference, original perspectives on Buddhism. Shimura was asked to give advice for a young audience by Shimura loved music very much. His writings about

688 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 music are scattered in many places in his books. In my of Math. (2) 89 (1969), 99–124, DOI 10.2307/1970811. first visit to his home, he played the well-tempered clavier MR272758 on a record player. In later years, he regularly visited the [11] Goro Shimura, On modular forms of half integral Metropolitan Opera with his wife. weight, Ann. of Math. (2) 97 (1973), 440–481, DOI Shimura had an interest in antiques. He published a 10.2307/1970831. MR332663 [12] Goro Shimura, On some arithmetic properties of modular book on Imari porcelains. He had this interest since he forms of one and several variables, Ann. of Math. (2) 102 was young; he wrote so in his book written in Japanese. (1975), no. 3, 491–515, DOI 10.2307/1971041. MR491519 But it was enhanced by a famous professor of oriental art at [13] Goro Shimura, On the holomorphy of certain Dirichlet se- Princeton University, I think. Shimura’s wife told me that ries, Proc. London Math. Soc. (3) 31 (1975), no. 1, 79–98, her husband had a very strong memory of shapes, perhaps DOI 10.1112/plms/s3-31.1.79. MR0382176 stronger than that professor. [14] Goro Shimura, The special values of the zeta functions Final recollections. Writing this article, I felt strongly the associated with cusp forms, Comm. Pure Appl. Math. 29 (1976), no. 6, 783–804, DOI 10.1002/cpa.3160290618. coherency of Shimura’s character. Time has passed, and I MR434962 am now much older than Shimura was when we first met. [15] Goro Shimura, On abelian varieties with complex multipli- Nobody can stop aging and change their ultimate destiny. cation, Proc. London Math. Soc. (3) 34 (1977), no. 1, 65– What I can be proud of, if anything, is that, after 1987, I in- 86, DOI 10.1112/plms/s3-34.1.65. MR0572987 vited Shimura about ten times to Kyoto University to give [16] Goro Shimura, On the periods of modular forms, intensive courses, which must benefit students and young Math. Ann. 229 (1977), no. 3, 211–221, DOI researchers. I also helped Shimura to edit his collected pa- 10.1007/BF01391466. MR463119 pers I–V with Alice Silverberg and Doi. But compared to [17] Goro Shimura, Automorphic forms and the periods of what Shimura gave me to live as a mathematician, they are abelian varieties, J. Math. Soc. Japan 31 (1979), no. 3, 561– 592, DOI 10.2969/jmsj/03130561. MR535097 very small contributions. I thank Professor Goro Shimura [18] Goro Shimura, The critical values of certain zeta func- again, this time with the deepest feeling of loss. Shimura is tions associated with modular forms of half-integral weight, not among us anymore and will not respond to our emails. J. Math. Soc. Japan 33 (1981), no. 4, 649–672, DOI But his mathematics will live like the music of Mozart. 10.2969/jmsj/03340649. MR630630 [19] Goro Shimura, On Eisenstein series, Duke Math. J. 50 References (1983), no. 2, 417–476. MR705034 [1] Goro Shimura, Reduction of algebraic varieties with respect [20] Goro Shimura, On Eisenstein series of half-integral to a discrete valuation of the basic field, Amer. J. Math. 77 weight, Duke Math. J. 52 (1985), no. 2, 281–314, DOI (1955), 134–176, DOI 10.2307/2372425. MR66687 10.1215/S0012-7094-85-05216-0. MR792175 [2] Goro Shimura, Sur les int´egrales attach´eesaux formes auto- [21] Goro Shimura, On Hilbert modular forms of half-integral morphes (French), J. Math. Soc. Japan 11 (1959), 291–311, weight, Duke Math. J. 55 (1987), no. 4, 765–838, DOI DOI 10.2969/jmsj/01140291. MR120372 10.1215/S0012-7094-87-05538-4. MR916119 [3] Goro Shimura, On analytic families of polarized abelian vari- [22] Goro Shimura, On the critical values of certain Dirichlet eties and automorphic functions, Ann. of Math. (2) 78 (1963), series and the periods of automorphic forms, Invent. Math. 149–192, DOI 10.2307/1970507. MR156001 94 (1988), no. 2, 245–305, DOI 10.1007/BF01394326. [4] Goro Shimura, Class-fields and automorphic functions, Ann. MR958833 of Math. (2) 80 (1964), 444–463, DOI 10.2307/1970658. [23] Goro Shimura, 퐿-functions and eigenvalue problems, Alge- MR169822 braic analysis, geometry, and number theory (Baltimore, [5] Goro Shimura, On the field of definition for a field ofauto- MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, morphic functions, Ann. of Math. (2) 80 (1964), 160–189, 1989, pp. 341–396. MR1463710 DOI 10.2307/1970497. MR167466 [24] Goro Shimura, Invariant differential operators on Hermit- [6] Goro Shimura, A reciprocity law in non-solvable exten- ian symmetric spaces, Ann. of Math. (2) 132 (1990), no. 2, sions, J. Reine Angew. Math. 221 (1966), 209–220, DOI 237–272, DOI 10.2307/1971523. MR1070598 10.1515/crll.1966.221.209. MR188198 [25] Goro Shimura, The representation of integers as sums of [7] Goro Shimura, Moduli and fibre systems of abelian va- squares, Amer. J. Math. 124 (2002), no. 5, 1059–1081. rieties, Ann. of Math. (2) 83 (1966), 294–338, DOI MR1925343 10.2307/1970434. MR199190 [8] Goro Shimura, Algebraic number fields and symplectic dis- Credits continuous groups, Ann. of Math. (2) 86 (1967), 503–592, Figures 1 and 2 are courtesy of Haru Shimura. DOI 10.2307/1970613. MR222048 Figure 3 is courtesy of Kamal Khuri-Makdisi. [9] Goro Shimura, Construction of class fields and zeta functions Figure 4 is courtesy of Ken Ribet. of algebraic curves, Ann. of Math. (2) 85 (1967), 58–159, Figure 5 is courtesy of Haru Shimura. DOI 10.2307/1970526. MR204426 [10] Goro Shimura, Local representation of Galois groups, Ann. Figure 6 is by Linda Kaye.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 689 The Legacy of J´ozefMarcinkiewicz: Four Hallmarks of Genius In Memoriam of an Extraordinary Analyst Nikolay Kuznetsov

This article is a tribute to one of the most prominent Pol- Considering what he did during his short life and ish mathematicians, J´ozefMarcinkiewicz, who perished what he might have done in normal circumstances eighty years ago in the Katy ´nmassacre. He was one of one may view his early death as a great blow to nearly 22,000 Polish officers interned by the Red Army Polish Mathematics, and probably its heaviest in- in September 1939 and executed in April–May 1940 in dividual loss during the second world war. the Katy ´nforest near Smolensk and at several locations elsewhere. One of these places was Kharkov (Ukraine), From the Marcinkiewicz Biography [9] where more than 3,800 Polish prisoners of war from On the occasion of the centenary of Marcinkiewicz’s birth, the Starobelsk camp were executed. One of them was a conference was held on 28 June–2 July 2010 in Pozna ´n. Marcinkiewicz; the plaque with his name (see Figure 1) In its proceedings, L. Maligranda published the detailed is on the Memorial Wall at the Polish War Cemetery article [9] about Marcinkiewicz’s life and mathematical re- in Kharkov.1 This industrial execution was authorized by sults; sixteen pages of this paper are devoted to his biog- Stalin’s secret order dated 5 March 1940 and organized by raphy, where one finds the following about his education Beria, who headed the People’s Commissariat for Internal and scientific career. Affairs (the interior ministry of the Soviet Union) known Education. Klemens Marcinkiewicz, J´ozef’s father, was a as NKVD. farmer well-to-do enough to afford private lessons for him Turning to the personality and mathematical achieve- at home (the reason was J´ozef’s poor health) before send- ments of Marcinkiewicz, it is appropriate to cite the article ing him to elementary school and then to gymnasium in [24] of his supervisor Antoni Zygmund (it is published in Białystok. After graduating in 1930, J´ozefenrolled in the the Collected Papers [13] of Marcinkiewicz; see p. 1): Department of Mathematics and Natural Science of the Stefan Batory University (USB) in Wilno (then in , Nikolay Kuznetsov is a principal research scientist in the Laboratory for Math- now Vilnius in Lithuania). ematical Modelling of Wave Phenomena at the Institute for Problems in Me- From the beginning of his university studies, J´ozefde- chanical Engineering, Russian Academy of Sciences, St. Petersburg. His email monstrated exceptional mathematical talent that attracted address is [email protected]. the attention of his professors, in particular, of A. Zyg- 1https://www.tracesofwar.com/sights/10355/Polish-War mund. Being just a second-year student, Marcinkiewicz -Cemetery-Kharkiv.htm. attended his lectures on orthogonal series, requiring some Communicated by Notices Associate Editor Daniela De Silva. erudition, in particular, knowledge of the Lebesgue in- For permission to reprint this article, please contact: tegral; this was the point where their collaboration be- [email protected] . gan. The first paper of Marcinkiewicz (see [13, p. 35]) DOI: https://doi.org/10.1090/noti2084

690 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 to the compiled in this caf´ewas substantial, taking into account that his stay in Lw´ow lasted only nine months. One finds the history of this book in [14, ch.I], whereas problems and their solutions, where applicable, are presented in ch. II. Marcinkiewicz posed his own prob- lem; it concerns the uniqueness of the solution for the in- tegral equation 1 ∫ 푦(푡)푓(푥 − 푡) d푡 = 0, 푥 ∈ [0, 1]. 0 He conjectured that if 푓(0) ≠ 0 and 푓 is continuous, then this equation has only the trivial solution 푦 ≡ 0 (see prob- lem no. 124 in [14, pp. 211and 212]). He also solved three problems; his negative answers to problems 83 and 106 posed by H. Auerbach and S. Banach, respectively, involve ingenious counterexamples. His positive solution of prob- lem 131 (it was formulated by Zygmund in a lecture given in Lw´ow in the early 1930s) was published in 1938; see [13, pp. 413–417]. During the next two academic years, Marcinkiewicz was a senior assistant at USB and after completing his habili- tation in June 1937 became the youngest docent at USB. Figure 1. Plaque for Marcinkiewicz on the Memorial Wall at The same year, he was awarded the J´ozefPiłsudski Scien- the Polish War Cemetery in Kharkov. tific Prize (the highest Polish distinction for achievements in science at that time). His last academic year 1938–1939, was published when he was still an undergraduate student. Marcinkiewicz was on leave from USB; a scholarship from It provides a half-page proof of Kolmogorov’s theorem the Polish Fund for National Culture afforded him op- (1924) guaranteeing the convergence almost everywhere portunity to travel. He spent October 1938–March 1939 for partial sums of lacunary Fourier series. Marcinkiewicz in Paris and moved to the University College London for completed his MSc and PhD theses (both supervised by April–August 1939, also visiting Cambridge and Oxford. Zygmund) in 1933 and 1935, respectively; to obtain his This period was very successful for Marcinkiewicz; he PhD degree he also passed a rather stiff examination. published several brief notes in the Comptes rendus de The second dissertation was the fourth of his almost five l’Acad´emiedes Sciences Paris. One of these, namely [12], dozen publications; it concerns interpolation by means became widely cited because the celebrated theorem con- of trigonometric polynomials and contains interesting re- cerning interpolation of operators was announced in it. sults (see [24, p. 17] for a discussion), but a long publica- Now this theorem is referred to as the Marcinkiewicz or tion history awaited this work. Part of it was published in Marcinkiewicz–Zygmund interpolation theorem (see be- the Studia Mathematica the next year after the thesis defense low). Moreover, an important notion was introduced in (these two papers in French are reproduced in [13, pp. 171– the same note: the so-called weak-퐿푝 spaces, known as 185 and 186–199]). The full, original text in Polish ap- Marcinkiewicz spaces now, are essential for the general peared in the Wiadomo´sci Matematyczne (the Mathemati- form of this theorem. cal News) in 1939. Finally, its English translation was in- Meanwhile, Marcinkiewicz was appointed to the posi- cluded in [13, pp. 45–70]. tion of Extraordinary Professor at the University of Pozna ´n Scientific career. During the two years between defending in June 1939. On his way to Paris, he delivered a lecture his MSc and PhD theses, Marcinkiewicz did the one year of there and this, probably, was related to this impending ap- mandatory military service and then was Zygmund’s assis- pointment. Also, this was the reason to decline an offer of tant at USB. The academic year 1935–1936 Marcinkiewicz professorship in the USA during his stay in Paris. spent as an assistant at the Jan Kazimierz University in Marcinkiewicz still was in England when the general Lw´ow. Despite twelve hours of teaching weekly, he was mobilization was announced in Poland in the second half an active participant in mathematical discussions at the fa- of August 1939; the outbreak of war became imminent. mous Scottish Caf´e(see [3, ch. 10], where this unique form His colleagues advised him to stay in England, but his of doing mathematics is described), and his contribution ill-fated decision was to go back to Poland. He regarded

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 691 of genius. One indication of the ingenuity of the idea be- hind these results is that the note [11], in which they first appeared, is the most cited work of Marcinkiewicz. Another important point about his work is that he skill- fully applied methods of real analysis to questions border- ing with complex analysis. A brilliant example of this mas- tery—one more hallmark of genius—is the Marcinkiewicz function 휇 introduced as an analogue of the Littlewood– Paley function 푔. It is worth mentioning that the short pa- per [10], in which 휇 first appeared, contains other fruitful ideas developed by many mathematicians subsequently. One more hallmark of genius one finds in the paper [11] entitled “Sur les multiplicateurs des s´eriesde Fourier.” There are many generalizations of its results because of their important applications. This work was the last of eight papers that Marcinkiewicz published in the Studia Mathematica; the first three he submitted during his stay in Lw´ow, and they appeared in 1936. Below, the above-mentioned results of Marcinkiewicz are outlined in their historical context together with some further developments. One can find a detailed presenta- tion of all these results in the excellent textbook [18] based on lectures of the eminent analyst Elias Stein, who made a considerable contribution to further development of ideas proposed by Marcinkiewicz. Figure 2. J ´ozef Marcinkiewicz. Marcinkiewicz Interpolation Theorem and Marcinkiewicz Spaces himself as a patriot of his homeland, which is easily ex- There are two pillars of the interpolation theory: the clas- plainable by the fact that he was just eight years old (very sical Riesz–Thorin and Marcinkiewicz theorems. Each of sensitive age in forming a personality) when the indepen- these serves as the basis for two essentially different ap- dence of Poland was restored. proaches to interpolation of operators known as the com- Contribution of Marcinkiewicz to Mathematics plex and real methods. The term “interpolation of opera- tors” was, presumably, coined by Marcinkiewicz in 1939, Marcinkiewicz was a prolific author, as demonstrated by because Riesz and Thorin, who published their results in the almost five dozen papers he wrote in just seven years 1926 and 1938, respectively, referred to their assertions as (1933–1939); see Collected Papers [13, pp. 31–33]. He was “convexity theorems.” open to collaboration; indeed, more than one third of his It is worth emphasizing again that a characteristic fea- papers (nineteen, to be exact) were written with five coau- ture of Marcinkiewicz’s work was applying real methods thors, of which the lion’s share belongs to his supervisor to problems that other authors treated with the help of Zygmund. complex analysis. It was mentioned above that in his Marcinkiewicz is known, primarily, as an outstanding paper [10] published in 1938, Marcinkiewicz introduced analyst, whose best results deal with various aspects of real the function 휇 without using complex variables but so analysis, in particular, theory of series (trigonometric and that it is analogous to the Littlewood–Paley function 푔, others), inequalities, and approximation theory. He also whose definition involves these variables. In the same year, published several papers concerning complex and func- 1938, Thorin published his extension of the Riesz con- tional analysis and probability theory. In the extensive pa- vexity theorem, which exemplifies the approach based on per [9] dedicated to the centenary of Marcinkiewicz’s birth, complex variables. Possibly this stimulated Marcinkiewicz one finds a detailed survey of all his results. to seek an analogous result with proof relying on real anal- This survey begins with the description of five topics ysis. Anyway, Marcinkiewicz found his interpolation the- concerning functional analysis ([9, pp. 153–175]). No orem and announced it in [12]; concurrently, a letter was doubt, the first two of them—the Marcinkiewicz interpo- sent to Zygmund that contained the proof concerning a lation theorem and Marcinkiewicz spaces—are hallmarks

692 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 particular case. Ten years after World War II, Zygmund re- Marcinkiewicz spaces. Another crucial step, made by constructed the general proof and published it in 1956; Marcinkiewicz in [12], was the introduction of the weak for this reason the theorem is sometimes referred to as the 퐿푝 spaces playing the essential role in his general interpola- Marcinkiewicz–Zygmund interpolation theorem. tion theorem. They are now called the Marcinkiewicz spaces An excellent introduction to the interpolation theory and usually denoted 퐿푝,∞. one finds in the book [1] based on the works of Jaak Pee- To give an idea of these spaces, let us consider a measure tre (he passed away on 1 April 2019 at age eighty-three), space (푈, Σ, 푚) over real scalars with a nonnegative mea- whose contribution to this theory cannot be overestimated. sure 푚 (just to be specific). For a real-valued 푓, which is In collaboration with Jacques-Louis Lions, he introduced finite almost everywhere and 푚-measurable, we introduce the “real method interpolation spaces” (see their funda- its distribution function mental article [8]), which can be considered as “descen- dants” of the Marcinkiewicz interpolation theorem. 푚({푥 ∶ |푓(푥)| > 휆}), 휆 ∈ (0, ∞) An important fact of Peetre’s biography is that his life and put was severely changed during World War II (another re- 1/푝 minder about that terrible time). With his parents, Jaak |푓|푝,∞ = sup 휆[푚({푥 ∶ |푓(푥)| > 휆})] for 푝 ∈ [1, ∞). escaped from Estonia in September 1944 just two days be- 휆>0 fore his home town of Pärnu was destroyed in an air raid 푝,∞ 푝 푝,∞ Then 퐿 = {푓 ∶ |푓|푝,∞ < ∞}, and it is clear that 퐿 ⊂ 퐿 of the Red Army. He was only ten years old when his fam- for 푝 ∈ [1, ∞), because |푓|푝,∞ ≤ ‖푓‖푝 in view of Cheby- ily settled in Lund (Sweden), where he spent most of his shev’s inequality. The Marcinkiewicz space for 푝 = ∞ is life. But let us turn to mathematics again. 퐿∞ by definition. The Marcinkiewicz interpolation theorem for operators It occurs that |푓|푝,∞ is not a norm for 푝 ∈ [1, ∞), but a 푝 푛 in 퐿 (ℝ ). We begin with this simple result because it has quasi-norm because numerous applications, being valid for subadditive opera- 푝 푛 tors mapping the Lebesgue spaces 퐿 (ℝ ) with 푝 ≥ 1 into |푓 + 푔|푝,∞ ≤ 2(|푓|푝,∞ + |푔|푝,∞) themselves (see, e.g., [18, ch. 1, sect. 4]). We recall that an (see, e.g., [1, p. 7]). However, it is possible to endow 퐿푝,∞, operator 푇 ∶ 퐿푝 → 퐿푝 is subadditive if 푝 ∈ (1, ∞), with a norm ‖⋅‖푝,∞, converting it into a Banach |푇(푓1 + 푓2)(푥)| = |푇(푓1)(푥)| + |푇(푓2)(푥)| for every 푓1, 푓2 . space. Moreover, the inequality

Furthermore, 푇 is of weak type (푟, 푟) if the inequality −1 |푓|푝,∞ ≤ ‖푓‖푝,∞ ≤ 푝(푝 − 1) |푓|푝,∞ 푟 푟 훼 mes{푥 ∶ |푇(푓)(푥)| > 훼} ≤ 퐴푟‖푓‖푟 holds for all 푓 ∈ 퐿푝,∞. It is worth mentioning that 퐿푝,∞ holds for all 훼 > 0 and all 푓 ∈ 퐿푟 with 퐴 independent of 푟 belongs (as a limiting case) to the class of Lorentz spaces 훼 and 푓. Here, mes{… } denotes the Lebesgue measure of 퐿푝,푞, 푞 ∈ [1, ∞] (see, e.g., [1, sect. 1.6] and references cited the corresponding set, and in this book). 1/푝 푝,∞ 푝 Another generalization of 퐿 , known as the Marcinkie- ‖푓‖푝 = [ ∫ |푓(푥)| d푥] ℝ푛 wicz space 푀휑, is defined with the help of a nonnegative, is the norm in 퐿푝(ℝ푛). Now, we are in a position to formu- concave function 휑 ∈ 퐶[0, ∞). This con- late the following. sists of all (equivalence classes of) measurable functions for which the norm Theorem 1. Let 1 ≤ 푟1 < 푟2 < ∞, and let 푇 be a subadditive 푡 operator acting simultaneously in 퐿푟푖 (ℝ푛), 푖 = 1, 2. If it is of 1 ‖푓‖ = sup ∫ 푓∗(푠) d푠 (푟, 푟) 푖 = 1, 2 푝 ∈ (푟 , 푟 ) 휑 weak type 푖 푖 for , then for every 1 2 the 푡>0 휑(푡) 0 푝 푛 inequality ‖푇(푓)‖푝 ≤ 퐵‖푓‖푝 holds for all 푓 ∈ 퐿 (ℝ ) with 퐵 is finite. Here 푓∗ denotes the nonincreasing rearrangement depending only on 퐴푟 , 퐴푟 , 푟1, 푟2, and 푝. 1 2 of 푓, i.e., When 퐵 is independent of 푓 in the last inequality, the operator 푇 is of strong type (푝, 푝); it is clear that 푇 is also of 푓∗(푠) = inf{휆 ∶ 푚({푥 ∶ |푓(푥)| > 휆}) ≤ 푠} for 푠 ≥ 0, 휆>0 weak type (푝, 푝) in this case. In the letter to Zygmund mentioned above, Marcinkie- and so is nonnegative and right-continuous. Moreover, its ∗ wicz included a proof of this theorem for the case 푟1 = 1 distribution function 푚({푥 ∶ |푓 (푥)| > 휆}) coincides with 1−1/푝 and 푟2 = 2. Presumably, it was rather simple; indeed, even that of 푓. If 휑(푡) = 푡 , then the corresponding Marcin- 푝,∞ 1 when 푟2 < ∞ is arbitrary, the proof is less than two pages kiewicz space is 퐿 , whereas 휑(푡) ≡ 1 and 휑(푡) = 푡 give 퐿 long in [18, ch. 1, sect. 4]. and 퐿∞, respectively.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 693 The Marcinkiewicz interpolation theorem for bounded See Vol. II, ch. XVI, sects. 2 and 3, where, in particular, linear operators. This kind of continuous operator is usu- it is demonstrated that 퐹, originally defined on a dense ally considered as mapping one normed space to another set in 퐿푝, 푝 ∈ [1, 2], is extensible to the whole space as a ′ one, in which case the operator’s norm is an important bounded operator 퐹 ∶ 퐿푝 → 퐿푝 , 푝′ = 푝/(푝 − 1), and so the ′ characteristic. However, the latter can be readily general- integral converges in 퐿푝 . To prove this assertion and its 푝 푝,∞ ized for a mapping of 퐿 to 퐿 . Indeed, if |푇푓|푝,∞ ≤ 푛-dimensional analogue one can use Theorem 2. Indeed, 1 ∞ 퐶‖푓‖푝, then it is natural to introduce the norm (or quasi- 퐹 ∶ 퐿 → 퐿 is bounded (this is straightforward to see), norm) of 푇 as the infimum over all possible values of 퐶. and by Plancherel’s theorem 퐹 is bounded on 퐿2, and so Now we are in a position to formulate the following. this theorem is applicable. On the other hand, the Riesz– Thorin theorem, which has no restriction 푝 ≤ 푞, yields a Theorem 2. Let 푝0, 푞0, 푝1, 푞1 ∈ [1, ∞] satisfy the inequali- −1 푝 ≤ 푞 푝 ≤ 푞 푞 ≠ 푞 푝, 푞 ∈ [1, ∞] more complete result valid for the inverse transform 퐹 as ties 0 0, 1 1, and 0 1, and let be 푝′ 푝 such that 푝 ≤ 푞 and the equalities well. The latter operator acting from 퐿 to 퐿 is bounded; here 푝′ ∈ [2, ∞), and so 푝 = 푝′/(푝′ − 1) ∈ (1, 2]. 1 1 − 휃 1 1 1 − 휃 1 = + and = + (2) In studies of conjugate Fourier series, the singular 푝 푝 푝 푞 푞 푞 0 1 0 1 integral operator (the periodic Hilbert transform) hold for some 휃 ∈ (0, 1). If 푇 is a linear operator that maps 퐿푝0 푞0,∞ 푝1 1 푡 into 퐿 and its norm is 푁0 and simultaneously 푇 ∶ 퐿 → 퐻(푓)(푠) = lim ∫ 푓(푠 − 푡) cot d푡 푞1,∞ 푝 푞 2휋 휖→0 2 퐿 has 푁1 as its norm, then 푇 maps 퐿 into 퐿 and its norm 휖≤|푡|≤휋 푁 satisfies the estimate plays an important role. Indeed, by linearity it is sufficient 1−휃 휃 2 푁 ≤ 퐶푁0 푁1 , (1) to define 퐻 on a basis in 퐿 (−휋, 휋), and the relations with 퐶 depending on 푝0, 푞0, 푝1, 푞1, and 휃. 퐻(cos 푛푡) = sin 푛푠 for 푛 ≥ 0, 퐻(sin 푛푡) = − cos 푛푠 for 푛 ≥ 1 The convexity inequality (1) is a characteristic feature show that it expresses passing from a trigonometric series of the interpolation theory. The general form of this theo- to its conjugate. Moreover, these formulae show that 퐻 is rem (it is valid for quasi-additive operators, whose special bounded on 퐿2(−휋, 휋) and its norm is equal to one. case are subadditive ones described prior to Theorem 1) is In the mid-1920s, Marcel Riesz obtained his celebrated proved in [23, ch. XII, sect. 4]. In particular, it is shown result about this operator; first, he announced it in a brief that one can take note in the Comptes rendus de l’Acad´emiedes Sciences Paris, 푞 푞 1/푞 푝(1−휃)/푝0 푝휃/푝1 퐶 = 2 + 0 1 ; and three years later published his rather long proof that ( ) 1/푝 푝 |푞 − 푞0| |푞 − 푞1| 푝 퐻 is bounded on 퐿 (−휋, 휋) for 푝 ∈ (1, ∞); i.e., for every see [23, Vol. II, p. 114, formula (4.18)], where, unfortu- finite 푝 > 1 there exists 퐴푝 > 0 such that nately, the notation differs from that adopted here. Special ‖퐻(푓)‖푝 ≤ 퐴푝‖푓‖푝 for all 푓 ∈ 퐿푝(−휋, 휋). (2) cases of Theorem 2 and diagrams illustrating them can be found in [9, pp. 155–156]. It should be emphasized that However, (2) does not hold for 푝 = 1 and ∞; see [23, Vol. I, the restriction 푝 ≤ 푞 is essential; indeed, as early as 1964, ch. VII, sect. 2] for the corresponding examples and a proof R. A. Hunt [6] constructed an example demonstrating that of this inequality. Theorem 2 is not true without it. For a description of this There are several different proofs of this theorem; the example see, e.g., [1, pp. 16–17]. original proof of M. Riesz was reproduced in the first edi- It was Marcinkiewicz himself who proposed an exten- tion of Zygmund’s monograph [23], which appeared in sion of his interpolation theorem to other function spaces; 1935. In the second edition published in 1959, this proof namely, the so-called diagonal case (when 푝0 = 푞0 and was replaced by that of Calder´onobtained in 1950. Let us 푝1 = 푞1) of his theorem is formulated for Orlicz spaces outline another proof based on the Marcinkiewicz inter- in [12]. References to papers containing further results on polation theorem analogous to Theorem 1 but involving interpolation in these and other spaces (e.g., Lorentz and 퐿푝-spaces on (−휋, 휋) instead of the spaces on ℝ. 푀휑) can be found in [1, pp. 128–129] and [9, pp. 163– First we notice that it is sufficient to prove (2) only for 166]. 푝 ∈ (1, 2]. Indeed, assuming that this is established, then Applications of the interpolation theorems. (1) In his for 푓 ∈ 퐿푝 and 푔 ∈ 퐿푝′ we have monograph [23], Zygmund gave a detailed study of the 휋 one-dimensional Fourier transform ∫ [퐻(푓)(푠)] 푔(−푠) d푠 ≤ 퐴푝‖푓‖푝 ‖푔‖푝′ 1 −휋 ∫ 퐹(푓)(휉) = 푓(푥) exp{−푖 휉푥} d푥, 휉 ∈ ℝ. ′ √2휋 ℝ by the Hölder inequality (as above 푝 = 푝/(푝 − 1), and so

694 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 푝′ ≥ 2 when 푝 ≤ 2). Since for which the norm ∞ 1/푝 푝 d푡 ‖푎‖ = {∫ [푡−휃 퐾(푡, 푎)] } 휋 휋 휃,푝 0 푡 ∫ [퐻(푓)(푠)] 푔(−푠) d푠 = ∫ 푓(−푠) [퐻(푔)(푠)] d푠 , −휋 −휋 is finite. Here 퐾(푡, 푎) is defined on 퐴0 + 퐴1 for 푡 ∈ (0, ∞) by

−1 inf {‖푎0‖퐴0 +푡‖푎1‖퐴1 ∶ 푎0 ∈ 퐴0, 푎1 ∈ 퐴1 and 푎0 +푎1 = 푎}. the inequality ‖퐻(푔)‖푝′ ≤ 퐴푝 ‖푔‖푝′ is a consequence of 푎0,푎1 the assertion converse to the Hölder inequality. 퐾 푝 = ∞ 2 This -functional was introduced by Peetre. If , then It was mentioned above that 퐻 is bounded in 퐿 . Hence, −휃 the expression sup푡>0{푡 퐾(푡, 푎)} gives the norm ‖푎‖휃,∞ in order to apply Theorem 1 for 푝 ∈ (1, 2], it is sufficient when finite. to show that this operator is of weak type (1, 1), and this is Every 퐴휃,푝 is an intermediate space with respect to the an essential part of Calder´on’s proof; see [23, Vol. I, ch. IV, pair (퐴0, 퐴1), i.e., sect. 3]. Moreover, an improvement of the latter proof al- lowed S. K. Pichorides [16] to obtain the least value of the 퐴0 ∩ 퐴1 ⊂ 퐴휃,푝 ⊂ 퐴0 + 퐴1. constant 퐴 in (2). It occurs that 퐴 = tan 휋/(2푝) and 푝 푝 Moreover, if 퐴0 ⊂ 퐴1, then cot 휋/(2푝) is this value for 푝 ∈ (1, 2] and 푝 ≥ 2, respec- 퐴 ⊂ 퐴 ⊂ 퐴 ⊂ 퐴 , tively. 0 휃0,푝0 휃1,푝1 1

There are many other applications of interpolation the- provided either 휃0 > 휃1 or 휃0 = 휃1 and 푝0 ≤ 푝1. For any orems in analysis; see, e.g., [1, ch. 1], [23, ch. XII], and 푝, it is convenient to put 퐴0,푝 = 퐴0 and 퐴1,푝 = 퐴1. Now references cited in these books. we are in a position to explain what the interpolation of an Further development of interpolation theorems. Results operator is in terms of the family {퐴휃,푝} and another family constituting the interpolation space theory were obtained of spaces {퐵휃,푝} constructed by using some Banach spaces in the early 1960s and are classical now. This theory 퐵0 and 퐵1 in the same way as 퐴0 and 퐴1. was created in the works of Nachman Aronszajn, Alberto Let 푇 ∶ 퐴0 + 퐴1 → 퐵0 + 퐵1 be a linear operator such Calder´on,Mischa Cotlar, Emilio Gagliardo, Selim Grig- that its norm as the operator mapping 퐴0 (퐴1) to 퐵0 (퐵1) orievich Krein, Jacques-Louis Lions, and Jaak Peetre, to is equal to 푀0 (푀1). Then the operator 푇 ∶ 퐴휃,푝 → list a few. We leave aside several versions of complex in- 퐵휃,푝 is also bounded, and its norm is less than or equal 1−휃 휃 terpolation spaces developed from the Riesz–Thorin the- to 푀0 푀1 . Along with the method based on the 퐾- orem (see, e.g., [1, ch. 4]) and concentrate on “espaces functional, there is an equivalent method (also developed de moyennes” introduced by Lions and Peetre in their by Peetre) involving the so-called 퐽-functional. Further de- celebrated article [8]. These “real method interpolation tails concerning this approach to interpolation theory can spaces,” usually denoted (퐴0, 퐴1)휃,푝, are often considered be found in [1, chs. 3 and 4]. as “descendants” of the Marcinkiewicz interpolation theo- rem. The Marcinkiewicz Function Prior to describing these spaces, it is worth mention- In the Annales de la Soci´et´ePolonaise de Math´ematique, vol- ing another germ of interpolation theory originating from ume 17 (1938), Marcinkiewicz published two short pa- Lw´ow. Problem 87 in the Scottish Book [14] posed by Ba- pers. Two remarkable integral operators were considered nach demonstrates his interest in nonlinear interpolation. in the first of these notes (see [10] and [13, pp. 444–451]); Presumably, it was formulated during Marcinkiewicz’s stay they and their numerous generalizations became indis- in Lw´ow. Indeed, he solved problems 83 and 106 in pensable tools in analysis. One of these operators is al- [14], which were posed before and after, respectively, Ba- ways called the “Marcinkiewicz integral”; see [23, ch. IV, nach’s problem on interpolation. A positive solution of sect. 2] for its definition and properties. In particular, it the latter problem (due to L. Maligranda) is presented in is used for investigation of the structure of a measurable [14, pp. 163–170]. set near an “almost arbitrary” point; see [18, sects. 2.3 and Let us turn to defining the family of spaces {(퐴0, 퐴1)휃,푝} 2.4], whereas further references to papers describing some involved in the real interpolation method; here 휃 ∈ (0, 1) of its generalizations can be found in the monographs [18] and 푝 ∈ [1, ∞]. In what follows, we write 퐴휃,푝 instead and [23]. The second operator is usually referred to as the of (퐴0, 퐴1)휃,푝 for the sake of brevity. Let 퐴0 and 퐴1 be “Marcinkiewicz function” (see, e.g., [9, pp. 192–194]), but two Banach spaces, both continuously embedded in some it also appears as the “Marcinkiewicz integral.” Presum- (larger) Hausdorff topological vector space. Then for a pair ably, the mess with names began as early as 1944, when (휃, 푝) the space 퐴휃,푝 with 푝 < ∞ consists of all 푎 ∈ 퐴0 + 퐴1 Zygmund published the extensive article [22], section 2 of

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 695 which was titled “On an Integral of Marcinkiewicz.” In Furthermore, Marcinkiewicz conjectured that for 푝 > 1 fact, this 14-page section is devoted to a detailed study of the inequalities the Marcinkiewicz function 휇, whose properties were just outlined by Marcinkiewicz himself in [10]. It is not clear 퐴푝‖푓‖푝 ≤ ‖휇‖푝 ≤ 퐵푝‖푓‖푝 (3) whether Zygmund had already received information about hold, where again 푓 must have the zero mean value in the Marcinkiewicz’s death when he decided to present in de- second inequality. Moreover, he foresaw that it would not tail the results from [10] (the discovery of mass graves in be easy to prove these inequalities; indeed, the proof given the Katy ´nforest was announced by the Nazi government by Zygmund in his article [22] is more than 11 pages long. in April 1943). The first step towards generalization of the Marcinkie- Zygmund begins his presentation with a definition of wicz function was made by Daniel Waterman; his paper the Littlewood–Paley function 푔(휃; 푓), which is a nonlin- [21] was published seven (!) years after presentation of the ear operator applied to an integrable, 2휋-periodic 푓. The work to the AMS. However, its abstract appeared in the purpose of introducing 푔(휃; 푓) was to provide a characteri- 푝 Proceedings of the International Congress of Mathematicians zation of the 퐿 -norm ‖푓‖푝 in terms of the Poisson integral held in 1954 in Amsterdam. Waterman considered the 휇- of 푓. After describing some properties of 푔(휃), Zygmund function notes: ∞ [퐹(휏 + 푡) + 퐹(휏 − 푡) − 2퐹(휏)]2 1/2 It is natural to look for functions analogous to 푔(휃) 휇(휏; 푓) = { ∫ d푡} , 푡3 but defined without entering the interior of the 0 unit circle. where 휏 ∈ (−∞, ∞) and 퐹 is a primitive of 푓 ∈ 퐿푝(−∞, ∞), After a reference to [10], Zygmund continues: 푝 > 1. His proof of inequalities (3) for 휇(휏; 푓) heavily re- Marcinkiewicz had the right idea of introducing lies on the M. Riesz theorem about conjugate functions on 1 the function ℝ (see [21, p. 130] for the formulation), and its proof in- volves the Marcinkiewicz interpolation theorem described 휇(휃) = 휇(휃; 푓) above. 휋 [퐹(휃 + 푡) + 퐹(휃 − 푡) − 2퐹(휃)]2 1/2 Another consequence of inequalities (3) for 휇(휏; 푓) is a 1,푝 = { ∫ 3 d푡} characterization of the Sobolev space 푊 (ℝ), 푝 ∈ (1, ∞). 0 푡 휋 Indeed, putting 퐹(휃 + 푡) + 퐹(휃 − 푡) − 2퐹(휃) 2 1/2 = { ∫ 푡[ ] d푡} ∞ 푡2 [푓(휏 + 푡) + 푓(휏 − 푡) − 2푓(휏)]2 1/2 0 M(휏; 푓) = { ∫ d푡} 푡3 where 퐹(휃) is the integral of 푓, 0 1,푝 ′ 휃 for 푓 ∈ 푊 (ℝ), we have that M(휏; 푓) = 휇(휏; 푓 ). Then (3) 퐹(휃) = 퐶 + ∫ 푓(푢) d푢 . can be written as 0 퐴 ‖푓′‖ ≤ ‖M(⋅; 푓)‖ ≤ 퐵 ‖푓′‖ , More generally, he considers the functions 푝 푝 푝 푝 푝

휋 which implies the following assertion. Let 푝 ∈ (1, ∞). |퐹(휃 + 푡) + 퐹(휃 − 푡) − 2퐹(휃)|푟 1/푟 휇 (휃) = { ∫ d푡} Then 푓 ∈ 푊 1,푝(ℝ) if and only if 푓 ∈ 퐿푝(ℝ) and M(⋅; 푓) ∈ 푟 푡푟+1 0 퐿푝(ℝ). 휋 푟 |퐹(휃 + 푡) + 퐹(휃 − 푡) − 2퐹(휃)| 1/푟 Stein extended these results to higher dimensions in the = { ∫ 푡푟−1 | | d푡} , | 2 | late 1950s and early 1960s (it is worth mentioning that 0 푡 휇 is referred to as the Marcinkiewicz integral in his paper so that 휇2(휃) = 휇(휃). He proves the following facts [17]). For this purpose he applied the real-variable tech- which are clearly analogues of the corresponding nique used in the generalization of the Hilbert transform properties of 푔(휃). ∞ 푓(푥 + 푡) − 푓(푥 − 푡) These facts are the estimates P.V. ∫ d푡 0 푡 ‖휇푞‖푞 ≤ 퐴푞‖푓‖푞 and ‖푓‖푝 ≤ 퐴푝‖휇푝‖푝 to higher dimensions. Indeed, this can be written as valid for 푞 ≥ 2 and 1 < 푝 ≤ 2, respectively, where 푓 ∞ has the zero mean value in the second inequality and the 퐹(푥 + 푡) + 퐹(푥 − 푡) − 2퐹(푥) assertion: For every 푝 ∈ (1, 2] there exists a continuous, 2휋- ∫ 2 d푡, 0 푡 periodic function 푓 such that 휇푝(휃; 푓) = ∞ for almost every 휃. which resembles the expression for 휇(휏; 푓), and so Stein, in

696 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 his own words, was Stein begins his investigation of properties of this func- guided by the techniques used by A. P. Calder´on tion by proving that ‖휇(⋅; 푓)‖2 ≤ 퐴‖푓‖2, where 퐴 is and A. Zygmund [2] in their study of the 푛-di- independent of 푓, and his proof involving Plancherel’s mensional generalizations of the Hilbert trans- theorem is not elementary at all. Even less elementary form; connected with this are some earlier ideas is his proof that 휇(⋅; 푓) is of weak type (1, 1). Then of Marcinkiewicz. the Marcinkiewicz interpolation theorem (see Theorem 1 above) implies that ‖휇(⋅; 푓)‖ ≤ 퐴‖푓‖ for 푝 ∈ (1, 2] pro- The definition of singular integral given in [2], to which 푝 푝 vided 푓 ∈ 퐿푝(ℝ푛). For all 푝 ∈ (1, ∞) this inequality is Stein refers, involves a function Ω(푥) defined for 푥 ∈ ℝ푛 proved in [17] with assumptions (i)–(iii) changed to the and assumed (i) to be homogeneous of degree zero, i.e., following ones: Ω(푥′) is absolutely integrable on the unit to depend only on 푥′ = 푥/|푥|; (ii) to satisfy the Hölder sphere and is odd there, i.e., Ω(−푥′) = −Ω(푥′). A few years condition with exponent 훼 ∈ (0, 1]; and (iii) to have the later, A. Benedek, A. P. Calder´on,and R. Panzone demon- zero mean value over the unit sphere in ℝ푛. Then strated that for a 퐶1-function Ω, condition (iii) implies the Ω(푦′) 푆(푓)(푥) = lim ∫ 푓(푥 − 푦) d푦 last inequality for all 푝 ∈ (1, ∞). 휖→0 푛 |푦|>휖 |푦| In another note, Stein obtained the following general- ization of the one-dimensional result. exists almost everywhere provided 푓 ∈ 퐿푝(ℝ푛), 푝 ∈ [1, ∞). Let 푝 ∈ (2푛/(푛 + 2), ∞) and 푛 ≥ 2. Then 푓 belongs to the Furthermore, this singular integral operator is bounded in 1,푝 푛 푝 푛 푝 푛 Sobolev space 푊 (ℝ ) if and only if 푓 ∈ 퐿 (ℝ ) and 퐿 (ℝ ) for 푝 > 1; i.e., the inequality ‖푆(푓)‖푝 ≤ 퐴푝‖푓‖푝 2 1/2 holds with 퐴푝 independent of 푓. [푓(⋅ + 푦) + 푓(⋅ − 푦) − 2푓(⋅)] 푝 푛 { ∫ 푛+2 d푦} ∈ 퐿 (ℝ ). Moreover, in the section dealing with background facts, ℝ푛 |푦| Stein notes that 휇 is a nonlinear operator and writes (see For 푛 > 2 this does not cover 푝 ∈ (1, 2푛/(푛 + 2)] and so is [17, p. 433]): weaker than the assertion formulated above for 푛 = 1. An “interpolation” theorem of Marcinkiewicz is In the survey article [9, pp. 193–194], one finds a list of very useful in this connection. papers concerning the Marcinkiewicz function. In particu- In quoting the result of Marcinkiewicz, [...] we lar, further properties of 휇 were considered by A. Torchin- shall not aim at generality. For the sake of sim- sky and S. Wang [19] in 1990, whereas T. Walsh [20] pro- plicity we shall limit ourselves to the special case posed a modification of the definition (4), (5) in 1972. that is needed. After that the required form of the interpolation theorem Multipliers of Fourier Series and Integrals (see Theorem 1 above) is formulated and used later in the During his stay in Lw´ow, Marcinkiewicz collaborated with paper, thus adding one of the first items in the now long Stefan Kaczmarz and Juliusz Schauder,2 who had awak- list of its applications. Since the term interpolation was ened his interest in multipliers of orthogonal series. Studies novel, quotation marks are used by Stein in the quoted in this area of analysis were initiated by in piece. Indeed, Zygmund’s proof of the Marcinkiewicz the- the 1920s; in its general form, the problem of multipliers orem had appeared in 1956, just two years earlier than is as follows. Let 퐵1 be a Banach space with a Schauder ∞ Stein’s article. basis {푔푛}푛=1. The (linear) operator 푇 is called a multiplier ∞ Stein begins his generalization of the Marcinkiewicz when there is a sequence {푚푛}푛=1 of scalars of this space function 휇(휏; 푓) with the case when 푓 ∈ 퐿푝(ℝ푛), 푝 ∈ [1, 2]. and 푇 acts as follows: Realizing the analogy described above, he puts ∞ ∞ 퐵 ∋ 푓 = ∑ 푐 푔 → 푇푓 ∼ ∑ 푚 푐 푔 . Ω(푦′) 1 푛 푛 푛 푛 푛 퐹 (푥) = ∫ 푓(푥 − 푦) d푦 , 푥 ∈ ℝ푛, (4) 푛=1 푛=1 푡 |푦|푛−1 |푦|≤푡 Here ∼ means that the second sum assigned as 푇푓 can be- where Ω satisfies conditions (i)–(iii), and notes that if 푛 = long to the same space 퐵1 or be an element of another Ba- 1 and Ω(푦) = sign 푦, then nach space 퐵2; this depends on properties of the sequence. 푥 Multipliers of Fourier series are of paramount interest, and

퐹푡(푥) = 퐹(푥+푡)+퐹(푥−푡)−2퐹(푥) with 퐹(푥) =∫ 푓(푠) d푠. this was the topic of the remarkable paper [11] published 0 by Marcinkiewicz in 1939. Therefore, it is natural to define the 푛-dimensional Marcin- kiewicz function as follows: 2Both perished in World War II. Being in the reserve, Kaczmarz was drafted and killed during the first week of war; the circumstances of his death are un- ∞ 2 1/2 [퐹푡(푥)] clear. Schauder was in hiding in occupied Lw´ow, and the Gestapo killed him in 휇(푥; 푓) = { ∫ 3 d푡} . (5) 0 푡 1943 while he was trying to escape arrest.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 697 Not long before Marcinkiewicz’s visit to Lw´ow started, The trigonometric form was used by Marcinkiewicz for Kaczmarz investigated some properties of multipliers in double Fourier series as well, and his sufficient condi- 푝 the function spaces (mainly 퐿 (0, 1) and 퐶[0, 1]) under tions on bounded real multipliers {휆푚푛} look rather awk- ∞ rather general assumptions about the system {푔푛}푛=1. Fur- ward. Now, the restrictions on {휆푚푛} ⊂ ℂ are usually ex- ther results about multiplier operators were obtained in pressed in a rather condensed form by using the so-called the joint paper [7] of Kaczmarz and Marcinkiewicz. It dyadic intervals; see, e.g., [18, sect. 5.1]. Applying these was submitted to the Studia Mathematica in June 1937; conditions to multipliers acting on the expansion i.e., their collaboration lasted for another year after ∞

Marcinkiewicz left Lw´ow. This paper has the same title as ∑ 푐푚푛 exp 푖{푚푥 + 푛푦} 푝 that of Kaczmarz and concerns the case when 퐿 (0, 1) with 푚,푛=−∞ 푝 ≠ ∞ is mapped to 퐿푞(0, 1), 푞 ∈ [1, ∞]; it occurs that the of 푓 ∈ 퐿푝((0, 2휋)2), 푝 ∈ (1, ∞), one obtains an updated case 푞 = ∞ is the simplest one. In this paper, it is assumed formulation of the multiplier theorem; see, e.g., [9, p. 201]. that every function 푔 is bounded, whereas the sequence 푛 A simple corollary derived by Marcinkiewicz from this {푔 }∞ is closed in 퐿1(0, 1). In each of four theorems that 푛 푛=1 theorem is as follows (see [11, p. 86]). The fractions differ by the ranges of 푝 and 푞 involved, certain conditions are imposed on {푚 }∞ , and these conditions are neces- 푚2 푛2 |푚푛| 푛 푛=1 , , (7) sary and sufficient for the sequence to define a multiplier 푚2 + 푛2 푚2 + 푛2 푚2 + 푛2 푝 푞 operator 푇 ∶ 퐿 → 퐿 . provide examples of multipliers in 퐿푝 for double Fourier After returning to Wilno, Marcinkiewicz kept on his series. The reason to include these examples was to an- studies of multipliers initiated in Lw´ow, and in May 1938, swer a question posed by Schauder, and this is specially he submitted (again to the Studia Mathematica) the semi- mentioned in a footnote. Moreover, after remarking that nal paper [11], in which the main results are presented in a his Theorem 2 admits an extension to multiple Fourier se- curious way. Namely, Theorems 1 and 2, concerning mul- ries, Marcinkiewicz added a straightforward generalization tipliers of Fourier series and double Fourier series, are for- of formulae (7) to higher dimensions, again referring to mulated in the reverse order. Presumably, the reason for Schauder’s question. This is evidence that the question was this is the importance of multiple Fourier series for appli- an important stimulus for Marcinkiewicz in his work. cations and generalizations. Let us formulate Theorem 1 A natural way to generalize Marcinkiewicz’s theorems is in a slightly updated form. to consider multipliers of Fourier integrals. Study of these 푝 Let 푓 ∈ 퐿 (0, 2휋), 푝 ∈ (1, ∞), be a real-valued function operators was initiated by Mikhlin in 1956; see note [15], and let its Fourier series be in which the first result of that kind was announced. Sev- ∞ eral years later, Mikhlin’s theorem was improved by Lars 푎0/2 + ∑ 퐴푛(푥), 푤ℎ푒푟푒 퐴푛(푥) = 푎푛 cos 푛푥 + 푏푛 sin 푛푥. Hörmander [5], and since then it has been widely used for 푛=1 various purposes. To formulate this theorem we need the ∞ If a bounded sequence {휆푛}푛=1 ⊂ ℝ is such that 푛-dimensional Fourier transform

2푘+1 퐹(푓)(휉) = (2휋)−푛/2 ∫ 푓(푥) exp{−푖 휉 ⋅ 푥} d푥, 휉 ∈ ℝ푛, 푛 ∑ |휆푛 − 휆푛+1| ≤ 푀 for all 푘 = 0, 1, 2, … , (6) ℝ 푘 푛=2 defined for 푓 ∈ 퐿2(ℝ푛) ∩ 퐿푝(ℝ푛), 푝 ∈ (1, ∞). It is clear where 푀 is a constant independent of 푘, then the mapping 푓 ↦ that any bounded measurable function Λ on ℝ푛 defines ∞ 푝 the mapping ∑푛=1 휆푛 퐴푛 is a bounded operator in 퐿 (0, 2휋). It is well known that for 푝 = 2 this theorem is true with 푇 (푓)(푥) = 퐹−1[Λ(휉)퐹(푓)(휉)](푥) , 푥 ∈ ℝ푛, condition (6) omitted, but this is not mentioned in [11]. Λ ∞ 2 푛 푝 푛 The assumptions that 푓 is real-valued and {휆푛}푛=1 ⊂ ℝ such that 푇Λ(푓) ∈ 퐿 (ℝ ). If 푇Λ(푓) is also in 퐿 (ℝ ) and 푇Λ were not stated in [11] explicitly but used in the proof. is a bounded operator, i.e., This was noted by Solomon Grigorievich Mikhlin [15], ‖푇 (푓)‖ ≤ 퐵 ‖푓‖ for all 푓 ∈ 퐿푝(ℝ푛) (8) who extended this theorem to complex-valued multipliers Λ 푝 푝,푛 푝 and functions. Also, he used the exponential form of the with 퐵푝 independent of 푓, then Λ is called a multiplier Fourier expansion: for 퐿푝. 2 ∞ The description of all multipliers for 퐿 is known as well for 퐿1 and 퐿∞ (it is the same for these two spaces); see [18, 푓(푥) = ∑푐푛 exp 푖푛푥 . 푛=−∞ pp. 94–95]. However, the question about characterization of the whole class of multipliers for other values of 푝 is far

698 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 from resolved. The following assertion gives widely used [4] Loukas Grafakos and Lenka Slav´ıkov´a, The Marcinkiewicz sufficient conditions. multiplier theorem revisited, Arch. Math. (Basel) 112 (2019), no. 2, 191–203, DOI 10.1007/s00013-018-1269-7. Theorem (Mikhlin, Hörmander). Let Λ be a function of the 푘 푛 MR3908837 퐶 -class in the complement of the origin of ℝ . Here 푘 is the [5] Lars Hörmander, Estimates for translation invariant oper- least integer greater than 푛/2. If there exists 퐵 > 0 such that ators in 퐿푝 spaces, Acta Math. 104 (1960), 93–140, DOI ℓ 10.1007/BF02547187. MR121655 ℓ | 휕 Λ(휉) | |휉| | | ≤ 퐵 , 1 ≤ 푗1 < 푗2 < ⋯ < 푗ℓ ≤ 푛 , [6] Richard A. Hunt, An extension of the Marcinkiewicz interpo- |휕휉 휕휉 ⋯ 휕휉 | 푗1 푗2 푗ℓ lation theorem to Lorentz spaces, Bull. Amer. Math. Soc. 70 for all 휉 ∈ ℝ푛, ℓ = 0, … , 푘, and all possible ℓ-tuples, then (1964), 803–807, DOI 10.1090/S0002-9904-1964-11242- inequality (8) holds; i.e., Λ is a multiplier for 퐿푝. 8. MR169037 [7] S. Kaczmarz and J. Marcinkiewicz, Sur les multiplicateurs In various versions of this theorem, different assump- des s´eriesorthogonales, Studia Math. 7 (1938), 73–81. Also tions are imposed on the differentiability of Λ. In par- [13, pp. 389–396]. ticular, Hörmander [5, pp. 120–121] replaced the point- [8] J.-L. Lions and J. Peetre, Sur une classe d’espaces wise inequality for weighted derivatives of Λ by a weaker d’interpolation (French), Inst. Hautes Etudes´ Sci. Publ. one involving certain integrals (see also [18, p. 96]). Re- Math. 19 (1964), 5–68. MR165343 cently, Loukas Grafakos and Lenka Slav´ikov´a[4] obtained [9] Lech Maligranda, J´ozef Marcinkiewicz (1910–1940)—on new sufficient conditions for Λ in the multiplier theorem, the centenary of his birth, Marcinkiewicz centenary vol- thus improving Hörmander’s result. Their conditions are ume, Banach Center Publ., vol. 95, Polish Acad. Sci. Inst. optimal in a certain sense explicitly described in [4]. Math., Warsaw, 2011, pp. 133–234, DOI 10.4064/bc95-0- 10. MR2918095 Corollary. Every function that is smooth everywhere except at [10] J. Marcinkiewicz, Sur quelques int´egrales du type de the origin and is homogeneous of degree zero is a Fourier multi- Dini, Ann. Soc. Polon. Math. 17 (1938), 42–50. Also plier for 퐿푝. [13, pp. 444–451]. [11] J. Marcinkiewicz, Sur les multiplicateurs des s´eries de Its immediate consequence is the Schauder estimate Fourier, Studia Math. 8 (1939), no. 2, 78–91. Also ‖ 휕2푢 ‖ [13, pp. 501–512]. ‖ ‖ ≤ 퐶푝,푛‖Δ 푢‖푝 , 1 ≤ 푗1, 푗2 ≤ 푛 , [12] J. Marcinkiewicz, Sur l’interpolation d’op´erations, C. R. ‖휕푥푗1 휕푥푗2 ‖ 푝 Acad. Sci. Paris 208 (1939), 1272–1273. Also [13, pp. 539– valid for 푢 belonging to the Schwartz space of rapidly de- 540]. caying infinitely differentiable functions. For this purpose [13] J´ozef Marcinkiewicz, Collected papers, edited by An- one has to use the equality toni Zygmund. With the collaboration of Stanislaw Lo- 2 jasiewicz, Julian Musielak, Kazimierz Urbanik, and An- 휕 푢 휉푗 휉푗 퐹 ( ) (휉) = 1 2 퐹(Δ 푢)(휉) , 1 ≤ 푗 , 푗 ≤ 푛 , toni Wiweger. Instytut Matematyczny Polskiej Akademii 2 1 2 휕푥푗1 휕푥푗2 |휉| Nauk, Pa ´nstwowe Wydawnictwo Naukowe, Warsaw, 1964. and the fact that the function 휉 휉 /|휉|2 is homogeneous MR0168434 푗1 푗2 [14] R. Daniel Mauldin (ed.), The Scottish Book: Mathematics of degree zero. from the Scottish Caf´e, Birkhäuser, Boston, Mass., 1981. In- cluding selected papers presented at the Scottish Book Con- ACKNOWLEDGMENTS. The author thanks Irina ference held at North Texas State University, Denton, Tex., Egorova for the photo of Marcinkiewicz’s plaque and May 1979. MR666400 Alex Eremenko, whose comments helped to improve [15] S. G. Mihlin, On the multipliers of Fourier integrals the original manuscript. (Russian), Dokl. Akad. Nauk SSSR (N.S.) 109 (1956), 701– 703. MR0080799 [16] S. K. Pichorides, On the best values of the constants in the References theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math. [1] Jöran Bergh and Jörgen Löfström, Interpolation spaces. 44 (1972), 165–179. (errata insert), DOI 10.4064/sm-44- An introduction, Grundlehren der Mathematischen Wis- 2-165-179. MR312140 senschaften, No. 223, Springer-Verlag, Berlin-New York, [17] E. M. Stein, On the functions of Littlewood-Paley, Lusin, and 1976. MR0482275 Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430– [2] A. P. Calderon and A. Zygmund, On the existence of cer- 466, DOI 10.2307/1993226. MR112932 tain singular integrals, Acta Math. 88 (1952), 85–139, DOI [18] Elias M. Stein, Singular integrals and differentiability 10.1007/BF02392130. MR52553 properties of functions, Princeton Mathematical Series, No. [3] Roman Duda, Pearls from a lost city: The Lvov school of math- 30, Princeton University Press, Princeton, N.J., 1970. ematics, History of Mathematics, vol. 40, American Mathe- MR0290095 matical Society, Providence, RI, 2014. Translated from the 2007 Polish original by Daniel Davies. MR3222779

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 699 [19] Alberto Torchinsky and Shi Lin Wang, A note on the Marcinkiewicz integral, Colloq. Math. 60/61 (1990), no. 1, 235–243, DOI 10.4064/cm-60-61-1-235-243. MR1096373 [20] T. Walsh, On the function of Marcinkiewicz, Studia Math. 44 (1972), 203–217. (errata insert), DOI 10.4064/sm-44- 3-203-217. MR315368 Why navigate your [21] Daniel Waterman, On an integral of Marcinkiewicz, Trans. Amer. Math. Soc. 91 (1959), 129–138, DOI graduate school 10.2307/1993149. MR103953 [22] A. Zygmund, On certain integrals, Trans. Amer. Math. Soc. experience alone? 55 (1944), 170–204, DOI 10.2307/1990189. MR9966 [23] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cam- bridge University Press, New York, 1959. MR0107776 [24] A. Zygmund, J´ozefMarcinkiewicz (Polish), Wiadom. Mat. (2) 4 (1960), 11–41. Also [13, pp. 1–33]. MR115885 “CROSSWORD! (or: Diversion as a vehicle “Solvitur for conversation on Ambulando” power and usage)”

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700 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 BOOK REVIEW American Mathematics 1890– 1913: Catching Up to Europe Reviewed by Deborah Kent

In 1994, the AMS published The The Emergence of the American Mathematical Research Emergence of the American Math- Community begins with an overview of American math- ematical Research Community, ematics during its first one hundred years. During this 1876–1900: J. J. Sylvester, Felix period, American mathematics evolved in the context of Klein, and E. H. Moore, written by general scientific structure building. Mathematical practi- and David Rowe. tioners in the United States worked alongside geologists, This now classic work laid care- physicists, botanists, and other self-identified scientists to fully documented groundwork organize societies, develop employment opportunities, and for subsequent study and artic- provide outlets for scientific publication. The notion of ulated a periodization for Amer- research in academic disciplines emerged as practitioners ican mathematics. By targeting looked to Europe as a model for scientific achievement. the years 1876–1900, Parshall Specifically, German educational institutions introduced and Rowe explicitly structure two seminar-style instruction to train graduate students as other periods in the history of American mathematics— future researchers. one from 1776 to 1876 and at least one after 1900. Such The foundation of The Johns Hopkins University in periodization is frequently utilized as a helpful analytical 1876 marked a major change in American higher educa- framework in historical scholarship, and justifications for tion. Before that, American colleges were designed primar- the selected start and end dates become part of a work’s ily for undergraduate instruction and did little to encourage historical argument.1 faculty research. Johns Hopkins, though, adopted the German vision for a university faculty charged to educate Deborah Kent is an associate professor of mathematics at Drake University. undergraduate students, conduct independent research, Her email address is [email protected]. and train graduate students to be future researchers. This Communicated by Notices Book Review Editor Stephan Ramon Garcia. influential German model likewise informed Clark Univer- 1Historical periodization sometimes presents challenges, particularly sity when it started in 1889 and the University of Chicago, in cross-cultural contexts. For an example, see work of Bernard Lewis, which opened for classes in 1892. The quarter century of https://www.gatestoneinstitute.org/323/the-periodization-of -history---excerpts or https://www.youtube.com/watch?v= central consideration in Parshall and Rowe thus begins in OS7-ZRFwVxM. For an overview of the postmodernist position on periodiza- the year shared by the US centennial and the start of a new tion, see especially Chapter 1 in Lawrence Besserman, ed., The Challenge kind of academic institution in America. In the detailed of Periodization: Old Paradigms and New Perspectives, 1996. A discussion in the meaty middle eight chapters, Parshall and sampling of other perspectives is available in J. H. Bentley, “Cross-Cultural Rowe focus on at Johns Hopkins; Interaction and Periodization in World History,” American Historical Re- view (June 1996), 749–770; and L. Grinin, “Periodization of History: A Felix Klein at Leipzig, then Göttingen; and Eliakim Hast- theoretic-.” In: History & Mathematics, Moscow: ings Moore (see Figure 1) at the University of Chicago. The KomKniga/URSS, pp. 10–38, 2007. authors articulate how their focus on students and research For permission to reprint this article, please contact: reprint-permission from these schools illustrates “the process of maturation @ams.org. of an American mathematical research community which DOI: https://dx.doi.org/10.1090/noti2072 had fully emerged by 1900” [Parshall and Rowe, p. xv].

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 701 Book Review

In an epilogue chapter, Parshall a range of existing literature, espe- and Rowe sketch developments cially relevant recent work, and then in the American mathematical re- either provide contextualized, care- search community during a period fully documented critical analysis of of consolidation and growth from previously unstudied primary source 1900 to 1933. The bookend year of material or articulate new, conclu- 1933 marks an influx of European sively argued insights or perspective mathematicians escaping the Nazi on known sources. Reviewing a work regime. Parshall and Rowe describe of historical scholarship, then, in- how, when scholars like Hermann volves considering to what extent the Weyl and Albert Einstein joined work adds to the existing scholarship the original faculty at the Institute and how effectively the author builds for Advanced Study, they entered a a historical argument in support of mathematical community of similar stated claims. quality to the one they’d left behind With the title American Mathemat- in Europe. ics 1890–1913: Catching Up to Europe, In title and content alike, The Emer- Batterson implies two claims. First, gence of the American Mathematical that his book will justify this partic- Research Community, 1876-1900: J. J. ular periodization in the history of Sylvester, Felix Klein, and E. H. Moore American mathematics. Why choose invited subsequent studies concern- these particular dates? What makes Figure 1. Eliakim Hastings Moore, 1902. ing the development of American 1890 and 1913 uniquely significant mathematics influenced by people within the larger picture of American beyond these “big three,” in places outside of Hopkins, mathematics? Second, the title indicates that the text will Göttingen, and Chicago, as well as in times before and after demonstrate that American mathematics reaches what the delineated quarter century. Batterson calls “parity with European nations” within the American Mathematics 1890–1913: Catching Up to Eu- delineated period [Batterson, p. 197]. The preface confirms rope aligns itself as exactly such scholarship. From the these expectations by stating that the book “examines the outset, Batterson states very accurately that his book “has 1890–1913 transformation in American mathematics” a nontrivial overlap” with the one by Parshall and Rowe [Batterson, p. xii] and claiming that American mathematics [Batterson, p. xii]. Batterson’s closing acknowledgments achieved “international standing” between the years 1900 further note that “[r]eaders familiar with their work will and 1913 [Batterson, p. xiii]. If a transformation occurred, undoubtedly recognize its impact” [Batterson, p. xii]. Such then there must be a before circumstance that is somehow familiar readers might also wonder why so much similar demonstrably different from the situation after said trans- material is revisited in this work. Batterson explains that his formation. So any thoughtful review of Catching Up to “1890–1913 periodization provides a different window for Europe must take into consideration the degree to which distinguishing the heroes” involved in “the coming of age the author situates his claims in the context of related liter- of American mathematics” [Batterson, p. xiii]. Batterson ature and how effectively evidence is marshaled in support especially wants to frame William Osgood and Maxime of these claims. Bôcher as “intellectual pioneers” who “overcame the ob- Some precedent exists for using Batterson’s chosen dates structions to change that were imposed by institutional as significant timestamps in the development of American culture and economic conditions at American universities” mathematics. At one end, AMS president Thomas Fiske gave [Batterson, xii]. Batterson starts his periodization when he a periodization of the development of pure mathematics does because Osgood began an instructorship at Harvard in America in an address at the AMS meeting in 1904. Like in 1890, and Bôcher joined him the following year. Then Parshall and Rowe, Fiske drew a line at the establishment E. H. Moore moved to the University of Chicago in 1892 of Johns Hopkins University in 1876 to demarcate the first and, says Batterson, “American mathematics began an ex- attempt made to “stimulate in a systematic manner research ponential rise” [Batterson, p. 89]. in the field of pure mathematics” [Fiske, p. 3].2 He ends the Building a historical argument is not unlike writing a second, and thus begins the third, period in 1891, when mathematical paper. One frames a research question in the the New York Mathematical Society (this would become context of existing scholarship, then provides substantial the American Mathematical Society in 1894) assumed a evidence for a sustained and focused argument that builds broader scope and began to publish the Bulletin. On the to the stated conclusion, a claim that will advance the boundary of knowledge in the field. To add a contribution 2Smith and Ginsberg also drew a line at the foundation of Johns Hopkins to historical scholarship, one first must be conversant with [Smith and Ginsberg, 1934].

702 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Book Review other end, Garret Birkhoff described the year 1912 as “a of Charles Eliot, president of Harvard from 1869 to 1909, milestone marking the transition from primary emphasis and Daniel Coit Gilman, president of Johns Hopkins on mathematical education at Harvard to primary emphasis from the founding in 1876 until 1901. Again, for minimal on research” [G.D. Birkhoff in Duren, p. 27]. It is too bad examples, Batterson references neither a directly relevant that Batterson seems unaware of both 2013 book on the history of math- Fiske’s and Birkhoff’s perspectives, ematics at Harvard [Nadis and Yau, when an appeal to either could have 2013] nor a standard source with helped his cause and connected his a critical perspective on Gilman’s work to foundational scholarship in contributions to American higher ed- the history of American mathematics. ucation [Cordasco, 1960]. Much like The opening chapter, “An Amer- mathematical knowledge, historical ican Colony in Göttingen,” revisits scholarship is built up by researchers narratives told elsewhere about the who tackle open questions—they group of American graduate stu- find and fill gaps in existing work or dents who studied in Europe (many illuminate entirely new directions, with Felix Klein in Göttingen) in perhaps with new sources or new the 1880s. The second chapter is a methodologies. Overall, Batterson somewhat uneven discussion labeled adds little to the existing coverage of “19th-Century American Notions of the efforts and impacts of both Eliot Scholarship.” Presumably, the pur- and Gilman. It is clear that he has pose of this chapter is to articulate a conducted extensive archival work, picture of American mathematics be- although unfortunate that such effort fore the transformation Batterson in- and expense was not applied more tends to highlight. A tighter narrative uniformly to new material. There is rooted in existing scholarship could no shortage of open questions or have sketched the contours more effi- Figure 2. William Fogg Osgood, 1896. unmined archival material relevant to ciently to allow more space and focus the history of American mathematics. on the primary stated argument of the Not until Chapter 4 does American book. For one example, in describing Mathematics 1890–1913: Catching the well-known Neptune controversy, Up to Europe turn to the events on Batterson ignores work that focuses which Batterson based the start date on how American mathematicians for his periodization. This “Harvard seized on this event as an opportu- and Chicago Hire Osgood, Bôcher, nity to assert their competence at a and Moore” chapter offers a few new time of perceived European scientific insights into the Harvard careers of superiority [Kent, 2011]. The long Osgood (see Figure 2) and Bôcher, digression instead becomes unnec- expanding a bit on Batterson’s own essarily detailed and loses the point previous work [Batterson, 2009]. It about American involvement in the is unfortunate that in his efforts to story [Batterson, pp. 30–36]. Batter- champion Osgood, he includes no son’s passages on Harvard and Yale reference to Diann Porter’s recent similarly exclude most of the volumi- biography, William Fogg Osgood at nous literature on nineteenth-century Harvard. Agent of a Transformation of higher education in America. For one Mathematics in the United States, even example, Batterson seems not to have the title of which indicates this could consulted Roger Geiger’s masterful be a rich source of evidence untapped treatise on the metamorphosis of by Batterson [Porter, 2013]. Whatever American research universities into Batterson’s analysis of Porter’s work, world leaders in scientific research be- current standards of historical schol- Figure 3. George David Birkhoff, 1913. tween the years 1900 and 1940 [Gei- arship dictate at least some mention ger, 2004]. Any relevant subsequent of such closely aligned contemporary work, such as Catching Up to Europe, would stand firmer on work, if for no other reason than to demonstrate the author the solid scholarly foundation provided by Geiger’s work. is aware of it. This trend of omitted sources continues in the third The most original part of the book comes in Chapter chapter, which focuses on the academic institution building 5, where Batterson delves into the archives of astronomer

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 703 Book Review

Simon Newcomb and mathematician Henry White to ex- There is no question that both the American-educated plore discussions about the possibility that the AMS might Birkhoff and Alexander achieved significant mathematical take over the American Journal of Mathematics. J. J. Sylvester, results. Even so, are two big results enough to “catch up” the first professor of mathematics at Johns Hopkins, had with Europe? It seems not, based on Veblen’s correspon- founded the American Journal of Mathematics in 1878 in dence during the period that depicts Hilbert alone at the keeping with Gilman’s research university mandate. After pinnacle of mathematics [Barrow-Green, 2011]. It must almost seventy-five years of sporadic private publication of be noted, too, that the 1912 ICM was hosted in Cam- mathematical periodicals in the US, the American Journal bridge, UK. Perhaps invitations to speak at a meeting in of Mathematics was the first mathematical journal designed English-speaking Britain might not say much about the to circulate original research and backed by institutional perception of Osgood and Bôcher and American mathe- support. It rapidly became the top publication outlet for matics more generally within continental Europe. Prior to increasingly research-minded American mathematicians. those 1912 invitations, Simon Newcomb gave an invited After Sylvester left America in 1883, “stories of sloppy plenary lecture in applied mathematics and had been a editing and poor refereeing” demoralized a young cohort vice president at the ICM in Rome in 1908.4 Batterson’s keen to publish original work [Batterson, p. 126]. In this statement about how American mathematics appeared well-written and carefully documented chapter, Batterson “to European eyes” heavily relies on a reference to a biog- details how the politics and personalities in this situation raphy [Reid, 1976] in which the author provides no actual led to a new research journal, the Transactions of the Amer- attribution to Courant himself [Batterson, pp. 184–185]. ican Mathematical Society. A revised version of this chapter Hence a key piece of Batterson’s claim is not substantiated also appeared as an article in the May 2019 issue of the by documented evidence. More, and more reliable, data Notices. would likely have presented a more accurate picture of The sixth chapter focuses on the preceptor program at the European perception of American mathematics. For Princeton. Woodrow Wilson introduced this program in example, it may have been helpful to include reports from 1905 to improve undergraduate education by introducing Poincaré or Darboux during a 1904 visit [Zitarelli, 2011] or some research-oriented instructors to the Princeton faculty. some other documented firsthand accounts. , Luther Eisenhart, and Joseph Wedderburn There is a striking lack of data in the book overall. Some all started as Princeton preceptors. The former two later quantification related to numbers and types of publica- were particularly influential in establishing mathematics as tions, PhDs granted, and academic trajectories might have a strength at Princeton, especially after National Research helped Batterson fortify his argument that a new era in Council Fellowships in mathematics started in 1923. In American mathematics indeed began in 1913. Rich material the seventh chapter, Batterson examines events that moved along these lines is available in recent articles publishing George Birkhoff (see Figure 3) from his Princeton precep- data about participants in the American mathematical torship to his longtime Harvard faculty position in 1913. community, their educational attainments, and scholarly This final chapter title, “The Verge of Parity with Europe,” output [Fenster and Parshall, 1994; Zitarelli, 2001]. Mod- seems like a bit of a disclaimer. Those words belong to ern search engines make it fairly straightforward to get a Veblen, who did not give a time frame or definition for sense of data for years that may be missing in these studies. parity with Europe [Batterson, p. xi]. Considered as a whole, work on the history of American Indeed, it is also not clear what Batterson means by mathematics known to this reviewer still suggests that, “reaching international standing” [Batterson, p. xiii]. The in fact, American mathematics had “earned standing” by evidence he presents for the claim that this happened for the time of the Second World War rather than the First as American mathematics by 1913 likewise invites closer Batterson claims to prove [Batterson, p. xi]. examination. The main claims of having “caught up” to The Catching Up to Europe book jacket claims that by Europe seem to be Birkhoff’s publication of his proof of the First World War, not only had the gap between math- Poincare’s Geometric Theorem,3 James Alexander’s proof ematical research in European and American universities that different constructions of the same manifold must “largely closed” but also that “[i]t would remain so.” Reality have the same Betti numbers and torsion coefficients [Bat- is not so tidy. The twenty years from 1913 to 1933 brought terson, p. 187], the 1912 ICM plenary speaking invitations internal tensions within the American mathematical com- for Osgood and Bôcher, and a memory from Courant that munity, significant institutional transitions, and a major in 1913 mathematicians in Göttingen looked admirably world war, all of which resulted in a trajectory that is not across the Atlantic for the first time. These claims invite a strictly monotone increasing. To demonstrate that parity more nuanced and critical analysis than what is provided. had, in fact, been sustained from 1913 onwards would

3Poincaré’s Geometric Theorem can be stated as follows: If T is an area-pre- 4Other invited American speakers at the Rome ICM included David Eugene serving and boundary-component-preserving homeomorphism of an annulus Smith and L. E. Dickson. Dickson had also been invited to speak at the that rotates the inner boundary and outer boundary in opposite directions, Paris ICM in 1900. Americans Artemas Martin, Harris Hancock, and then T has at least two fixed points [Batterson, p. 182]. Irving Stringham were also invited to speak in Paris.

704 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Book Review involve acknowledging, addressing, and minimizing Deborah Kent, The curious aftermath of Neptune’s discovery, known evidence to the contrary. Significant work by Re- Physics Today 64 (2011), 12, 46. inhard Siegmund-Schultze includes many sources that Steve Nadis and Shing-Tung Yau, A history in sum: 150 years of mathematics at Harvard (1825–1975), Harvard University contradict the claim of parity by 1913 [Siegmund-Schultze, Press, Cambridge, MA, 2013. MR3100544 2001 and 2009]. Batterson neither addresses these coun- Diann R. Porter, William Fogg Osgood at Harvard: Agent of a terexamples nor much discusses the impact of World War transformation of mathematics in the United States, Docent I on either American mathematical research or Europe’s Press, Boston, MA, 2013. MR3135333 view of it. The 2014 book The War of Guns and Mathematics: Constance Reid, Courant in Göttingen and New York: The sto- ry of an improbable mathematician, Springer-Verlag, New Mathematical Practices and Communities in France and Its York-Heidelberg, 1976. MR0532647 Western Allies around World War I could have shed some Reinhard Siegmund-Schultze, Mathematicians fleeing from light on these matters [Aubin and Goldstein, 2014]. Work Nazi Germany: Individual fates and global impact, Princeton completed since the publication of Catching Up to Europe University Press, Princeton, NJ, 2009. MR2522825 Reinhard Siegmund-Schultze, Rockefeller and the internation- likewise reveals rich archives that also support the much alization of mathematics between the two world wars: Docu- more complex picture suggested by earlier scholarship on ments and studies for the social history of mathematics in the American mathematics during the first third of the twen- 20th century, Science Networks. Historical Studies, vol. 25, tieth century. Birkhäuser Verlag, Basel, 2001. MR1826906 As a contribution to historical scholarship, Catching David Eugene Smith and Jekuthiel Ginsburg, A history of mathematics in America before 1900: Three Centuries of Sci- Up to Europe leaves some things to be desired. A broader ence in America, Arno Press, New York, 1980. Reprint of familiarity with relevant scholarship, both classic and the 1934 original. MR749578 current, could have both enriched and focused the signif- David E. Zitarelli, The 1904 St. Louis Congress and westward ex- icant archival endeavors of this project. Connecting with pansion of American mathematics, Notices Amer. Math. Soc. descendants of historical actors of interest can be a thrill 58 (2011), no. 8, 1100–1111. MR2856141 David E. Zitarelli, Towering figures in American mathematics, for many historians—Batterson notably accomplished 1890–1950, Amer. Math. Monthly 108 (2001), no. 7, 606– contact with a grandson of Osgood and a great-grandson 635, DOI 10.2307/2695269. MR1862099 of Bôcher. Beyond these efforts, more careful attention to building a historical argument may have altered the outcome of insufficient evidence necessary to justify the claim that American mathematics had indeed caught up to Europe by 1913. Or it might have altered the claim. As it stands, readers unfamiliar with much of the history of mathematics in the United States may nonetheless find in this well-written book some items of interest about “how life has evolved for research mathematicians in the United States” [Batterson, p. xiii]. Deborah Kent References David Aubin and Catherine Goldstein (eds.), The war of guns Credits and mathematics: Mathematical practices and communities Figures 1–3 are courtesy of the American Mathematical So- in France and its western allies around World War I, History ciety. of Mathematics, vol. 42, American Mathematical Society, Author photo is courtesy of Radina Droumeva. Providence, RI, 2014. MR3308762 June Barrow-Green, An American goes to Europe: three letters from Oswald Veblen to George Birkhoff in 1913/1914, Math. Intelligencer 33 (2011), no. 4, 37–47, DOI 10.1007/ s00283-011-9248-x. MR2865297 Steve Batterson, Bôcher, Osgood, and the ascendance of Ameri- can mathematics at Harvard, Notices Amer. Math. Soc. 56 (2009), no. 8, 916–928. MR2554178 Francesco Cordasco, Daniel Coit Gilman and the Protean PhD: The Shaping of American Graduate Education, E. J. Brill, Leiden, 1960. Della Dumbaugh Fenster and Karen Hunger Parshall, A pro- file of the American mathematical research community: 1891– 1906, The history of modern mathematics, Vol. III, Aca- demic Press, Boston, MA, 1994, pp. 179–227. MR1282505 Roger L. Geiger, To Advance Knowledge: The Growth of American Research Universities, 1900–1940, Transaction Publishers, 2004.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 705 BOOKSHELF New and Noteworthy Titles on our Bookshelf May 2020

The Shape of a Life 99 Variations on a Proof One Mathematician’s Search by Philip Ording for the Universe’s Hidden Geometry by Shing-Tung Yau and Steve Nadis This entire book is de- voted to 99 proofs, lib- The Shape of a Life is an autobiogra- erally interpreted, of the phy of Shing-Tung Yau, winner of following statement: “If the 1982 Fields Medal and many x3-6x2+ 11 x-6 = 2x-2, then other prestigious awards. The x=1 or x=4.” It seems im- book is coauthored with science probable that one could writer and Discover contributing write more than a few editor Steve Nadis, who previously pages on such a modest Yale, 2019, 328 pages 2019, Yale, collaborated with Yau in 2010 on 272 pages Princeton, 2019, topic, but Ording has the popular science book The Shape of Inner Space: String managed to parlay this simple question into a perceptive Theory and the Geometry of the Universe’s Hidden Dimensions. reflection on mathematics and its culture. Students and Although there is no explicit mathematics in the book, the professors alike will enjoy this unusual book. writing style is clear enough that the mathematically literate The inspiration for 99 Variations on a Proof comes from reader will be able to understand, at an intuitive level, the Raymond Queneau’s Exercises in Style, a 1947 work that stunning breakthroughs and remarkable discoveries that retells the same story in 99 strikingly different ways. The Yau and his collaborators contributed to. book is divided into 99 short chapters, each of which ex- The Shape of a Life is an interesting and engaging read, plores a different “proof” of the main result. The word written in a detailed yet lively style. The book vividly docu- “proof” here appears in quotes since most would not pass ments Yau’s trials as a desperately poor child in Hong Kong; muster in an undergraduate course. Each comes with a short his unlikely path to the UC Berkeley graduate program; parenthetical description. For example, Proof 36 (Social his deteriorating relationship with his doctoral advisor, Media) appears in the form of a fictitious tweet by Girol- Shiing-Shen Chern; the development of geometric analysis; amo Cardano: “Cube & 9 times first power equals 6 times the discovery of Calabi-Yau manifolds; and more. square & 4 solved by reduction to @delferro’s equation The Shape of a Life maintains an appropriate pace, never arxiv.org/abs/4307.1160 #cubic #tartaglia”. Some ap- dragging its feet nor skipping important details. Yau is proaches seem uncomfortably familiar, such as Proof 44 always frank and forthright. He has strong opinions and (Omitted with Condescension): “There is a simply beauti- does not hesitate to share them. Whether one agrees with ful theorem which provides all solutions of the equation him or not, one must conclude that The Shape of a Life is a x3-6x2+ 11 x-6 = 2x-2. Alas, any further explanation would deeply personal reflection that provides a keen insight into deny you the satisfaction of discovering it on your own…” the life and mind of one of the world’s top mathematicians. and Proof 94 (Authority): “Of course, if x3-6x2+ 11 x-6 = 2x-2, then it follows from Euler that the real number in question must be 1 or 4.” There are also some serious mathematical proofs, graphical proofs, proofs by experiment, and several unique expositions that demonstrate a great deal of artistry. For example, there are proofs in dialogue form and dog- The Bookshelf is prepared monthly by Notices Associate Editor Stephan gerel, along with screenplays and blog entries. Fittingly, the Ramon Garcia. book ends with Proof 99 (Prescribed): “The proof is left to Appearance of a book in the Notices Bookshelf does not represent an en- the reader.” dorsement by the Notices or by the AMS. Suggestions for the Bookshelf can be sent to notices-booklist @ams.org.

706 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 AMS BOOKSHELF

The AMS Book Program serves the mathematical community by publishing books that further mathematical research, awareness, education, and the profession while generating resources that support other Society programs and activities. As a professional society of mathematicians and one of the world’s leading publishers of mathematical literature, we publish books that meet the highest standards for their content and production. Visit bookstore.ams.org to explore the entire collection of AMS titles.

A Passage to Modern Analysis Real Analysis by William J. Terrell A Constructive Approach Through Interval Arithmetic The author’s title reveals his pur- by Mark Bridger pose—he wishes to give readers This book contains the clearest a thorough grounding in the exposition of real analysis using rudiments of analysis so they constructivist principles that ex- can pursue further study in those ists in English. The author un- areas of pure and applied math- derstands how undergraduates ematics that build upon it. The think and meets them where first third of the book is a careful they are. If you seek to avoid development of the analysis of

Pure and Applied Undergraduate Texts, Volume 41 (AMSTEXT/41) 41 Volume Pure and Applied Undergraduate Texts, 38 (AMSTEXT/38) Volume Pure and Applied Undergraduate Texts, proofs by contradiction in your real-valued functions of a sin- teaching of analysis, this text gle real variable. The middle third generalizes to Rn and makes it possible. Every proof is constructive, in particular, introduces metric spaces and normed vector spaces. The existence is never established by proving that the assump- final third presents some tools of advanced analysis: Fou- tion of nonexistence leads to a contradiction. rier series, ordinary and partial differential equations, the But the constructivist philosophy is not the most in- Lebesgue integral, and function spaces. teresting feature of the book. Bridger develops the real The exposition is careful and clear throughout. The numbers out of intervals of rational numbers using interval defining feature is the author’s clear-eyed vision of where arithmetic. (This is an approach due to Gabriel Stolzen- he wants his readers to go and his foresight in planning berg.) The appealing feature of this development is that the path along this passage. Everything feels completely it yields a clear analogy to scientific measurement. Any natural and inevitable because the ground in R and Rn scientific measurement of known accuracy falls within an has been so thoroughly prepared. Instructors will observe interval with rational endpoints because any measuring Terrell carefully laying this groundwork, students will just instrument yields an approximation. This makes the theory feel a natural progression to higher levels of abstraction. particularly appealing, and accessible, to students looking The nearly 600 exercises illuminate the exposition and give at further study in computer science or physical science. instructors plenty of opportunities to push and stretch their Computer scientists especially will realize that interval students. There is more than enough material here for three arithmetic, with its error bounds, is precisely what they semesters of coursework, or instructors can pick and choose need to trap an exact value inside a range of computed topics to construct a year-long course. The prerequisites are values. The constructivist law of -trichotomy will especially low—a few semesters of calculus, some linear algebra, and resonate with them: Given real numbers x, y, and > 0, exposure to the notion of proof—but the end goal is lofty. either x > y, x < y, or x and y are within of one another. Readers who study the entire book will be exceedingly well There is a strong argument to be made that this approach prepared for graduate study leading to research in pure or through intervals will feel much more natural and satis- applied topics in analysis. factory to physicists, engineers, applied mathematicians, and computer scientists than the standard mathematical approach. These students realize that every number they compute is an approximation and interval arithmetic tracks the error bounds for them. Even pure mathematicians who The AMS Bookshelf is prepared bimonthly by AMS Acquisitions Specialist might not sympathize with the underlying philosophical for MAA Press titles Stephen Kennedy. His email address is skennedy principles of constructivism will be gratified by the satis- @amsbooks.org. fying solidity and practical utility of interval arithmetic.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 707 2020 SECTIONAL SAMPLER and the probability that a wild type cell will divide is Mathematics of given by 푃(w.t. div.) = 1 − 푃(mut. div.). A division is fol- lowed by a death, with probabilities proportional to cells’ Cellular Evolution death rates (formulated similarly). The newly produced offspring of the cell that divided then replaces the dead and Some Biomedical cell, and the total population remains at 푁. In the sim- plest case, the progeny cell retains the type of the parent Applications cells. If mutations are included, the progeny cell may be of a different type compared to the parent cell. If we are interested in oncogenes (or, more generally, Natalia L. Komarova the production of one-hit mutants), we can study the sta- tistics and dynamics of mutant generation and spread; we Background: Two Types of Genes can assume that the mutants can be advantageous (e.g. There are two types of genes whose evolution we would 푟퐵 > 푟퐴, 푑퐵 = 푑퐴), neutral (푟퐵 = 푟퐴, 푑퐵 = 푑퐴), or dis- like to understand because of these genes’ important role advantageous (e.g. 푟퐵 < 푟퐴, 푑퐵 = 푑퐴). In the context of in cancer. Tumor suppressor genes protect cells from acquir- tumor suppressor genes (two consecutive mutations) we ing malignant properties. If a genetic mutation inactivates can study the timing of the generation of two-hit mutants. one of the copies of such a gene, the other copy will still This basic process is as well understood as it is unreal- protect the cell from becoming malignant. Therefore, in istic [4, 7]. Adding more complexities leads to interesting the context of this pathway, it takes two independent ge- mathematical problems and opens up possibilities to an- netic “hits” to transform a cell. Oncogenes work differently. swer some biologically relevant questions. A specific mutation in any of the two copies of such agene can turn on a malignant function, making the cell cancer- Spatial Moran Process ous. It takes only a single genetic hit to turn on this mech- One important modification is to include a network of in- anism of cancerous transformation. teractions. A reproducing cell can only replace a dying cell Many cancers evolve as a chain of several events that can if they are connected by an edge of a network. In particu- include mutations in one or both of these types of genes. lar, a geometric network can be used to describe 1D, 2D In this talk we will look at the mathematical methods to (Figure 1), or 3D spatial systems. Do such spatial interac- study the evolution of cancerous populations and concen- tions speed up or slow down evolution? In the context of trate on these two gene types as building blocks of cancer. tumor-suppressor genes, the result is that under geomet- ric networks, double-hit mutants are produced faster (and The Moran Process of Cellular Turnover this can be explained intuitively) [1, 5]. When it comes Cellular populations are sometimes described by means to one-hit mutants (such as mutations in oncogenes), the of the so-called Moran process. Assume that there are 푁 answer is less straightforward: spatial systems may have cells that undergo discrete rounds of divisions and deaths. fewer or more mutants compared to nonspatial systems of In each round, one division and one death occur. There the same size. are two types of cells, which we will refer to as “wild type” (type A) and “mutant” (type B) cells. All wild type cells are characterized with a constant division rate, 푟퐴, and a constant death rate, 푑퐴. Similarly, mutants have a constant division rate, 푟퐵, and a constant death rate, 푑퐵, where in general, 푟퐴 ≠ 푟퐵 and/or 푑퐴 ≠ 푑퐵. For each update, a cell is selected for division with a probability that is proportional to its division rate. For example, if there are 푚 mutants and 푁 − 푚 wild type cells in the system, the probability that a mutant cell will divide is given by 푚푟 푃(mut. div.) = 퐵 , (푁 − 푚)푟퐴 + 푚푟퐵

Natalia L. Komarova is a professor of mathematics at the University of Califor- Figure 1. Nonspatial (a) and spatial (b) Moran process. nia Irvine. Her email address is [email protected]. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti2083

708 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 2020 Sectional Sampler

Cellular Hierarchies [4] Natalia L. Komarova, Anirvan Sengupta, and Martin A. Even if we don’t consider mutations, not all cells are cre- Nowak, Mutation-selection networks of cancer initiation: Tu- mor suppressor genes and chromosomal instability, J. Theoret. ated equal. In real tissues, we can distinguish the so-called Biol. 223 (2003), no. 4, 433–450, DOI 10.1016/S0022- stem cells (SCs) and differentiated cells (DCs). Oversim- 5193(03)00120-6. MR2067856 plifying, we can say that SCs can divide into two SCs, two [5] Natalia L. Komarova, Leili Shahriyari, and Dominik DCs, or one of each. Differentiated cells do not divide at Wodarz, Complex role of space in the crossing of fitness val- all. What happens if we add cellular hierarchies to the ba- leys by asexual populations, J. Roy. Soc. Interface 11 (2014), sic Moran process (so, in the presence of a single type of no. 95, 20140014. mutation, there are now four types of cells)? Does this hi- [6] Leili Shahriyari and Natalia L. Komarova, Symmetric vs. erarchical structure change the speed of evolution? It turns asymmetric stem cell divisions: an adaptation against cancer?, out that it has an effect that is opposite of that described PloS One 8 (2013), no. 10, e76195. [7] Daniel B. Weissman, Michael M. Desai, Daniel S. Fisher, above. Cellular hierarchies slow down evolution consider- and Marcus W. Feldman, The rate at which asexual pop- ably [6]. Perhaps this is one of the reasons the organs of ulations cross fitness valleys, Theoret. Population Biol. 75 multicellular organisms often have a hierarchical structure: (2009), no. 4, 286–300. this is a type of protection against cancer. Cooperation and Defection Cells in organs can cooperate. For example, suppose two consecutive mutations that occur in the same cell make this cell very advantageous. It could happen that two cells with complementary mutations, if they are found in each other’s proximity, can share the products of the mutated genes, and both act as a very advantageous cell. Does co- operation speed up evolution? Yes, absolutely [3]. But cheating (defection) can speed up evolution even more! Natalia L. Komarova Random Environments Credits So far, all the properties conferred by genetic mutations were fixed. For example, a disadvantageous mutant would Figure 1 is from [5]. always have a reduced division rate. This is a simplification Photo of the author is courtesy of the author. which does not take into account the randomness of the environment. Depending on time and/or space, a mutant could have a higher or lower division rate compared to the wild type. What consequences does this have for the evo- lutionary process? Some very counterintuitive results can be found in this area [2], which will be presented in the lecture.

References [1] Richard Durrett and Stephen Moseley, Spatial Moran mod- els I. Stochastic tunneling in the neutral case, Ann. Appl. Probab. 25 (2015), no. 1, 104–115, DOI 10.1214/13- AAP989. MR3297767 [2] Suzan Farhang-Sardroodi, Amir H. Darooneh, Moham- mad Kohandel, and Natalia L. Komarova, Environmental spatial and temporal variability and its role in non-favoured mutant dynamics, J. Roy. Soc. Interface 16 (2019), no. 157, 20180781. [3] Natalia L. Komarova, Spatial interactions and cooperation can change the speed of evolution of complex phenotypes, Proc. Nat. Acad. Sci. 111 (2014), Supplement 3, 10789–10795.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 709 KEYSTONES

Probabilistic Inference for Trust Andres Molina-Markham and Joseph J. Rushanan

Computer systems that need to account for uncertainty— counterthreat behaviors and valid behavior of a system such as those enabled by artificial intelligence (AI)—require to designing robust protection mechanisms and Bayesian novel approaches to evaluate their trustworthiness. These inference engines needed to estimate trust. systems exhibit probabilistic behavior that depends on The MITRE Corporation is addressing the trust inference data from sources (sensors and computation) with varying question in two specific AI applications. First, we have de- degrees of trust, uncertain availability, and complex sto- veloped a trust inference engine for the widely important chastic relations. Hence, describing the valid behavior of a problem of trust in positioning, navigation, and timing probabilistic system and verifying that an implementation (PNT) information. Such inference considers multiple satisfies a formal specification require new approaches [1]. sources of information, such as the Global Positioning Sys- Furthermore, the problem becomes more challenging when tem (GPS) and other sensor inputs, situational awareness adversaries can influence decisions (to induce erroneous information, and auxiliary sources (e.g., network data). detections or misinterpretation of the state of operations The challenge is to fuse trust assumptions and assessments and more generally to force wrong decisions) by manip- of these sources into useful assurance metrics that can be ulating observations or the results of actions taken by an scrutinized and refined. Our solution, PNTTING (PNT Trust AI-enabled system (a system designed to perform tasks that Inference Engine) facilitates this trust fusion according to normally require human intelligence, such as visual per- probabilistic models with rigorous semantics, leveraging ception, threat assessment, or complex decision making). the emerging work in probabilistic programming lan- Therefore, evaluating trust in the presence of adversaries guages (e.g., languages such as Anglican [2] or Gen [3]). requires the triad of describing the relevant threats, estab- PNTTING’s probabilistic models describe relations between lishing a set of assurance mechanisms to counter those inputs and outputs and how inputs are transformed and threats, and characterizing valid behavior in a way that is combined. PNTTING encodes input transformations and amenable to verification. While a comprehensive solution other relations via probabilistic models developed by as- surance model designers and PNT engineers. to the problem of evaluating trustworthiness in adversar- Second, the work on PNTTING is being extended to ial settings is an open problem, its solution will rely on address the problem of evaluating the extent to which we mathematics at various levels, from modeling threat and can trust the decisions made by autonomous vehicles (AVs) Andres Molina-Markham is a lead cyber security researcher at The MITRE in adversarial environments. The goal of this new effort is Corporation. His email address is [email protected]. to develop a similar formal framework and a probabilistic Joseph J. Rushanan is a principal mathematician at The MITRE Corpora- inference engine to assess AV decision-making platforms. tion. His email address is [email protected]. Vehicles belong to a class of cyber-physical systems that Communicated by Notices Associate Editor Emilie Purvine. can be exploited to cause serious harm. Thus, assessing the For permission to reprint this article, please contact: reprint-permission validity of an AV decision-making platform in adversarial @ams.org. conditions is a critical problem that requires modern prob- DOI: https://dx.doi.org/10.1090/noti2081 abilistic modeling concepts [4], as well as state-of-the-art

710 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Keystones probabilistic computing concepts [3], to attain rigor and [5] T. Rainforth et al., Interacting Particle Markov Chain Monte practicality. Carlo, Proceedings of the 33rd International Conference Our methods to evaluate trust leverage the vast research on Machine Learning, 48, 2016. [6] B. Milch and S. Russell, General-purpose MCMC inference done in probabilistic modeling and inference [2], [3], [5], over relational structures, arXiv:1206.6849, 2012. [6]. Adaptation of these methods leads to many open math- ematical questions, including: What is the effect of our prior assumptions of trust? How much computation is needed for accurate trust inference? What is the dependence of this inference on the possible metrics for validity? Answering these questions requires logic, statistical methods, theory of algorithms, and advanced programming language concepts.

References [1] S. Russell, D. Dewey, and M. Tegmark, Research priorities for robust and beneficial artificial intelligence, AI Mag., 36 (2015), no. 4, pp. 105–114. [2] D. Tolpin, J.-W. van de Meent, H. Yang, and F. Wood, Andres Joseph J. Rushanan Design and implementation of probabilistic programming lan- Molina-Markham guage Anglican, Proceedings of the 28th Symposium on the Implementation and Application of Functional Pro- gramming Languages, 2016, p. 6. [3] M. F. Cusumano-Towner, F. A. Saad, A. K. Lew, and V. K. Credits Mansinghka, Gen: A General-purpose Probabilistic Program- Author photos are courtesy of the authors. ming System with Programmable Inference, Proceedings of the 40th ACM SIGPLAN Conference on Programming Language Design and Implementation, New York, NY, 2019, pp. 221–236, DOI 10.1145/3314221.3314642. Approved for Public Release; Distribution Unlimited. Public Release Case [4] S. Russell, Unifying logic and probability, Commun. ACM, Number 19-3582. 58 (2015), no. 7, pp. 88–97. ©2019 The MITRE Corporation. ALL RIGHTS RESERVED.

Graduation: an intersection of who you are and who you will become.

Graduate Student Members: Is this your big year? If you are graduating this year, take advantage of your discounted AMS Introductory Membership rate next year! This rate is available for five continuous years as you transition through your early career. For more information, watch for your 2021 membership renewal coming this August and receive 25 AMS “points”* when you renew as an Intro member!

* Apply points toward the purchase of AMS products, including book purchases, and even as a partial payment toward your next membership.

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MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 711 FROM THE AMS SECRETARY

AMS Prizes & Awards Cole Prize in Algebra This Prize recognizes a notable research work in algebra Levi L. Conant Prize that has appeared in the last six years. The work must be published in a recognized, peer-reviewed venue. This prize was established in 2000 in honor of Levi L. Conant to recognize the best expository paper published About this Prize in either the Notices of the AMS or the Bulletin of the AMS This prize (and the Frank Nelson Cole Prize in Number in the preceding five years. Theory) was founded in honor of Professor Frank Nelson Cole upon his retirement after twenty-five years as secretary About this Prize of the American Mathematical Society. Cole also served Levi L. Conant was a mathematician and educator who as editor-in-chief of the Bulletin for twenty-one years. The spent most of his career as a faculty member at Worcester original fund was donated by Professor Cole from moneys Polytechnic Institute. He was head of the mathematics de- presented to him on his retirement, and was augmented by partment from 1908 until his death and served as interim contributions from members of the Society. The fund was president of WPI from 1911 to 1913. Conant was noted as later doubled by his son, Charles A. Cole, and supported an outstanding teacher and an active scholar. He published by family members. It has been further supplemented by a number of articles in scientific journals and wrote four and by an anonymous donor. textbooks. His will provided for funds to be donated to the The current prize amount is US$5,000 and the prize is AMS upon his wife’s death. awarded every three years. Prize winners are invited to present a public lecture at Worcester Polytechnic Institute as part of their Levi L. Next Prize: January 2021 Conant Lecture Series, which was established in 2006. Find and download videos of previous Conant Lec- Nomination Period: March 1–June 30, 2020 tures at www.wpi.edu/academics/math/news/lconant -series.html. To make a nomination, go to https://www.ams.org The Conant Prize is awarded annually in the amount /cole-prize-algebra. of US$1,000.

Next Prize: January 2021 Mary P. Dolciani Prize

Nomination Period: March 1–June 30, 2020 for Excellence in Research

To make a nomination, go to https://www.ams.org The AMS Mary P. Dolciani Prize for Excellence in Research /conant-prize. recognizes a mathematician from a department that does not grant a PhD who has an active research program in mathematics and a distinguished record of scholarship. The primary criterion for the prize is an active research program as evidenced by a strong record of peer-reviewed publications.

712 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Calls for Nominations FROM THE AMS SECRETARY

Additional selection criteria may include the following: Mathematics, Grenander received many honors. He was a • Evidence of a robust research program involving Fellow of the American Academy of Arts and Sciences and undergraduate students in mathematics; the National Academy of Sciences and was a member of • Demonstrated success in mentoring undergradu- the Royal Swedish Academy of Sciences. ates whose work leads to peer-reviewed publica- The current prize amount is US$5,000 and the prize is tion, poster presentations, or conference presen- awarded every three years. tations; • Membership in the AMS at the time of nomina- Next Prize: January 2021 tion and receipt of the award is preferred but not required. Nomination Period: March 1–June 30, 2020 About this Prize This prize is funded by a grant from the Mary P. Dolciani To make a nomination, go to https://www.ams.org Halloran Foundation. Mary P. Dolciani Halloran was a /grenander-prize. gifted mathematician, educator, and author. She devoted her life to developing excellence in mathematics education and was a leading author in the field of mathematical Bertrand Russell Prize textbooks at the college and secondary school levels. Read more about her and the Foundation at www.dolciani of the AMS halloranfoundation.org/meet-mary/. The Bertrand Russell Prize honors research or service con- The prize amount is US$5,000, awarded every other year tributions of mathematicians or related professionals to for five award cycles. promoting good in the world and recognizes the various ways that mathematics furthers human values. Next Prize: January 2021 About this Prize Nomination Period: March 1–June 30, 2020 The Bertrand Russell Prize of the AMS was established in To make a nomination, go to https://www.ams.org 2016 by Thomas Hales. The prize looks beyond the con- /dolciani-prize. fines of the profession to research or service contributions of mathematicians or related professionals to promoting good in the world. It recognizes the various ways that math- Ulf Grenander Prize ematics furthers fundamental human values. Mathematical contributions that further world health, our understanding in Stochastic Theory of climate change, digital privacy, or education in develop- ing countries are some examples of the type of work that and Modeling might be considered for the prize. The Grenander Prize recognizes exceptional theoretical and The current prize amount is US$5,000, awarded every applied contributions in stochastic theory and modeling. three years. It is awarded for seminal work, theoretical or applied, in the areas of probabilistic modeling, statistical inference, Next Prize: January 2021 or related computational algorithms, especially for the analysis of complex or high-dimensional systems. Nomination Period: March 1–June 30, 2020 About this Prize To make a nomination, go to https://www.ams.org This prize was established in 2016 by colleagues of Ulf /russell-prize. Grenander (1923–2016). Professor Grenander was an in- fluential scholar in stochastic processes, abstract inference, and pattern theory. He published landmark works through- Ruth Lyttle Satter Prize out his career, notably his 1950 dissertation, Stochastic Processes and Statistical Interference at Stockholm University, in Mathematics Abstract Inference, his seminal Pattern Theory: From represen- tation to inference, and General Pattern Theory. A long-time The Satter Prize recognizes an outstanding contribution to faculty member of Brown University’s Division of Applied mathematics research by a woman in the previous six years.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 713 Calls for Nominations FROM THE AMS SECRETARY

About this Prize About this Award This prize was established in 1990 using funds donated by One program is selected each year by the AMS Committee Joan S. Birman in memory of her sister, Ruth Lyttle Satter. on the Profession and is awarded US$1,000 provided by Professor Birman requested that the prize be established to the Mark Green and Kathryn Kert Green Fund for Inclusion honor her sister’s commitment to research and to encour- and Diversity. age women in science. An anonymous benefactor added Preference is given to programs with significant partici- to the endowment in 2008. pation by underrepresented minorities. Note that programs The current prize amount is US$5,000 and the prize is aimed at pre-college students are eligible only if there is awarded every two years. a significant component of the program benefiting indi- viduals from underrepresented groups at or beyond the Next Prize: January 2021 undergraduate level.

Nomination Period: March 1–June 30, 2020 Next Award: 2021

To make a nomination, go to https://www.ams.org Nomination Deadline: September 15, 2020 /satter-prize. The nomination procedure can be found at https://www .ams.org/make-a-diff-award. For questions, contact Albert Leon Whiteman [email protected]. Memorial Prize Joint Prizes & Awards The Whiteman Prize recognizes notable exposition and exceptional scholarship in the history of mathematics. Birkhoff Prize in About this Prize This prize was established in 1998 using funds donated by Applied Mathematics Mrs. Sally Whiteman in memory of her husband, Albert Leon Whiteman. (AMS-SIAM George David The US$5,000 prize is awarded every three years. Birkhoff Prize) Next Prize: January 2021 The Birkhoff Prize is awarded for an outstanding contribu- Nomination Period: March 1–June 30, 2020 tion to applied mathematics in the highest and broadest sense. To make a nomination, go to https://www.ams.org About this Prize /whiteman-prize. The prize was established in 1967 in honor of Professor George David Birkhoff, with an initial endowment con- Mathematics Programs tributed by the Birkhoff family and subsequent additions by others. The American Mathematical Society (AMS) and that Make a Difference the Society for Industrial and Applied Mathematics (SIAM) award the Birkhoff Prize jointly. This Award for Mathematics Programs that Make a Differ- The current prize amount is US$5,000, awarded every ence was established in 2005 by the AMS's Committee on three years to a member of AMS or SIAM. the Profession to compile and publish a series of profiles of programs that: Next Prize: January 2021 1. aim to bring more persons from underrepresented backgrounds into some portion of the pipeline be- Nomination Period: March 1–June 30, 2020 ginning at the undergraduate level and leading to advanced degrees in mathematics and professional To make a nomination, go to https://www.ams.org success, or retain them once in the pipeline; /birkhoff-prize. 2. have achieved documentable success in doing so; and 3. are replicable models.

714 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Calls for Nominations FROM THE AMS SECRETARY

About this Prize Frank and Brennie Morgan The prize was established in 1995. It is entirely endowed by Prize for Outstanding a gift from Mrs. Frank (Brennie) Morgan. It is made jointly by the American Mathematical Society, the Mathematical Research in Mathematics by Association of America, and the Society for Industrial and Applied Mathematics. an Undergraduate Student The current prize amount is US$1,200, awarded annu- ally. The Morgan Prize is awarded each year to an undergrad- uate student (or students for joint work) for outstanding Next Prize: January 2021 research in mathematics. Any student who was enrolled as an undergraduate in December at a college or university in Nomination Period: March 1–June 30, 2020 the United States or its possessions, Canada, or Mexico is eligible for the prize. To make a nomination go to https://www.ams.org The prize recipient’s research need not be confined to a /morgan-prize. single paper; it may be contained in several papers. How- ever, the paper (or papers) to be considered for the prize must be completed while the student is an undergraduate. Publication of research is not required.

With MathPrograms.Org you can: • Receive, read, rate, and respond to electronic applications for your mathematical sciences programs, such as undergraduate summer research programs and travel grant competitions. • Customize your settings and control the application form; also set secure access for the admissions committee. • Enter program announcements for public display. • Download data to personal computers for use in word processing and spreadsheets or as a full permanent storage le.

SERVICE IS FREE Institutions pay annually for one program TO APPLICANTS or for multiple programs.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 715 Cal fo Nomination

Frank and Brennie Morgan AMS-MAA-SIAM Prize 20 for Outstanding Research 21 in Mathematics by an Undergraduate Student

The prize is awarded each year to an undergraduate student (or students for joint work) for outstanding research in mathematics. Any student who is an undergraduate in a college or university in the Questions may be directed to: United States or its possessions, Canada, or Mexico is eligible to be James Sellers, Secretary Mathematical Association considered for this prize. of America The prize recipient’s research need not be confined to a single paper; Penn State University it may be contained in several papers. However, the paper (or papers) University Park, PA 16802 to be considered for the prize must be completed while the student Telephone: 814-865-7528 is an undergraduate; they cannot be written after the student’s Email: [email protected] graduation. The research paper (or papers) may be submitted for the committee’s consideration by the student or a nominator. Each Nominations and submissions should be sent to: submission for the prize must include at least one letter of support from a person, usually a faculty member, familiar with the student’s Carla Savage, Secretary American Mathematical Society research. Publication of research is not required. Computer Science Department The recipients of the prize are to be selected by a standing joint North Carolina State University committee of the AMS, MAA, and SIAM. The decisions of this Raleigh, NC 27695-8206 committee are final. Nominations for the 2021 Morgan Prize are due or uploaded via the form no later than June 30, 2020. Those eligible for the 2021 prize must available at: www.mathprograms.org have been undergraduates in December 2019. FROM THE AMS SECRETARY

2020 Award for an Exemplary Program or Achievement in a Mathematics Department

The Department of Mathematics at the Massachusetts Institute of Technology is the recipient of the 2020 AMS Award for an Exemplary Program or Achievement in a Mathematics Department.

Figure 1. PRIMES students at an annual conference at MIT. Citation innovative, year-long model for guiding high school The 2020 Award for an Exemplary Program or Achieve- student research is being used also in computer science, ment in a Mathematics Department is presented to the bioinformatics, computational and physical biology, Department of Mathematics at the Massachusetts Institute genomics, and neuroscience. PRIMES students use their knowledge of mathematics and computer programming to of Technology. The MIT Mathematics Department is being solve problems related to cancer research, Internet security, honored for its Program for Research in Mathematics, En- traffic control, refugee migration, brain research, laser en- gineering, and Science for High School Students (PRIMES), gineering, and many other applied fields. PRIMES Circle which provides significant research experiences and and MathROOTS help attract members of underrepresented mathematics enrichment to high school students locally groups to pursue careers in the STEM fields. A 2015 article and globally, with particular attention to increasing the about PRIMES, which sums up its experience and provides representation of women and underrepresented minorities. advice for setting up similar programs in other institutions, PRIMES is a free, year-long research and enrichment published in the Notices of the AMS (Pavel Etingof, Slava program for high school students, created in the MIT Gerovitch, and Tanya Khovanova, “Mathematical Research Mathematics Department in October 2010. The program’s in High School: The PRIMES Experience,” Notices of the AMS

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 717 FROM THE AMS SECRETARY

62, no. 8 (2015)), has been translated into Mandarin and least 31 have been published in such journals as Representa- summarized in Spanish. In 2017, the PRIMES-Switzerland tion Theory, Journal of Algebra, Journal of Algebraic Combina- program was established at the University of Geneva and torics, Journal of Combinatorics, Journal of Integer Sequences, ETH in Zürich. Electronic Journal of Combinatorics, International Journal of PRIMES includes five sections. MIT PRIMES offers Game Theory, Transactions of the AMS, College Mathematics research projects and guided reading to students living Journal, Topology and Its Applications, Involve, Math Horizons, within driving distance from Boston. Program participants Cell Reports, Letters in Biomathematics, Physical Review E, and work with MIT researchers on exciting unsolved problems PLoS Computational Biology. Several PRIMES students have in mathematics, computer science, and computational won prizes and awards at the MAA Undergraduate Student biology. PRIMES-USA is a distance mentoring math re- Poster sessions, at the Intel International Science and Engi- search program for high school juniors from across the neering Fair, in the Siemens Competition in Math, Science, United States (outside of Greater Boston). PRIMES Cir- and Technology, and in the Intel/Regeneron Science Talent cle and MathROOTS are math enrichment programs for Search. Four Davidson Fellow Laureates and eight Davidson high-potential high school students from underrepresented Fellows have been PRIMES participants. backgrounds or underserved communities. PRIMES Circle From 2013 to 2019, eighty-four students completed the is a spring-term program for local students from Greater PRIMES Circle program, including sixty female, fourteen Boston. MathROOTS is a free two-week residential summer African American, and nine Latino students. From 2015 to program. CrowdMath, run jointly with Art of Problem Solving, is a massive collaborative year-long online research 2019, one hundred students completed the MathROOTS forum open to all high school and college students around program, including forty-four female, forty-seven African the world. American, and forty-nine Latino students. Among the PRIMES has three main goals: eighty participants in the 2015 to 2018 summer programs, 1. To give talented high school students a unique oppor- forty-two were admitted to MIT, and twenty-eight enrolled. tunity to experience the joy and beauty of mathematical In 2019, forty out of 112 PRIMES students were female, and research twenty-one were minority students 2. To inspire them to pursue careers in the mathematical sciences About the Program 3. To diversify the pool of students interested in mathemat- In October of 2020 MIT PRIMES, founded by Pavel Etingof ics by providing additional opportunities for promising and Slava Gerovitch, will celebrate its tenth anniversary. It young women and underrepresented groups. started as an experiment in year-long math research by high Between 2011 and 2019, 281 students participated in MIT school students with just twenty-one local participants. PRIMES and PRIMES-USA. All of the 276 research projects The experiment proved very successful, with the program completed were presented at nine annual PRIMES confer- growing more than fivefold in ten years and expanding ences, 179 research papers have been posted online, and at both nationally and internationally.

Figure 2. PRIMES Circle students Laura Clervil and Figure 3. MathROOTS students at a study session on Sekai Carr giving a talk at an annual conference at the roof of the MIT Math Department’s building. MIT.

718 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 FROM THE AMS SECRETARY

PRIMES offers real, not toy, research projects to high Female and minority students from PRIMES Circle school students and provides academic mentorship for a commented in their survey: “My experience in MIT PRIMES full year. The program builds collaborative teams that in- Circle has allowed me to appreciate mathematics from clude faculty, postdoctoral researchers, graduate students, a new perspective and become fascinated by how beau- undergraduates, and high school students, promoting part- tifully simple a complex idea can become. The best part nership and wider outreach in the mathematical sciences was walking out of the program with better developed community. critical thinking skills and a mathematical toolbox that I Keys to the PRIMES success are thorough preparation, could apply to the real world.” “To me, PRIMES Circle is a continuous review of research projects, and effective men- wonderful opportunity to learn what it is like to be a real torship techniques. Choosing a research project for a high mathematician.” school student is no easy task. PRIMES’s experience shows Minority students attending MathROOTS also com- that most fruitful are the projects that have an accessible mented on their experience: “Personally, I thought it was beginning with relatively simple initial steps; flexibility in awesome seeing people from a multitude of backgrounds switching among several related questions; computer-assisted all interested and passionate about math.” “Before Math- ROOTS, I was unsure about MIT, but now, I feel like it's my exploration aimed at finding patterns and making conjec- home. I hope I get to return there soon.” tures; faculty advisor involvement; relation to the mentor’s own research area; understanding of the big picture and motivation; About the Award a learning component that encourages the student to study The Award for an Exemplary Program or Achievement in advanced mathematics; and doability within a year-long a Mathematics Department was established by the AMS time frame. Council in 2004 and was given for the first time in 2006. Effective mentorship involves striking a balance between This award recognizes a department that has distin- guiding the student and allowing independent thinking, guished itself by undertaking an unusual or particularly being attuned to the learning and research style of every effective program of value to the mathematics community, student, and regularly reviewing the project progress and internally or in relation to the rest of society. Departments adjusting its scope, if needed. Head mentor Tanya Khova- of mathematical sciences in North America that offer at nova regularly meets with students to gather their feedback least a bachelor’s degree in mathematical sciences are and help with communication and motivation issues. eligible. Through the generous support of an anonymous PRIMES has accumulated experience in supervising donor, the award carries a cash prize of US$5,000. The both individual and group (two to three students) research award is presented by the AMS Council acting on the projects, guided reading groups, a larger research group recommendation of a selection committee. The members (a six-student computer algebra lab), and an open online of the 2020 selection committee were George E. Andrews, forum with a varying group of participants (CrowdMath). Maria M. Klawe, Richard S. Laugesen, Brea Ratliffe (Chair), The program regularly conducts both internal and exter- and Ulrica Y. Wilson. nal evaluations via surveys and interviews with student participants. Credits Figures 1 and 2 are by Slava Gerovitch. An award-winning student wrote in her testimonial: Figure 3 is by Sandi Miller. “PRIMES is an incredible opportunity that allows high schoolers to do what they would never normally have the chance to do: research, while also providing the guidance and encouragement that is crucial for success. Ultimately, PRIMES has truly cemented my interest in math, and it is for this reason that I would definitely encourage any student similarly passionate about mathematics to apply!”

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 719 FROM THE AMS SECRETARY

2020 Award for Impact on the Teaching and Learning of Mathematics

Darryl Yong of Harvey Mudd College has been named the recipient of the 2020 AMS Award for Impact on the Teach- ing and Learning of Mathematics.

Citation AMS containing teacher development materials using a Dr. Darryl Yong is a profes- problem-based approach. sor of mathematics at Harvey At the college level, Dr. Yong has become an expert Mudd College, where he also on inquiry-based learning methods and participated in a serves as the program director four-year controlled study of flipped classroom instruction for the Mathematics Clinic. An supported by the NSF, which led to several research articles accomplished mathematician in conference proceedings and peer-reviewed journals. Dr. who has written six books and Yong is regarded by his colleagues at Harvey Mudd and several research papers that the other Claremont Colleges as a gifted teacher who will continue to have a profound influence on how students have appeared in top applied and teachers perceive mathematics. In particular, he was math and physics journals, Dr. the founding director of the Claremont Colleges Center for Yong is also a prominent re- Teaching and Learning and served as the associate dean for Darryl Yong searcher in math education, diversity at Harvey Mudd from 2011 to 2016. with a scholarly focus on active For his many sustainable and replicable contributions and inquiry-based learning, inclusive pedagogy, and train- to mathematics and mathematics education at both the ing of high school math teachers. precollege and college levels, the AMS Committee on Edu- In 2007, Dr. Yong started a nonprofit professional de- cation is delighted to award Dr. Darryl Yong the AMS Award velopment organization for math teachers called Math for for Impact on the Teaching and Learning of Mathematics. America Los Angeles (MfA LA). This program has supported over 200 high school math and computer science teachers Biographical Sketch with multiyear fellowships for salary supplements, in addi- Darryl Yong earned his BS in mathematics at Harvey Mudd tion to providing professional development opportunities College, where he was also a piano performance major. He and a supportive community. He is the primary author of then earned a PhD in applied mathematics at the University four NSF Robert Noyce Scholarship Grants that have raised of Washington under the supervision of Jirair Kevorkian. over US$12 million for MfA LA. Dr. Yong spent a sabbatical He served as a von Kármán Instructor at the California In- year teaching high school mathematics in the Los Angeles stitute of Technology from 2001 to 2003 and has worked Unified School District, which he wrote about in a 2012 at Harvey Mudd College since 2003. At Mudd, Dr. Yong is AMS Notices article entitled “Adventures in Teaching: A Pro- currently a professor of mathematics, associate dean for fessor Goes to High School to Learn about Teaching Math.” diversity and faculty development, and Mathematics Clinic He has also worked with the Teacher Leadership Program program director. at the IAS/Park City Mathematics Institute since 2007 and His previous applied mathematics research focused on has cotaught a math course for elementary and secondary multiple-scale analysis of hyperbolic partial differential math teachers that led to a book series published by the equations. His scholarly activities now focus on the retention

720 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 FROM THE AMS SECRETARY and professional development of secondary school mathe- of students in the first two years of college. The US$1,000 matics teachers and improving undergraduate mathematics award is given annually, and the recipient is selected by education. He is regularly invited to speak and provide the COE. The endowment fund that supports the award professional development for mathematics faculty on the was established in 2012 by a contribution from Kenneth teaching and learning of mathematics, broadening partic- I. and Mary Lou Gross in honor of their daughters Laura ipation in STEM, and helping institutions build capacity and Karen. The award is presented by the AMS COE acting for increasing diversity and inclusion. on the recommendation of a selection subcommittee. For He received a Caltech ASCIT Teaching Award in 2002, a the 2020 award, the members of the subcommittee were: Pomona Unified School District Community Service Award • Douglas Ensley in 2012, and the Mathematical Association of America • Michael Dorff Southern California–Nevada Section Teaching Award in • Jon Wilkening (Chair) 2017. A listing of the previous recipients of the Impact Award can be found on the AMS website at: https://www.ams Response .org/ams-awards/impact. I am truly honored to receive this award and humbled to be included among the other award recipients. All of Credit the different efforts that were mentioned above (Math for Photo of Darryl Yong is courtesy of Harvey Mudd College. America Los Angeles, IAS/Park City Mathematics Institute, the Claremont Colleges Center for Teaching and Learning, and research projects) are/were collaborative efforts made possible by supportive and wonderful colleagues across many institutions. I am especially grateful to my colleagues at Harvey Mudd College for nominating me and supporting me in my work. And I am grateful to Kenneth I. and Mary Lou Gross and the AMS for their support of mathematics education at the precollege and college levels. About the Award The Award for Impact on the Teaching and Learning of Mathematics was established by the AMS Committee on Education (COE) in 2013. The award is given annually to a mathematician (or group of mathematicians) who has made significant contributions of lasting value to mathematics education. Priorities of the award include recognition of (a) accomplished mathematicians who have worked directly with precollege teachers to enhance DATA ON THE teachers’ impact on mathematics achievement for all stu- dents, or (b) sustainable and replicable contributions by COMMUNITY mathematicians to improve the mathematics education DOCTORAL RECIPIENTS New PhD graduates, their employment plans, demographics, and starting salaries

DOCTORAL DEGREES & THESIS TITLES PhD graduates, their thesis titles, and where they earned their degrees

FACULTY SALARIES By rank and employment status

RECRUITMENT & HIRING The academic job market

DEPARTMENTAL PROFILE The number of—faculty, their employment statuses and demographics; course enrollments; graduate students; masters and bachelors degrees awarded

www.ams.org/ Sponsored by: annual-survey AMS | ASA | IMS | MAA | SIAM

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 721 FROM THE AMS SECRETARY

2020 Mathematics Programs That Make a Difference Award

The Graduate Research Opportunities for Women program is the recipient of the 2020 AMS Mathematics Programs That Make a Difference Award.

Citation about preparation, the importance The American Mathematical Society, (or not) of GRE scores, how to ap- through its Committee on the Pro- proach letter writers, and so forth. fession, is pleased to recognize the Other meet-and-greet activities aim Graduate Research Opportunities for to connect participants to experts in Women (GROW) Program with the a potential field of interest. The com- 2020 Mathematics Programs That munity-building aspect of the gath- Make a Difference Award. GROW is an ering is crucial for female students in annual series of conferences that nur- ture, mentor and expose undergradu- a majority-male profession; the feel- ate women to the opportunities that ing that one is not alone can boost await a career in mathematics. Funded confidence. The program includes by the National Science Foundation inspirational talks by iconic female and participating universities, the speakers that make a big impression GROW Program is in its fifth year and on the participants. As one confer- GROW 2017. Left to right: has served hundreds of participants. ence attendee who is now in graduate Emmy Murphy, Ben Antieau, Bryna Kra. Over this short span, GROW has built school writes in her support letter: “I a community which, as much as the gained confidence, personal and pro- conference programming itself, has helped to make the mathematics fessional connections, and exposure profession a more appealing place to various careers in mathematics.… for women to live and work. Through I met many women who assuaged feedback, GROW steadily improves my mounting fears about applying and creates best practices for future and succeeding in graduate school.… iterations as well as for replication. Sharing my fears and concerns about Activities at GROW include research graduate school with other women talks where scholars discuss not only who were either entering or attending their results but their varied routes graduate school was one of the most through academics, giving a personal helpful aspects of GROW.” The AMS touch and dispelling the straight-and- narrow myth around career paths. commends the GROW Program for There are also panel discussions its success in bringing more persons about graduate admissions. Con- GROW 2019 participants. from underrepresented backgrounds ference-goers come with questions into the mathematical profession.

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About the Program About the Award The GROW workshop series encourages female-identifying In 2005, the American Mathematical Society, acting upon undergraduates to consider research in mathematics as a the recommendation of its Committee on the Profession discipline and a career. For the past five years, approxi- (CoProf), established the Mathematics Programs That mately eighty students have gathered over a weekend in Make a Difference Award in order to profile those programs October for a mixture of research talks, panel discussions, that are succeeding and could serve as a model for others. and opportunities to meet students and scholars from Specifically, the committee seeks to honor programs that: across the country. The participants share meals and have 1. aim to bring more persons from underrepresented mi- numerous opportunities for networking and mentoring. nority backgrounds into some portion of the pipeline GROW is designed to encourage women to think and beginning at the undergraduate level and leading to know about mathematics and to feel confident about the an advanced degree in mathematics and professional options for graduate students, as well as providing them success, or retain them once in the pipeline; with resources for their future success. 2. have achieved documentable success in doing so; and The first iteration of the GROW series was designed by 3. are replicable models. Bryna Kra at Northwestern University, and she, along with Preference is given to programs with significant partici- more than forty volunteers, led organization of the con- pation by underrepresented minorities. ferences at Northwestern from 2015 to 2017. Since then, This recognition includes an award of US$1,000 pro- GROW has moved first to the University of Michigan in vided by the Mark Green and Kathryn Kert Green Fund for 2018, with Sarah Koch and Karen Smith as lead organizers, Inclusion and Diversity. and then to the University of Illinois at Urbana-Champaign For a list of previous recipients of the Mathematics Pro- in 2019, with Zoi Rapti as the lead organizer. The next it- grams That Make a Difference Award, see the AMS website eration will be at the University of Chicago in 2020. Over at https://www.ams.org/make-a-diff-award. 350 students from across the United States have already participated in GROW, and participants have represented —Elaine Kehoe with information from Bryna Kra more than seventy-five undergraduate institutions. The panel discussions cover what constitutes research Credits in mathematics, with panelists sharing stories of their tra- Photo from GROW 2017 is courtesy of the Department of Mathematics, Northwestern University. jectories, an introduction to the varied options for careers Photo from GROW 2019 is courtesy of Zoi Rapti. in academia, and a wealth of information on the nuts and bolts of applying to graduate school in mathematics. A highlight of each meeting has been the Saturday evening Read more about the Graduate Research Opportunities dinner, with leading figures (including Alexandra Bellow, for Women Program on page 724. Dusa McDuff, Ingrid Daubechies, Kristin Lauter, and Marisa Eisenberg) giving inspirational talks.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 723 AMS COMMUNICATION

GROWing a Graduate Cohort Evelyn Lamb

Bryna Kra was frustrated. It was early 2015, and the University of Chicago for the sixth iteration. Hundreds math professor at Northwestern University had seen the of young women have participated in GROW and been number of women applying to her program decline for encouraged to consider graduate school in mathematics. several years, even as the total number of applications was “Ultimately, the main point of GROW is to increase the increasing. Anecdotally, her colleagues at other institu- comfort that bright women undergraduates have with the tions were reporting similar problems. She wanted to do idea of doing a doctorate in the mathematical sciences, something to reverse the trend. After talking with other that it is something available to them,” says Ezra Getzler, mathematicians at Northwestern, she decided to organize one of Kra’s colleagues at Northwestern and co-organizer a conference specifically focused on encouraging women of the conference when it took place there. Kra says that to apply for graduate school in math. A few months later, many alumnae have told her GROW gave them the nudge fifty undergraduate women descended on Evanston for the they needed to apply to graduate school. first iteration of the Graduate Research Opportunities for Genesis Women, or GROW, conference in October 2015. This year, it received the Programs that Make a Difference award from Putting on a conference for dozens of undergraduates in the American Mathematical Society. “It’s an honor and just a few months was no small feat. Kra and her co-orga- rewarding to know that the program has been recognized nizers Getzler and Laura De Marco had the resources and only five years after it was established, and it makes it clear connections to make GROW happen on their condensed timeline. They used funds from a Research Training Group how valuable such programs are,” Kra says. “My hope is (RTG) grant, in addition to university-level funding that that this will give the impetus for other institutions to take they could secure on short notice. “Part of the purpose of charge of the program in future years, hosting GROW and an RTG is to create a vertical conveyor belt for mathematics building more diverse departments.” in the United States, and so here we’re dealing with a key In the five years since the first conference, GROW has transition point from undergraduate to graduate school,” taken place three times at Northwestern and once each at Getzler says. “It’s a continual struggle to make sure that the University of Michigan and the University of Illinois people are trained at the graduate level in mathematical at Urbana-Champaign. In October, it will move to the sciences in the United States, and [GROW] is a very good way of increasing awareness among an important and to some degree underserved group.” Once the plans were in place, the organizers had to get the word out. “I sent information about the conference to everybody we knew,” Kra says. “In two months, we had 120 applications from students at 40 schools.” The first year, they accepted 50 students. In subsequent years, they increased attendance to 80 undergraduates to balance their Figure 1. GROW participants at the 2017 conference. goals of reaching as many people as possible while allow- ing as much one-on-one interaction as possible between Evelyn Lamb is a freelance math and science writer based in Salt Lake City. attendees. Her email address is [email protected]. Organizers wanted to make sure to welcome students For permission to reprint this article, please contact: reprint-permission from communities that do not have good access to infor- @ams.org. mation about graduate school, including first-generation DOI: https://dx.doi.org/10.1090/noti2088 college students, women from underrepresented racial or

724 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 AMS COMMUNICATION ethnic groups, and those who come from socioeconomi- cally disadvantaged backgrounds. “There are a lot of middle class assumptions baked into the PhD, which means there’s a cultural divide for people not coming from that back- ground. So I would hope that part of the role of GROW is to bridge that divide as well as the gender divide,” Getzler says. They advertised to departments all over the country, in- cluding historically Black colleges and universities, schools without graduate programs in math, and schools that rarely Figure 2. Sarah Koch describes her work to participants at GROW. send students to graduate school in mathematics. They wanted to reach as broad a group as possible of interested undergraduates who could benefit from the conference. “I was extremely impressed at what a cross section of America these students were,” Getzler says of the women who par- ticipated. (There have been some international participants as well, but the majority have been American women.) The Conference Weekend GROW is a weekend event with a mix of talks from invited Figure 3. Bryna Kra talks with students at GROW. speakers, panels about topics such as how to apply to grad- As the daughter of an academic mathematician, Kra uate school and what to expect from a career in research knew plenty about how graduate school worked and what mathematics, and less-structured time for one-on-one a career as a researcher could look like. But when she was interactions. The organizers invited a large number of men- an undergraduate, she was nevertheless uncertain if going tors—about one for every two students—to participate over to graduate school would be the right choice for her. “It cer- the course of the weekend. Some mentors were on panels, tainly didn’t feel so welcoming that I’d never seen a class by and some were local professors and graduate students who were available for informal chats with undergraduates. a woman,” she said. “I think it’s important that people see “I think here in this conference, anyone can approach that there are other women out there.” She hopes that the anyone else,” says Zoi Rapti, one of the organizers of GROW conference gives women, whether they are already graduate when it went to UIUC. She says the atmosphere at GROW, school savvy or not, a way to see themselves as having the especially the amount of time given for undergraduates to potential to have fulfilling careers in mathematics. talk with older students and professors, made them feel The organizers chose not to include presentations of comfortable approaching speakers and asking questions. undergraduate student research during the weekend in “One thing that I have always done in my career, and order to keep the focus more on the decision of whether will continue to do, is if I have to do something new, like to go to graduate school and how to get there. “It’s not an apply to grad school, or apply for this job, or apply for this undergraduate conference. It’s a conference about going to fellowship, I will always seek the counsel of older people graduate school in math,” Koch says. “If we had the under- who have done it before me,” says Sarah Koch, who co-or- graduates present their work, I think that would have kind ganized the conference when it moved to Michigan. “I think of detracted from getting together and thinking about going these conversations are really special and can make a huge to graduate school. What does that look like mathemati- difference in the direction that somebody wants to go.” cally? What does that look like practically?” Undergraduate One of the key events was the “nuts and bolts” panel presentations add a layer of stress for students, distracting about how to apply to graduate school. (The panel was them from those questions. “We wanted them to be looking so popular that the organizers made it significantly lon- forward and looking at the graduate students, looking at ger after the first year in response to feedback after the the postdocs, looking at the math talks,” Koch says. “It’s conference.) Panelists gave students information about how to apply and answered specific questions about ev- really about the next step for them instead of what they’re erything from GRE scores to graduate stipends. A panel already working on.” like that would have been helpful to Karen Smith, another A highlight of each conference for Kra was the Saturday Michigan conference organizer, as a student. “When I was evening banquet, which featured a plenary talk by an august an undergraduate, I had never heard of graduate school,” senior woman mathematician about her career journey. In she says. “Even after I had heard of it, I thought I would 2015, the speaker was Alexandra Bellow, the first tenured have to pay for it. I had never heard of getting funding for woman mathematics professor at Northwestern. Eighty graduate school.” years old, she spoke softly, even with the microphone. “The

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 725 AMS COMMUNICATION room was so silent,” Kra says. “Everybody was just glued to her stories. It was quite moving.” Evolution and GROWth As the conference moves around to different universities, each organizing committee makes their own tweaks to GROW, and the experience will continue to evolve. “I think it’s taken on its own life,” Kra says. In 2018, after hosting three GROW conferences, she and her colleagues handed the duties off to Koch, Smith, and Mel Hochster at the University of Michigan. Koch was a panelist and mentor at the 2017 conference. “It was such a positive, hopeful atmosphere,” she says. She went back to Michigan enthusiastic about continuing the tradition. She and her co-organizers adapted the struc- ture of previous GROW conferences to their own setting, modifying some parts of it to match their goals more closely. They kept Northwestern’s emphasis on one-on- one mentoring opportunities and expanded the idea by creating booklets with pictures and short biographies of the allies and mentors who were attending the conference so students could get some relevant information, such as Figure 4. Ingrid Daubechies and Emmy Murphy serve on a their institutions and research areas, before meeting them panel at the 2017 GROW conference. in person. “We only had a weekend with all these students, and we wanted them to get the most out of their experi- ence,” Koch says. The Michigan organizers also decided to bring other scientists on board as they were planning the meeting. They consulted with researchers in the university’s psychology department who study retention in STEM PhD programs. They wanted to get advice from the experts on what they could do to make the conference as effective as possible in encouraging interested women to apply for graduate school and helping them succeed when they got there. The psychologists made a few recommendations. First, they recommended that the conference include people in a variety of roles on career panels, not just professors. The organizers included panelists from government agencies, Figure 5. Dusa McDuff speaks to the 2016 GROW conference. the tech industry, and a nonprofit math education organi- zation to broaden the perspectives offered. “It wasn’t just about becoming a professor,” Smith says. They also broadened the scope of the colloquium-style talks, incorporating more applications of math to biology and social justice. In addition to standard theoretical math talks, they had a colloquium by an entrepreneur who de- signed an app that uses math to help people manage or prevent jet lag and one about gerrymandering, a hot-button political issue with connections to interesting problems and solutions in theoretical mathematics. Second, the psychologists recommended an even greater focus on early career role models, “making sure that the participants saw people that were just a little bit older than they were succeeding in those roles,” Smith says. “So instead of having a lot of programming where professors, Figure 6. Participants at the 2015 GROW conference.

726 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 AMS COMMUNICATION even female professors, were presenting their work, it was continues in her career. “It has a community effect, and it very important to have first-year graduate students present- shows you that the community can be diverse.” ing their work, people who would be just one step ahead.” Emilee Cardin knew she wanted to go to graduate school When mathematicians at UIUC were approached about when she applied for GROW in 2018. She was a student hosting GROW 2019, “of course we jumped on it,” Rapti at the College of William and Mary and had participated says. Jeremy Tyson, their department chair, attended the in an REU at the University of Michigan–Dearborn the 2018 conference as a panelist and to help prepare for summer before she attended the conference. She was eager hosting the next year. The team at UIUC adopted the same to continue studying math, but she worried that she might conference structure and added more of an emphasis on not get into the graduate programs she was interested in. helping undergraduate students learn about potential “I felt like I was here to get as much advice and learn from departments to apply to for graduate school. “They got as many people as possible,” she says. to think about particular graduate programs, and maybe Friday afternoon, before GROW even officially started, compare them and see what would be a good or a bad fit she ended up in a conversation with Koch about recom- for them,” Rapti says. mendation letters. “I had a moment with Sarah, where she The UIUC GROW conference also continued Michigan’s helped me take a breath and not be stressed out about it,” emphasis on graduate students mentoring undergraduates. she says. “She took the time to be helpful, even though I One of the most popular events that year was a mentoring wasn’t even a student there yet.” Talking with Koch, Smith, session run by the local Association for Women in Mathe- and other Michigan faculty and students at GROW influ- matics (AWM) chapter. Faculty members left the room so enced her graduate application decisions. “I think it made undergraduate and graduate students could talk privately me more confident in applying to Michigan,” Cardin says. about the graduate application process. “It was a huge “I felt like it would be a more welcoming place after I went success,” Rapti says. “People just wouldn’t leave the room.” to GROW. It seemed less unapproachable.” Cardin is now in her first year of graduate work at Mich- The Student Experience igan. When we talked on the phone for this article, she was Priyanka Nanayakkara attended the first GROW conference standing in the atrium where another favorite moment at as a sophomore at UCLA. She enjoys both math and writing GROW took place. She was sitting with a Boston College and wanted to find a way to combine those interests, even- graduate student for lunch. “I remember her talking about tually deciding on statistics. “I was so focused on figuring how she loved it and she was so happy to have the people out my major in undergrad, but the time to apply for grad she was with [in graduate school], and I thought, ‘Yes, that’s school comes pretty quickly, so I think it was helpful to what I’d like.’” She says the graduate school application hear, ‘You could really do this. It’s not out of your reach. process had felt like a constant competition, but talking You’re capable of it,’” she says. with graduate students and faculty members about their Nanayakkara was impressed at the generosity of the experiences helped her prioritize the community she would mentors she talked with at the conference and believes they find in graduate school and beyond. The Boston College helped her see a future as an academic. “I knew it existed. I student told her about how she was working to prioritize knew I wanted to go to grad school, but GROW helped me mental health in graduate school and how her cohort had actually do it,” she says. “That space to really feel confident helped her out when she needed it. “I hear those words was so useful, and I don’t think it can be understated how ringing in my ears a lot,” Cardin says. valuable that is.” Nanayakkara is now working on a PhD in technology Work/Life and social behavior at Northwestern as part of a joint pro- “Work-life balance” can be a bit of a cliché, but the fact is gram between the computer science department and com- that people of all genders have to consider how they will munication studies. “I go to Northwestern now, and every manage the demands of their careers, their other interests time I walk past Tech [Technological Institute, a building on and hobbies, and the needs of their partners or family campus], I always think about eating lunch right there with members when they are considering their potential career all the people from GROW,” she says. “I think about that paths. Despite changes in American society in the past few conference as a turning point in my early college career.” decades, women still spend more time on childcare and She also feels like GROW helped her understand the caregiving for infirm relatives than men do, and they are importance of community in her academic discipline. “It more likely to face negative repercussions for it in profes- was a really important step in introducing the idea of a co- sional settings. hort: these are the people who will be progressing with you It’s no surprise, then, that conversations about family throughout your life and career, and you’ll see them again and career are common at GROW. “More 20-year-old and again,” Nanayakkara says. GROW’s demographics give women think about having a family than 20-year old men, her hope that she can be part of an inclusive cohort as she or at least that is my impression,” Getzler says. GROW

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Figure 7. Students participate in a workshop at the 2015 GROW Figure 8. Laura De Marco talks about her work at the 2015 conference. GROW conference. provides a safe environment for women to talk with other Moreover, men need to be part of the solution if the women, both fellow students and women who are further proportion of women mathematicians is going to change. along in their careers, about how they have managed or “By not including male grad students, we would miss out plan to manage the sometimes competing demands of on future vectors for this kind of program,” Getzler says. career and family. “These things have to be thought over “But we indeed have male grad students—who have now a bit more deeply when we’re discussing women grad stu- become junior faculty—who have taken this level of activ- dents,” Getzler says. ism across the country.” Dusa McDuff and Ingrid Daubechies gave plenary ad- Kra also hoped the conference would provide a valuable dresses at GROW in 2016 and 2017, respectively. McDuff change of perspective to men who participated. “The men talked about taking time off from her career to raise chil- came away with a lot from it,” she says. “A colleague said dren, and Daubechies spoke about weathering unexpected to me, ‘You know, I never knew what it felt like to be in events in life that may mean you have to put your work on the minority, and that was a great feeling to understand.’” the back burner. “I think the students were really impressed Getzler agrees that GROW has encouraged this kind of by listening to these senior women talking about things introspection among men who have participated. “I think a that they also worry about,” Kra says. huge problem is that women have a tendency to be silenced if there are men in the room. And part of the power of the Cardin got to the conference preoccupied with getting program is that now, since women are the vast majority, it’s into graduate school but says the panels and career talks a corrective for the men in the room, to remind them that helped her think about the process more holistically. “It being a little bit quieter could be a good fit for everybody, was so good to hear you can be a human as well as being that voices which aren’t always encouraged to take charge in a math graduate program,” she says. do so,” Getzler says. What About the Men? One of the criticisms Kra read in a post-conference survey after the first conference was about the number of Kra deliberately included men as panelists and mentors men involved. “For the future, I addressed that straight when she started GROW. Women are disproportionately up from the beginning, saying men are involved because asked to participate in service and outreach events in STEM there are many men actively working to change who’s in fields. Even if the events are important and the women the room. We can’t do it on our own, and it shouldn’t just enthusiastic about participating, the volume of service be on women,” she says. “And we never got another com- requests can be a burden. “I did not want the conference plaint about it.” to be more of a ‘service tax’ on women and other people from underrepresented groups,” she says. Broader Impacts The organizers also wanted students to be able to inter- GROW is still a young conference. As such, there are no act with a diverse group of panelists and mentors. “One of longitudinal studies tracking its effect on participants, the things [Kra] was very firm about is that these programs though surveys done before and after each conference do should expose prospective graduate students to a realistic show positive effects. Kra and other organizers believe it atmosphere of what graduate school will be like,” Getzler has already made a difference, not only to participants, but says. “For that reason, she did not want the weekend to be also to the schools that have hosted the conference and the heavily dominated by contact only with other women.” broader profession.

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As is the case with most conferences, person-to-person schools. “This is not any one department’s problem; it’s a networking is one of the most valuable aspects. “GROW is problem in the field,” Kra says. “If we fix it at Northwestern exactly what it sounds like. It’s GROW-ing a network, which and get to parity, well that’s great, but it doesn’t do much I think is really awesome,” Koch says. for the field. I would rather see things fixed more broadly.” Attendees have expanded their professional networks, creating connections with both their peers and more senior women mathematicians. “I’m still friends with some of the people I met at GROW,” Cardin says. GROW has also helped connect attendees to other ex- isting programs and conferences for mathematicians. Nan- ayakkara learned about the AWM at GROW. When she got back to LA, she started looking up more information about the organization and learned about its annual essay contest for students. Her essay about Loyola Marymount University mathematician Alissa Crans won the undergraduate contest Evelyn Lamb in 2016. “Before GROW, I don’t think I really knew about the Association for Women in Mathematics. That was a Credits direct result [of attending the conference].” Figures 1–4 are courtesy of Mike Jue. Cardin met mathematician Ruthi Hortsch, who works Figures 5–8 are courtesy of the Department of Mathematics, Northwestern University. at Bridge to Enter Advanced Mathematics (BEAM), when Author photo is courtesy of the author. she attended GROW. BEAM is a math education nonprofit organization for middle school students from underserved See the official announcement and citation for the communities in New York City and Los Angeles. It provides 2020 AMS Award for Mathematics Programs that Make enrichment for math-interested students and helps them a Difference on page 722. progress in mathematics as they continue their education, primarily through summer programs. “BEAM is probably one of the coolest math-related things I’ve ever done,” Cardin says. After learning about BEAM from Hortsch, she spent part of the summer before starting graduate school in New York working at BEAM, and she has plans to join them again this year. Meeting When was the last time you visited the Hortsch at GROW gave her “an immediate connection to something I fell in love with,” she says. Kra and other GROW organizers have also found that AMS Bookstore? the conference has a positive effect on their departments. “I think it really did change the culture of the department,” Kra says. Northwestern is creating a post-baccalaureate Spend smart. program for students from underrepresented groups in mathematics. “As much as it’s a problem that there are very few women, it’s a much bigger problem that there are so few people from other underrepresented groups,” she Search better. says. A year-long program is very different from a weekend conference, but they are using some of their lessons from GROW as they shape the new program. Stay informed. Both Northwestern and Michigan have seen the number of women who have applied to graduate school increase. “Our department got a lot out of it in many different ways, one of which was that we had this large number of really bookstore.ams.org strong, really qualified female graduate student applicants last year,” Koch says. “I wasn’t really thinking of it as re- cruiting for Michigan per se,” Smith says. “That was more of a side benefit for us.” Kra says the effect the conference has had at Northwest- ern is a reason it should continue to travel to different

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Mathematics People

renamed to honor the late Maryam Mirzakhani. It honors Guth Awarded exceptional contributions to the mathematical sciences by a midcareer mathematician. It carries a cash award of Mirzakhani Prize US$20,000. Larry Guth of the Massachusetts Institute of Technology has been —From an NAS announcement awarded the newly named Maryam Mirzakhani Prize in Mathematics “for developing surprising, origi- 2020 AWM Prizes Awarded nal, and deep connections between The Association for Women in Mathematics (AWM) pre- geometry, analysis, topology, and sented several awards at the Joint Mathematics Meetings combinatorics, which have led to held in Denver, Colorado, in January 2020. the solution of, or major advances Erika Camacho of Arizona State Larry Guth on, many outstanding problems University was honored with the Lou- in these fields.” The citation reads: ise Hay Award for Contribution to “Guth has made spectacular contributions to many areas Mathematics Education “in recogni- of mathematics, including systolic geometry, analysis, and tion of her leadership and contribu- combinatorics. He has developed surprising, original, and tions as a mathematical scholar and deep connections between geometry, analysis, topology, educator.” The prize citation reads: and combinatorics, leading to major advances or solutions “Dr. Camacho has a passion for men- for many outstanding problems in these fields. His accom- toring, especially the mentoring of plishments include the introduction of a new cell decom- underrepresented students. Her men- Erika Camacho position of Euclidean space, writing the authoritative book toring begins with her excitement for on the polynomial method, and creating a new induction mathematics based in her research in on scales algorithm called the Bourgain-Guth method.” mathematical physiology. This research involves develop- Guth received his PhD from MIT in 2005 under the ing mathematical models that describe the interactions of supervision of Tomasz Mrowka. He held a postdoctoral photoreceptors in the retina. Dr. Camacho brings graduate position (2005–2006) and an assistant professorship and undergraduate students into her research and also finds (2006–2008) at . He was assistant opportunities for students with other researchers. professor (2008–2011) at the University of Toronto, a “She created the Applied Mathematical Sciences Summer member of the Institute for Advanced Study (2010–2011), Institute and has codirected both this institute (2004–2007) and professor at the Courant Institute of New York Uni- and the Mathematical and Theoretical Biology Institute versity before joining MIT in 2012. His honors and awards (2011–2013). Through these institutes and her other men- include an NSF Postdoctoral Fellowship (2006–2008), a toring programs she has impacted over 600 undergraduates, Sloan Fellowship (2010), the Salem Prize (2013), the MIT including supervising the research of 89 of these students, School of Science Prize for Excellence in Graduate Teaching with 30 receiving conference award recognitions. (2015), a Clay Research Prize (with Nets Katz, 2015), the “Through her work Dr. Camacho changes perceptions. New Horizons in Mathematics Prize (2015), and the AMS Her own story is an existence proof that someone from an Bôcher Prize (2020). He was named a Simons Investigator underprivileged and Latina background can earn a PhD in 2014 and is a Fellow of the AMS and of the American in mathematics and be a successful mathematician. In Academy of Arts and Sciences. He is the author of the book over sixty-five plenary and panel presentations, she uses Polynomial Methods in Combinatorics. her story to inspire students to persevere and succeed in The prize, awarded by the National Academy of Sciences mathematics. Beyond presenting, Dr. Camacho meets and formerly called the NAS Award in Mathematics, was

730 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Mathematics People NEWS with attendees individually afterwards to learn about their stories and give them advice based on their own interests 2020 MAA Awards and passions. By inspiring more women and members of The Mathematical Association of America (MAA) awarded underrepresented groups to continue in their mathematical several prizes at the Joint Mathematics Meetings in Denver, pursuits, she enlarges the scope of what we see as successful Colorado, in January 2020. mathematicians.” Camacho received her PhD in 2003 from Vladimir Pozdnyakov of the Uni- Cornell University under the direction of Richard H. Rand. versity of Connecticut and J. Michael Among her many recognitions are the SACNAS Distin- Steele of the University of Pennsyl- guished Undergraduate Institution Mentor Award (2012), vania were awarded the Chauvenet the Outstanding Latino/a Faculty in Higher Education: Prize for their article “Buses, Bullies, Research/Teaching in Higher Education (Research Insti- and Bijections,” Mathematics Maga- tutions) (2018), the Presidential Award for Excellence in zine 89 (2016), no. 3. The prize cita- Science, Mathematics, and Engineering Mentoring (2014), tion reads in part: “Pozdnyakov and and the American Association for the Advancement of Sci- Steele show the remarkable utility ence Mentor Award (2019). She tells the Notices: “I grew Vladimir Pozdnyakov of bijections by considering seating up in East Los Angeles, the fourth of five children, where I assignments on a bus. Everyone has was taught by Jaime Escalante (of Stand and Deliver fame). I a designated seat, but all except the enjoy spending time with my husband and three children.” last passenger take seats at random. Margaret Robinson of Mount Then the final passenger—a bit of a Holyoke College has been named bully—boards, not only wanting his the recipient of the 2020 M. Gweneth own seat, but demanding that each Humphreys Award for Mentorship subsequently displaced person finds of Undergraduate Women in Math- his correct seat as well. What is the ematics. According to the prize cita- probability that the first person to board will need to change seats? tion, “Margaret Robinson has been J. Michael Steele a mainstay of caring and thoughtful “The authors obtain the answer teaching and mentoring for many via a brute-force combinatorial argu- years at Mount Holyoke College, ment, but then find the solution in an easier, more reveal- Margaret Robinson an institution whose mission is to ing way by making elegant use of permutation cycles. The educate women. Her focus is not authors then use bijections to derive even more surprising and beautiful results including the mean and variance of just on the top students but on making a meaningful (and the number of cycles in a random permutation. This well- joyful) mathematical intervention for all the generations crafted paper, which introduces the reader to the theory of of learners that have crossed her path. As one student put permutation patterns, flows naturally and easily, providing it, ‘she saw me in a way that no mathematics teacher had a journey that is interesting and insightful. This bus is before.’ Her impactful involvement in the Carleton Summer available for all—professor and student alike—delighting Math Program and the resounding response from a range of the rider with the simple power of bijections.” former mentees speak to her effectiveness and her ability to Pozdnyakov received his PhD from the University of forge personal connections.” Robinson received her PhD in Pennsylvania in 2001 under the supervision of J. Michael 1986 from Johns Hopkins University under the supervision Steele. He is currently professor of statistics and director of Jun-Ichi Igusa. Her honors include the Mount Holyoke of the Applied Financial Mathematics graduate program at Faculty Teaching Award (2010), the NES/MAA Award for Connecticut. He tells the Notices: “I’m an enthusiastic soccer Distinguished Teaching of Mathematics (2012), and the player—an old one.” J. Michael Steele received his PhD MAA Haimo Award for Distinguished Teaching of Math- from Stanford University in 1975 under Kai Lai Chung. ematics (2013). Robinson tells the Notices: “My favorite He is currently professor emeritus at the University of quote (that I tell my students and children) comes from Pennsylvania. Steele and Pozdnyakov were jointly awarded The Once and Future King by T. H. White: ‘“The best thing the MAA’s Carl Allendorfer Award in 2017. Steele tells the for being sad,” replied Merlin, beginning to puff and blow, Notices: “I’m now retired from teaching, but I am still in- “is to learn something. That’s the only thing that never fails. volved in writing. I am also passionate about languages … Learning is the only thing for you. Look what a lot of and language learning. French is in focus for the moment, things there are to learn.”’” and most recently it absorbs four or more hours of my day.” Aubrey D. N. J. de Grey of SENS Research Foundation —From AWM announcements and AgeX Therapeutics has been awarded the 2020 David P.

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Robbins Prize for his article “The Chromatic Number of the across the United States, as well as in Panama, South Korea, Plane Is at Least 5,” Geombinatorics 28 (2018), no. 1, which Japan, and Holland.” addresses the question, What is the minimum number of The 2020 Deborah and Franklin Tepper Haimo Awards colors needed to color the points of a Euclidean plane so for Distinguished College or University Teaching of Math- that no two points at distance exactly 1 have the same color? ematics were awarded to Federico Ardila of San Francisco This is “often known as the Hadwiger–Nelson problem; State University, Mark Tomforde of the University of Hadwiger, several years earlier and for other reasons, had Houston, and Suzanne L. Weekes of Worcester Polytechnic been the first to discuss the simplest coloring of the plane Institute. that demonstrates the upper bound.” De Grey received Ardila was recognized for inspiring students “from all his PhD in biology from the . His walks of life to recognize and realize their potential in research interests encompass the characterization of all the mathematics.” He is a “leader in the movement to broaden types of self-inflicted cellular and molecular damage that and deepen diversity in research mathematics.” He is constitute mammalian aging and the design of interven- director of the Mathematical Sciences Research Institute– tions to repair and/or obviate that damage. He is particu- Undergraduate Program (MSRI-UP), the largest Research larly interested in combinatorics, especially graph theory. Experiences for Undergraduates (REU) program in the Tim Chartier of Davidson College United States and the one that serves the largest number was awarded the 2020 Euler Book of students from underrepresented groups. He conceived Prize for Math Bytes (Princeton Uni- the SFSU-Colombia Combinatorics Initiative, through versity Press, 2014). The prize citation which he developed seven new courses to promote interna- reads: “Math Bytes gives readers a taste tional scholarly collaboration among undergraduates and of the mathematics and computing master’s students, including many from underrepresented applications that underlie many as- groups, at SFSU and the Universidad de Los Andes. He has pects of everyday life. With a wide published a wide range of expository articles in English, array of topics—including fractals, Spanish, and German. His YouTube channel contains more fonts, tweets, basketball, Google, dig- than 240 hours of freely available advanced mathematics, Tim Chartier ital images, movies, and more—the and his viewers come from over 150 countries. His article book exposes readers to a satisfying “Todos Cuentan: Cultivating Diversity in Combinatorics” assortment of mathematical ideas, many of which will be was published in the Notices in November 2016. Ardila new to nonmathematical audiences. That said, even more received his PhD from the Massachusetts Institute of Tech- mathematically inclined readers should find plenty of in- nology in 2003 under the direction of Richard P. Stanley. teresting material, including new ways of thinking about He is a Fellow of the AMS and of the Simons Foundation and applying familiar mathematical concepts. Chartier’s and the recipient of an NSF CAREER Award and of the exposition is clear, accessible, and fun. Regular challenge Premio Nacional de Ciencias and the Premio Nacional de problems encourage readers to explore for themselves Matemáticas in Colombia. His research is in combinatorics the ideas introduced in the text. All in all, Math Bytes is and its connections to geometry, algebra, topology, and an engaging and stimulating read that is sure to broaden applications. He enjoys reading, fútbol, playing records, or horizons and increase appreciation for the ubiquitous and playing marimba de chonta. invaluable role of computational mathematics in modern Tomforde “has had a deep and society.” Chartier received his PhD from the University positive impact at all levels of math- of Colorado, Boulder. He specializes in numerical linear ematics education.” According to algebra, with his recent work focusing on data science. He the prize citation, he has “recruited, has been a consultant on data analytics for ESPN, the New retained, and mentored members York Times, the US Olympic Committee, and teams in the of underrepresented groups spec- NBA, NFL, and NASCAR. He was the first chair of the Ad- tacularly” at all levels, including by visory Council for the National Museum of Mathematics. enrolling over seventy University In K–12 education, he has worked with Google and Pixar of Houston students in the Math on their educational initiatives. He was the recipient of the Alliance, the goal of which is to en- Mark Tomforde MAA Daniel Solow Author’s Award in 2019. Chartier tells sure that every underrepresented or the Notices: “My wife and I have professional training in underserved American student with mime, which includes master classes with Marcel Marceau. talent and ambition has the opportunity to earn a doctoral In fact, we paid for our wedding just over twenty-five years degree in a mathematical science. He is a cofounder and ago with a performance tour along the East Coast of the coorganizer of Gulf States Math Alliance (GSMath), one United States. We also have developed a mime show that of seven regional alliances, composed of members of the introduces mathematical ideas and have performed it Math Alliance in Texas, Louisiana, and Mississippi. He

732 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Mathematics People NEWS facilitates and promotes associated opportunities in the community to mathematics and statistics work done in Gulf Coast region. He developed the Cougars and Houston business, industry, and government, and with broadening Area Math Program (CHAMP), working in collaboration the participation and success of students in mathematical with neighborhood high schools and middle schools to sciences. provide a wide variety of mathematical activities. CHAMP Gerald J. Porter of the Univer- received the AMS Award for Mathematics Programs That sity of Pennsylvania received the Make a Difference in 2018, as well as a Phi Beta Kappa 2020 Gung and Hu Award for Dis- award for broadening participation in STEM. He developed tinguished Service to Mathematics a multifaceted collaboration between the University of for his service in “teaching, teacher Houston and Texas Southern University, recruited faculty education, research, MAA adminis- members from Houston as Math Alliance members, has tration, and, most importantly of all, served as a Project NExT consultant, and maintains mul- in leading the profession, especially tiple websites with a wide variety of materials for faculty the MAA, to value racial and gender and students. Tomforde received his PhD from Dartmouth diversity in all activities.” The prize College under the supervision of Dana P. Williams in 2002, Gerald J. Porter citation reads in part: “Jerry Porter held a postdoctoral fellowship at the University of Iowa, and has also taught at the College of William and Mary. has spent decades in service to the MAA. His service in He was a Project NExT Fellow in 2002. He tells the Notices: terms of years and variety at the national level is extensive “I am a cinephile, and in my spare time I enjoy watching a but his service and care for the organization goes far be- wide variety of movies from different genres.” yond the lengthy list of committees on which he served Weekes “has had an extraordinary and positions he has held. His is the service that, while impact on the mathematics commu- not appearing on any list, has made the difference in the nity via superlative teaching, advis- MAA and our profession. He pursued this service while ing, and mentoring of students and providing strong support to Executive Directors, learning faculty at Worcester Polytechnic In- and sharing his great expertise, and being a change agent stitute (WPI), regionally, and nation- in the areas in which he was involved. He has been a men- ally.” She designed and organized the tor to many young mathematicians and has nominated Applied and Industrial Mathematics them for awards and committees, welcomed them at both Institute for Secondary Teaching at section and national meetings, and shown by example WPI, which offers workshops for Suzanne L. Weekes the importance of inclusivity. For many years he was the high school mathematics teachers. only male member of the Joint Committee on Women; as As a member of the Math Advisory always, Jerry strengthened MAA’s role on this committee. Group of Transforming Postsecondary Education in Math- Jerry welcomed the women and minorities who attend our ematics (TPSE Math), she coorganized and hosted the New meetings and encouraged them to take an active role in the England Regional Meeting on Upper-Division Pathways at Association.” With Jim White, he directed the Interactive WPI, now a model for such workshops in other regions. Mathematics Text Project, which funded the creation of She directed the Center for Industrial Mathematics and computer laboratories in six colleges to encourage the Statistics and also directed its WPI Research Experiences for creation of computer-based algebra materials in teaching. Undergraduates (REU) Program in Industrial Mathemat- Porter received his PhD from Cornell University in 1963 ics and Statistics. She is a cofounder of the Mathematical under the direction of William Browder. He is a life mem- Sciences Research Institute Undergraduate Program (MS- RI-UP) and of the MAA’s Preparation for Industrial Careers ber of the AMS and the MAA. Porter has been retired from in Mathematical Sciences (PIC Math) program, which has teaching since 2006. He enjoys traveling and has visited increased awareness of nonacademic career options and about seventy-five countries and all fifty states in the United preparing students for industrial careers. Weekes grew up States. He is an avid photographer and has had six photo in the Republic of Trinidad and Tobago, received her BS in shows at Penn. Since his retirement he has audited courses mathematics from Indiana University, and earned her PhD and seminars at Penn, including courses in ethnomusi- in 1995 from the University of Michigan. She was awarded cology, art, and literature. In June, he and his wife, Judy, the M. Gweneth Humphries Award of the Association for will celebrate their sixtieth wedding anniversary. Together Women in Mathematics (AWM) in 2019. Her research they funded the public lecture given each year at the Joint work is in numerical methods for differential equations, Mathematics Meetings with the goal of increasing public including applications to spatiotemporal composites/ awareness and appreciation of mathematics. dynamic materials and cancer growth. She is involved in several initiatives connecting the academic mathematics —From MAA announcements

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 733 Mathematics People NEWS Bombieri Awarded Crafoord Legatiuk Awarded Prize in Mathematics Clifford Prize

Enrico Bombieri of the Institute for Dmitrii Legatiuk of Bauhaus-Univer- Advanced Study has been awarded sität Weimar has been selected as the the 2020 Crafoord Prize in Mathe- recipient of the 2020 W. K. Clifford Prize for his “significant contribu- matics “for outstanding and influ- tions in Clifford analysis, including ential contributions in all the major interpolation of monogenic func- areas of mathematics, particularly tions, quaternionic operator calculus, number theory, analysis and alge- and construction of advanced numer- braic geometry.” The prize is awarded ical methods.” Legatiuk’s “interest by the Royal Swedish Academy of in Clifford analysis, particularly its Enrico Bombieri Dmitrii Legatiuk Sciences and the Crafoord Founda- potential to solve difficult applied tion; the disciplines rotate every year. The prize carries a problems, has led him to such advances as using quaterni- cash award of 6 million Swedish krona (approximately onic operator calculus to construct representation formulas US$618,000). for solutions of boundary value problems in advanced elasticity theories, interpolation of monogenic functions The prize citation reads in part: “Enrico Bombieri be- by various tools, and developing a finite element exterior longs to an increasingly rare group of mathematicians who calculus based on script geometry. His interests span math- can solve problems in almost all areas of mathematics. ematics, computer science, and engineering, reflecting the However, his greatest passion has always been number broad applicability of Clifford algebras and echoing the theory, which is the study of integers. He was just sixteen wide-ranging interests of W. K. Clifford himself.” Legatiuk years old when he published his first work in number the- earned his PhD from Bauhaus-Universität Weimar, where ory and, among other things, he is a leading expert on the his doctoral research earned him the 2015 University Prize Riemann hypothesis on the distribution of prime numbers. for Young Scientists. “Enrico Bombieri has made significant contributions in The W. K. Clifford Prize is an international scientific algebra, advanced geometry, and complex analysis. He has prize intended to encourage young researchers to compete for excellence in research in theoretical and applied Clifford also contributed to solving Bernstein’s problem. This is a algebras and their analysis and geometry. It is awarded every variation of Plateau’s problem, about how to mathemati- three years at the International Conference on Clifford cally describe the shape of the soap film that forms when Algebras and Their Applications in Mathematical Physics, a wire frame is dipped into a soap solution.” held this year in Hefei, China. Bombieri was born in 1940 in Milan, Italy, and received his PhD in 1963 from the Università degli Studi di Milano. —G. Stacey Staples He has been professor at the University of Pisa (1966– Southern Illinois University 1974) and Scuola Normale Superiore, Pisa (1974–1977). He joined the faculty at the IAS in 1977, where he is now Chuzhoy Awarded professor emeritus. He was awarded the Fields Medal in 1974. His honors also include the Feltrinelli Prize (1976), NAS Held Prize the Balzan Prize (1980), the Cavaliere di Gran Croce al Merito della Repubblica, Italy (2002), the Premio Internazi- Julia Chuzhoy of Toyota Technological Institute, Chicago, onale Pitagora (2006), the AMS Joseph Doob Prize (2008), has been named the recipient of the 2020 Michael and the King Faisal International Prize (2010), and the Lifetime Sheila Held Prize of the National Academy of Sciences. According to the prize citation, Chuzhoy “has conducted Achievement Award of the Italian Scientists and Scholars influential work in the fields of graph algorithms, hardness of North America Foundation (2015). He is a member of of approximation, and structural graph theory, which have the American Academy of Arts and Sciences, the National introduced powerful new techniques and resolved deep Academy of Sciences, and the Royal Swedish Academy of open questions. Arts and Sciences, among many others. “Chuzhoy and her coauthors achieved remarkable re- sults in designing algorithms for graph routing problems, —From a Royal Swedish Academy announcement which are among the most studied and important problems

734 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Mathematics People NEWS in optimization. Insights from this work led to further made major contributions. The prize committee consisted significant impacts on structural graph theory, including of A. Basmajian, Y. Movsisyan, and V. Pambuccian. an exponential strengthening of the parameters of the Excluded Grid theorem. —Victor Pambuccian “Chuzhoy’s work on graph routing problems settled cen- New College, Arizona State University tral open questions in graph optimization and introduced powerful new graph decomposition and routing tech- niques, opening up the potential for future applications in Borodin and Petrov Awarded algorithm design and structural graph theory. The improved parameters for the Excluded Grid theorem led to faster al- 2020 Bernoulli Prize gorithms for a host of graph optimization problems, and stronger bounds for a number of graph theoretic results.” Alexei Borodin of the Massachusetts Institute of Technol- The prize carries a cash award of US$100,000. It honors ogy and Leonid Petrov of the University of Virginia have outstanding, innovative, creative, and influential research been awarded the 2020 Bernoulli Prize for an Outstand- in the areas of combinatorial and discrete optimization, or ing Survey Article in Probability or Statistics. They were related parts of computer science, such as the design and honored for their article “Integrable Probability: From analysis of algorithms and complexity theory. Representation Theory to Macdonald Processes,” Probabil- ity Surveys 11 (2014). The prize recognizes authors of an —From an NAS announcement influential survey publication in the areas of probability and statistics.

Borodin and Viazovska —Bernoulli Society announcement Awarded Fermat Prize National Academy of Alexei Borodin of the Massachusetts Institute of Technol- ogy and Maryna Viazovska of the École Polytechnique Engineering Elections Fédérale de Lausanne have been awarded the 2019 Fermat Prize for research in mathematics. Borodin was honored for The National Academy of Engineering (NAE) has elected the invention of integrable probability theory, a new area at eighty-seven new members and eighteen international the interface of representation theory, combinatorics, and members for 2020. Below are the new members whose statistical physics. Viazovska was honored for her original work involves the mathematical sciences. solution of the famous sphere packing problem in dimen- • Graham V. Candler, University of Minnesota, Minneap- sions 8 and 24. The prize rewards mathematicians under olis, for development and validation of computational forty-five years old whose research works are in number models for high-fidelity simulation of supersonic and theory, analytic geometry, probability, and research related hypersonic interactions. to the variational principles. • Kenneth C. Hall, Duke University, for development of —Fermat Prize announcement unsteady aerodynamic and aeromechanics theories and analysis for internal and external aerodynamic flows. • Mrdjan Jankovic, Ford Motor Company, for contri- Haykazyan Awarded butions to nonlinear control theory and automotive technology. Junior Prize • Sallie Ann Keller, University of Virginia, Charlottesville, for development and application of engineering and Levon Haykazyan of Oxford Asset Management has been awarded the 2020 Emil Artin Junior Prize in Mathematics. statistical techniques in support of national security Haykazyan was chosen for his paper “Spaces of Types in and industry. Positive Model Theory,” Journal of Symbolic Logic 84 (2019). • Ioannis G. Kevrekidis, Johns Hopkins University, for Established in 2001, the Emil Artin Junior Prize in Math- research on multiscale mathematical modeling and ematics carries a cash award of US$1,000 and is presented scientific computation for complex, nonlinear reaction, usually every year to a student or former student of an Ar- and transport processes. menian educational institution under the age of thirty-five • Tamara G. Kolda, Sandia National Laboratories, for con- for outstanding contributions to algebra, geometry, topol- tributions to the design of scientific software, including ogy, and number theory—the fields in which Emil Artin tensor decompositions and multilinear algebra.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 735 Mathematics People NEWS

• Muriel Médard, Massachusetts Institute of Technology, • Ila Varma, University of Toronto for contributions to the theory and practice of network • Cynthia Vinzant, North Carolina State University coding. • Alexander Wright, University of Michigan • Jorge Nocedal, Northwestern University, for contri- • Yao Yao, Georgia Institute of Technology butions to the theory, design, and implementation of • Zhizhen Zhao, University of Illinois, Urbana-Cham- optimization algorithms and machine learning software. paign • Alexander A. Shapiro, Georgia Institute of Technology, for contributions to the theory, computation, and ap- —From a Sloan Foundation announcement plication of stochastic programming. • Peter W. Shor, Massachusetts Institute of Technology, Credits for pioneering contributions to quantum computation. Photo of Larry Guth is courtesy of the MIT Department of • Charles W. Wampler II, General Motors Corporation, Mathematics. Photo of Vladimir Pozdnyakov is courtesy of Zhanna Pozd- for leadership in robotic systems in manufacturing, nyakova. mathematical methods for robot motion and machine Photo of J. Michael Steele is courtesy of Catalin Starica. design, and traction battery modeling. Photo of Tim Chartier is courtesy of Davidson College. • Elected as an international member was Wolfgang Mar- Photo of Suzanne Weekes is courtesy of Worcester Polytech- quardt, Forschungszentrum Jülich GmbH, Germany, nic Institute. Photo of Gerald J. Porter is courtesy of Felice Macera. for contributions to process systems engineering and Photo of Enrico Bombieri is courtesy of Cliff Moore. large-scale computations and for national leadership in Photo of Dmitrii Legatiuk is courtesy of Dmitrii Legatiuk. science/technology policy and management.

—From an NAE announcement 2020 Sloan Fellows The Alfred P. Sloan Foundation has announced the names of 126 recipients of the 2020 Sloan Research Fellowships. Each year the foundation awards fellowships in the fields of mathematics, chemistry, computational and evolutionary molecular biology, computer science, economics, neurosci- JOIN THE ence, physics, and ocean sciences. Grants of US$75,000 for 1 a two-year period are administered by each Fellow’s insti- AMS tution. Once chosen, Fellows are free to pursue whatever lines of inquiry most interest them, and they are permitted BOOKSTORE to employ fellowship funds in a wide variety of ways to further their research aims. Following are the names and institutions of the 2020 EMAIL LIST awardees in the mathematical sciences. Be the fi rst to learn about: • Jeff Calder, University of Minnesota • Publishing highlights • Roger Casals, University of California, Davis • Otis Chodosh, Stanford University • New titles being released • Damek Davis, Cornell University • Monthly sales • Tarek M. Elgindi, University of California, San Diego • Special discounts on AMS publications • Peter Hintz, Massachusetts Institute of Technology • Robert Hough, Stony Brook University • Hao Huang, Emory University SIGN UP AT • Sebastian Hurtado-Salazar, University of Chicago • Aleksandr Logunov, Princeton University ams.org/bookstore-email-list. • Linquan Ma, Purdue University • Sung-Jin Oh, University of California, Berkeley • Weijie Su, University of Pennsylvania • Omer Tamuz, California Institute of Technology • Samuel Taylor, Temple University

736 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 NEWS

Community Updates

AMS Department Every Generation Helps Chairs Workshop the Next In 2017, an energetic group of twenty mathematicians at the American Mathematical Society launched a special fundraising campaign to create a new endowment called the Next Generation Fund. The goal of the Next Generation Fund is to support hundreds of early career mathematicians each year at modest but impactful levels. Many donors contributed toward the $1.5 million matching gift challenge with donations that ranged from one dollar to more than two hundred thousand. Thanks to their personal commitment and community spirit, the campaign raised over three million dollars to inaugurate the Fund. The AMS held its annual Department Chairs Workshop on Initially, the Next Generation Fund will support pro- January 14, 2020, in Denver, Colorado, just prior to the grams such as travel grants to AMS meetings, Mathematics start of the 2020 Joint Mathematics Meetings. Research Communities, Child Care Grants, and Student Workshop leaders were: Luca Capogna, Head, Depart- Chapters. It is designed to be flexible over time to meet ment of Mathematical Sciences, Worcester Polytechnic the changing needs of mathematicians as they begin their Institute (WPI); Kevin Knudson, Chair, Department of professional careers. Mathematics, University of Florida; Gloria Marí-Beffa, As- With the inaugural campaign now accomplished, the sociate Dean for the Natural, Physical and Mathematical Next Generation Fund will continue to be a fundraising Sciences, University of Wisconsin–Madison; and Jennifer priority at the AMS. All members of the mathematics Zhao, Associate Dean for the College of Arts, Sciences and community are welcomed and encouraged to support it Letters, University of Michigan–Dearborn. through their ongoing donations. For more information, What makes a chair different from any other engaged contact the Development staff at 401-455-4111, or visit www faculty member in the department? This workshop exam- .ams.org/support. ined the chair’s role in leading a department. The day was structured to include and encourage networking and shar- —Robin Marek ing of ideas among participants. There were four sessions: • Modernizing mathematics and mathematicians • Evaluating teaching Deaths of AMS Members • Difficult conversations K. CHANDRASEKHARAN, of Switzerland, died on April 13, • The “entrepreneurial” mathematics department. 2017. Born on November 21, 1920, he was a member of The 2020 workshop was attended by sixty-eight department the Society for 69 years. chairs and leaders from across the country. HARRY D. HUSKEY, of Santa Cruz, California, died on April 9, 2017. Born on January 19, 1916, he was a member —Anita Benjamin of the Society for 74 years. AMS Office of Government Relations Credit Photo is courtesy of Anita Benjamin.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 737 NEWS

Mathematics Opportunities

Listings for upcoming mathematics opportunities to appear in Notices may be submitted to [email protected].

Early Career Opportunity Early Career Opportunity NSF Postdoctoral NSF Research Training Groups Research Fellowships in the Mathematical Sciences

The National Science Foundation (NSF) awards Mathemat- The National Science Foundation (NSF) Research Training ical Sciences Postdoctoral Research Fellowships (MSPRF) Groups in the Mathematical Sciences (RTG) program pro- in all areas of the mathematical sciences, including appli- vides funds for the training of US students and postdoctoral cations to other disciplines. Awards are either Research associates through structured research groups that include Fellowships or Instructorships. The Research Fellowship vertically integrated activities spanning the entire spectrum of educational levels from undergraduate through postdoc- provides full-time support for any eighteen academic-year toral. The deadline for full proposals is June 2, 2020. See months in a three-year period. The Research Instructorship www.nsf.gov/funding/pgm_summ.jsp?pims_id=5732. provides either two academic years of full-time support or one academic year of full-time and two academic years —NSF announcement of half-time support. The deadline for proposals is Oc- tober 21, 2020. See www.nsf.gov/funding/pgm_summ Early Career Opportunity .jsp?pims_id=5301&org=NSF. International Mathematics —NSF announcement Competition for Early Career Opportunity University Students

NRC Research Associateship Programs The Twenty-seventh International Mathematics Competi- tion for University Students will be held July 24–30, 2020, The National Academy of Sciences, Engineering, and Med- at American University in Blagoevgrad, Bulgaria. Students icine offers postdoctoral and senior research awards on completing their first, second, third, or fourth years of behalf of twenty-three US federal research agencies and university education are eligible. See www.imc-math.org affiliated institutions with facilities at over 100 locations .uk/. throughout the United States and abroad. Applications are sought from highly qualified candidates, including re- —John Jayne, University College London cent doctoral recipients and senior researchers. Upcoming deadlines are May 1, 2020, and August 1, 2020. See sites IPAM Call for Proposals .nationalacademies.org/pga/rap/. The Institute for Pure and Applied Mathematics (IPAM) —NRC announcement seeks program proposals from the mathematical, statistical, and scientific communities for long programs and work- The most up-to-date listing of NSF funding opportunities from the Division shops, to be reviewed at IPAM’s Science Advisory Board of Mathematical Sciences can be found online at www.nsf.gov/dms meeting in November. For more information, go to www and for the Directorate of Education and Human Resources at www.nsf .ipam.ucla.edu/propose-a-program/ .gov/dir/index.jsp?org=ehr. To receive periodic updates, subscribe or contact the to the DMSNEWS listserv by following the directions at www.nsf.gov IPAM director at [email protected]. Proposals /mps/dms/about.jsp. should also address the inclusion of women and members

738 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Mathematics Opportunities NEWS of underrepresented minorities as speakers, organizers, and place once a month in different museums, parks, and malls participants. of the Republic of Panama. The Panamanian Foundation for the Promotion of —IPAM announcement Mathematics (FUNDAPROMAT) invites you to get in- volved. You can volunteer to visit our beautiful country and Call for Papers for Haifa Workshop give a presentation to a general audience or participate in one of our math outreach events. You can donate funds to The Twentieth Haifa Workshop on Interdisciplinary Appli- cover the expenses of running the Foundation. You can mail cations of Graphs, Combinatorics, and Algorithms will be us math puzzles to use in our activities. You can share ideas held June 7–9, 2020, at the University of Haifa, Caesarea of new ways to have a greater impact on a general audience Rothschild Institute. Contributed talks are invited. The and to successfully reach those who might not consider workshop emphasizes the diversity of the use of combi- themselves math enthusiasts. If you know of someone who natorial algorithms and graph theory in application areas. might be interested in sponsoring our educational efforts Abstracts of one to two pages should be sent by April 27, in Panama and would benefit from having an international 2020, to [email protected]. See the web- presence, please do not hesitate to contact us. site cri.hevra.haifa.ac.il. You can follow us on Instagram as @fundapromat and through our public Facebook page known as “Fundapro- —Martin Golumbic, General Chair mat.” You can also access more information on our website University of Haifa www.fundapromat.org or you can email us directly at [email protected]. A New Panamanian Foundation for Mathematics —Jeanette Shakalli, PhD

The Panamanian Foundation for the Promotion of Mathe- matics (FUNDAPROMAT) is a private nonprofit foundation whose mission is to promote the study of mathematics in the Republic of Panama. The official launch of the Foun- dation was on April 14, 2020, in Panama City and on April 15 in the province of Chiriquí, outside of the capital city. The keynote speaker of the event was Dr. Michael Dorff, president of the Mathematical Association of America AMS AUTHOR (MAA), who gave his public presentation on “Math on RESOURCE CENTER Soap Bubbles.” The presentations to a general audience of the Program The Author Resource Center is a collection of on Math Outreach are among the most popular events information and tools available to assist you to organized by the Foundation. The purpose of these math presentations given by prominent international mathema- successfully write, edit, illustrate, and publish ticians is to convince kids and adults of all ages that math is your mathematical works. not only fun but it also has many interesting applications. We also want to inspire Panamanian youth to study math To begin utilizing or to follow a scientific career. Talented public speakers these important like Eugenia Cheng, Robert Lang, and Colin Wright have travelled to Panama to share their passion for mathematics resources, visit: with the Panamanian population. The Math Carnivals that highlight female Panamanian mathematicians are also in demand. The goal of these www.ams.org/authors events is to inspire Panamanian youth, in particular girls, to study math by showing them real-life examples of female Panamanian mathematicians who are successful in their careers. The first Math Carnival took place on May 12, 2019, in the Biomuseum to celebrate the International Day for Women in Mathematics. These events are currently taking

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 739 The automated job application database sponsored by the AMS.

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CONTACT: AMS Professional Programs American Mathematical Society 201 Charles Street | Providence, RI 02904-2213 USA 800.321.4267, ext. 4096 | [email protected] Classified Advertising Employment Opportunities

CANADA algorithmic and computational optimization, analysis of complex networks, etc.) by the time of the appointment, Brock University a record of or a strong potential for excellent and inde- Department of Mathematics & Statistics pendent research that will attract external funding and an Faculty Position in Mathematics of Data Science active program of fundamental and applied research in mathematical methods in data and information processing The Department of Mathematics and Statistics at Brock Uni- with applications to science and engineering problems. The versity invites applications for a probationary tenure-track successful candidate must demonstrate the ability to teach position as Assistant Professor in the mathematics of data both undergraduate and graduate courses in theoretical and science, effective July 1, 2020. computational discrete mathematics with applications to About the position the data sciences, and the ability and the commitment to The Department currently offers a BSc and an MSc in supervise graduate students. Mathematics and Statistics and is preparing to launch a Applicants should submit through the Brock Careers new trans-disciplinary BSc in Data Sciences and Analytics as website at the link below by March 31, 2020 (indicating well as a new PhD Program in Computational, Mathemat- file number stated above), a cover letter, a curriculum ical, and Statistical Sciences jointly with the Department vitae, samples of recent publications, statements of re- of Computer Science. The Department will also be con- search and teaching interests: https://brocku.wd3 tributing to the development and delivery of engineering .myworkdayjobs.com/brocku_careers/job/St programs at Brock University. The successful candidate is -Catharines-Main-Campus/Assistant-Professor expected to have a primary role in the development of these --Mathematics-of-Data-Science_JR-1005178. current and new programs in mathematics, data sciences Learn more about Brock University by visiting www and engineering. .brocku.ca. Qualifications The successful candidate must have recently completed 08 a PhD (since 2015) in an area of mathematics related to data science (e.g., computational discrete mathematics,

The Notices Classified Advertising section is devoted to listings of current employment opportunities. The publisher reserves the right to reject any listing not in keeping with the Society's standards. Acceptance shall not be construed as approval of the accuracy or the legality of any information therein. Advertis- ers are neither screened nor recommended by the publisher. The publisher is not responsible for agreements or transactions executed in part or in full based on classi ed advertisements. The 2020 rate is $3.65 per word. Advertisements will be set with a minimum one-line headline, consisting of the institution name above body copy, unless additional headline copy is speci ed by the advertiser. Headlines will be centered in boldface at no extra charge. Ads will appear in the language in which they are submitted. There are no member discounts for classi ed ads. Dictation over the telephone will not be accepted for classi ed ads. Upcoming deadlines for classi ed advertising are as follows: August 2020—May 15, 2020; September 2020—June 17, 2020; October 2020—July 17, 2020; November 2020—August 19, 2020; December 2020—September 16, 2020. US laws prohibit discrimination in employment on the basis of color, age, sex, race, religion, or national origin. Advertisements from institutions outside the US cannot be published unless they are accompanied by a statement that the institution does not discriminate on these grounds whether or not it is subject to US laws. Submission: Send email to [email protected].

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Brock University Brock University Department of Mathematics & Statistics Faculty Position in Data Science and Statistical Theory Faculty Position in Financial Mathematics The Department of Mathematics and Statistics at Brock Uni- and Mathematical Modeling versity invites applications for a probationary tenure-track The Department of Mathematics and Statistics at Brock Uni- position as Assistant Professor in data science and statistical versity invites applications for a probationary tenure-track theory, effective July 1, 2020. position as Assistant Professor in Financial Mathematics About the Position and Mathematical Modeling, effective July 1, 2020. The Department currently offers a BSc and MSc in About the position Mathematics and Statistics and is preparing to launch a The Department currently offers a BSc and an MSc in new trans-disciplinary BSc in Data Sciences and Analytics Mathematics and Statistics. The successful candidate is as well as a new PhD Program in Computational, Math- expected to have a primary role in the development of a ematical, and Statistical Sciences jointly with the Depart- new undergraduate program in Financial Mathematics, ment of Computer Science. Statistics graduates within sustain and expand existing financial math courses, and the department have an outstanding placement record as teach undergraduate courses in the MICA (Mathematics professional statisticians with high reputation, working in Integrated with Computers and Applications) concentra- diverse fields. The successful candidate will bring expertise tion. The successful candidate will also build a modern in Data Science and Statistical Theory, providing a bridge transdisciplinary research program which will strengthen between the existing Statistics program and the new Data and broaden the Department's MSc program, as well as Science program and will help to translate the former's contribute to a proposed joint PhD program with the De- success to the latter. partment of Computer Science. Commitment to develop Qualifications new graduate courses and supervise graduate students The successful candidate must have recently completed a in the areas of financial mathematics and mathematical PhD (since 2015) in statistics (or a related field) by the time modeling is anticipated. of the appointment, a proven record of research excellence, Qualifications and an active research program that will attract external The successful candidate must have recently com- funding. The successful candidate must demonstrate the pleted a PhD (since 2015) with expertise in analytical ability to teach both undergraduate and graduate courses and computational aspects of Financial Mathematics and in statistical theory (e.g., experiment design, sampling computer-based Mathematical Modeling. A strong record theory) and will be tasked with modernizing these courses of independent research in these areas which will attract and integrating them with the new Data Science program. external funding is desired. The successful candidate must The candidate must demonstrate the ability to supervise have a demonstrated teaching ability. graduate students and will have a primary role in extending Applicants should submit through the Brock Careers our existing MSc in Statistics to include a stream in Data Website at the link below by March 31, 2020, (indicating Science. file number stated above), a cover letter, a curriculum Applicants should submit through the Brock Careers vitae, samples of recent publications, statements of re- website at the link below by March 31, 2020 (indicating search and teaching interests: https://brocku.wd3 file number stated above), a cover letter, a curriculum .myworkdayjobs.com/brocku_careers/job/St vitae, samples of recent publications, statements of re- -Catharines-Main-Campus/Assistant-Professor search and teaching interests: https://brocku.wd3 --Financial-Mathematics---Mathematical .myworkdayjobs.com/brocku_careers/job/St -Modeling_JR-1005179-1. -Catharines-Main-Campus/Assistant-Professor Learn more about Brock University by visiting www --Data-Science---Statistical-Theory_JR-1005177. .brocku.ca. Learn more about Brock University by visiting www

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CHINA The Center for Applied Mathematics, also known as the Tianjin Center for Applied Mathematics (TCAM), located Tianjin University, China by a lake in the central campus in a building protected as Tenured/Tenure-Track/Postdoctoral Positions at historical architecture, is jointly sponsored by the Tianjin the Center for Applied Mathematics municipal government and the university. The initiative Dozens of positions at all levels are available at the recently to establish this center was taken by Professor S. S. Chern. founded Center for Applied Mathematics, Tianjin Univer- Professor Molin Ge is the Honorary Director, Professor sity, China. We welcome applicants with backgrounds in Zhiming Ma is the Director of the Advisory Board. Professor pure mathematics, applied mathematics, statistics, com- William Y. C. Chen serves as the Director. puter science, bioinformatics, and other related fields. We TCAM plans to fill in fifty or more permanent faculty also welcome applicants who are interested in practical positions in the next few years. In addition, there are a projects with industries. Despite its name attached with number of temporary and visiting positions. We look for- an accent of applied mathematics, we also aim to create a ward to receiving your application or inquiry at any time. strong presence of pure mathematics. There are no deadlines. Light or no teaching load, adequate facilities, spacious Please send your resume to [email protected]. office environment and strong research support. We are For more information, please visit cam.tju.edu.cn or prepared to make quick and competitive offers to self-mo- contact Ms. Erica Liu at [email protected], telephone: tivated hard workers, and to potential stars, rising stars, as 86-22-2740-6039. well as shining stars. 01

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37C70, 35B20, 37D15, List US$140, AMS members Analysis US$112, MAA members US$126, Order code SURV/246

Attractors Under bookstore.ams.org/surv-246 Autonomous and Non-autonomous Perturbations Geometry and Topology Matheus C. Bortolan, Universi- dade Federal de Santa Catarina, Topological Persistence in Florianópolis SC, Brazil, Alexan- Geometry and Analysis dre N. Carvalho, Universidade de Leonid Polterovich, Tel Aviv São Paulo, São Carlos SP, Brazil, University, Israel, Daniel Rosen, and José A. Langa, Universidad Ruhr-Universität Bochum, Ger- de Sevilla, Seville, Spain many, Karina Samvelyan, Tel Aviv University, Israel, and Jun This book provides a compre- Zhang, Université de Montréal, hensive study of how attractors behave under perturbations Canada for both autonomous and non-autonomous problems. Furthermore, the forward asymptotics of non-autonomous The theory of persistence mod- dynamical systems is presented here for the first time in a ules originated in topological unified manner. data analysis and became an When modelling real world phenomena imprecisions active area of research in algebraic topology. This book are unavoidable. On the other hand, it is paramount that provides a concise and self-contained introduction to mathematical models reflect the modelled phenomenon, persistence modules and focuses on its interactions with in spite of unimportant neglectable influences discounted pure mathematics, bringing the reader to the cutting edge by simplifications, small errors introduced by empirical of current research. In particular, the authors present appli- laws or measurements, among others. cations of persistence to symplectic topology, including the The authors deal with this issue by investigating the per- geometry of symplectomorphism groups and embedding manence of dynamical structures and continuity properties problems. Furthermore, they discuss topological function of the attractor. This is done in both the autonomous (time theory, which provides new insight into oscillation of independent) and non-autonomous (time dependent) functions. The book is accessible to readers with a basic framework in four distinct levels of approximation: the background in algebraic and differential topology. upper semicontinuity, lower semicontinuity, topological structural stability and geometrical structural stability. This item will also be of interest to those working in applications. This book is aimed at graduate students and researchers interested in dissipative dynamical systems and stability University Lecture Series, Volume 74 theory, and requires only a basic background in metric June 2020, 140 pages, Softcover, ISBN: 978-1-4704-5495- spaces, functional analysis and, for the applications, tech- 1, LC 2019059052, 2010 Mathematics Subject Classification: niques of ordinary and partial differential equations. 55U99, 58Cxx, 53Dxx, List US$55, AMS members US$44, MAA members US$49.50, Order code ULECT/74 Mathematical Surveys and Monographs, Volume 246 July 2020, 254 pages, Hardcover, ISBN: 978-1-4704-5308- bookstore.ams.org/ulect-74 4, 2010 Mathematics Subject Classification: 34D45, 35B41;

744 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 NEW BOOKS

GRADUATE STUDIES Extrinsic Geometric Flows IN MATHEMATICS 206 Ben Andrews, The Australian Na- New in Contemporary tional University, Canberra, Aus- Extrinsic Mathematics Geometric Flows tralia, Bennett Chow, University

Ben Andrews of California, San Diego, La Jolla, Bennett Chow Christine Guenther CA, Christine Guenther, Pacific Mat Langford Applications University, Forrest Grove, OR, and Mat Langford, University of Ten- nessee, Knoxville, TN Topological Phases C ONTEMPORARY of Matter and MATHEMATICS Quantum Computation Extrinsic geometric flows are 747 characterized by a submanifold Paul Bruillard, Pacific North- evolving in an ambient space Topological Phases west National Laboratory, Rich- with velocity determined by its extrinsic curvature. The of Matter and land, WA, Carlos Ortiz Mar- goal of this book is to give an extensive introduction to Quantum Computation rero, Pacific Northwest National a few of the most prominent extrinsic flows, namely, the Paul Bruillard Laboratory, Richland, WA, and Carlos Ortiz Marrero curve shortening flow, the mean curvature flow, the Gauß Julia Plavnik Julia Plavnik, Indiana University, curvature flow, the inverse-mean curvature flow, and fully Editors Bloomington, IN, Editors nonlinear flows of mean curvature and inverse-mean curva- ture type. The authors highlight techniques and behaviors This volume contains the pro- that frequently arise in the study of these (and other) flows. ceedings of the AMS Special Ses- To illustrate the broad applicability of the techniques devel- sion on Topological Phases of Matter and Quantum Com- oped, they also consider general classes of fully nonlinear putation, held from September 24–25, 2016, at Bowdoin curvature flows. College, Brunswick, Maine. The book is written at the level of a graduate student Topological quantum computing has exploded in pop- who has had a basic course in differential geometry and ularity in recent years. Sitting at the triple point between has some familiarity with partial differential equations. It mathematics, physics, and computer science, it has the is intended also to be useful as a reference for specialists. potential to revolutionize sub-disciplines in these fields. In general, the authors provide detailed proofs, although The academic importance of this field has been recognized for some more specialized results they may only present in physics through the 2016 Nobel Prize. In mathematics, the main ideas; in such cases, they provide references for some of the 1990 Fields Medals were awarded for develop- complete proofs. A brief survey of additional topics, with ments in topics that nowadays are fundamental tools for extensive references, can be found in the notes and com- the study of topological quantum computation. Moreover, mentary at the end of each chapter. the practical importance of this discipline has been under- scored by recent industry investments. Graduate Studies in Mathematics, Volume 206 The relative youth of this field combined with a high May 2020, 790 pages, Hardcover, ISBN: 978-1-4704-5596- degree of interest in it makes now an excellent time to 5, LC 2019059835, 2010 Mathematics Subject Classification: get involved. Furthermore, the cross-disciplinary nature 53C44, 58J35, 53A07, 52A20, 35K20, List US$98, AMS of topological quantum computing provides an unprec- members US$78.40, MAA members US$88.20, Order edented number of opportunities for cross-pollination of code GSM/206 mathematics, physics, and computer science. This can be seen in the variety of works contained in this volume. With bookstore.ams.org/gsm-206 articles coming from mathematics, physics, and computer science, this volume aims to provide a taste of different sub-disciplines for novices and a wealth of new perspectives for veteran researchers. Regardless of your point of entry into topological quantum computing or your experience level, this volume has something for you.

Contemporary Mathematics, Volume 747 May 2020, 240 pages, Softcover, ISBN: 978-1-4704-4074- 9, LC 2019040079, 2010 Mathematics Subject Classification: 81R50, 16D90, 81T05, 20G42, 18D10, 19D23, List US$120,

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 745 NEW BOOKS

AMS members US$96, MAA members US$108, Order code CONM/747 New AMS-Distributed bookstore.ams.org/conm-747 Publications Differential Equations Algebra and Algebraic Geometry Advances in Harmonic C ONTEMPORARY Analysis and Partial MATHEMATICS Differential Equations Algebraic Combinatorics, 748 Resurgence, Moulds Donatella Danielli, Purdue Uni- and Applications Advances in versity, West Lafayette, IN, and Harmonic Analysis (CARMA): Volume 1 Irina Mitrea, Temple University, and Partial Frédéric Chapoton, Université Differential Equations Philadelphia, PA, Editors de Strasbourg, France, Frédéric Donatella Danielli Irina Mitrea Fauvet, Université de Strasbourg, Editors This volume contains the pro- ceedings of the AMS Special Ses- France, Claudia Malvenuto, Uni- sion on Harmonic Analysis and versitá di Roma La Sapienza, Italy, Partial Differential Equations, and Jean-Yves Thibon, Université held from April 21–22, 2018, at Paris-Est Marne-la-Vallée, France, Northeastern University, Boston, Massachusetts. Editors The book features a series of recent developments at the interface between harmonic analysis and partial differential This is volume 1 of a 2-volume work comprising a total of equations and is aimed toward the theoretical and applied 14 refereed research articles which stem from the CARMA communities of researchers working in real, complex, and Conference (Algebraic Combinatorics, Resurgence, Moulds harmonic analysis, partial differential equations, and their and Applications), held at the Centre International de applications. Rencontres Mathématiques in Luminy, France, from June The topics covered belong to the general areas of the 26–30, 2017. The conference did notably emphasize the theory of function spaces, partial differential equations of role of Hopf algebraic techniques and related concepts (e.g. elliptic, parabolic, and dissipative types, geometric optics, Rota-Baxter algebras, operads, and Ecalle’s mould calculus) free boundary problems, and ergodic theory, and the em- which have lately proved pervasive in combinatorics, but phasis is on a host of new concepts, methods, and results. also in many other fields, from multiple zeta values to the algebraic study of control systems and the theory of rough This item will also be of interest to those working in analysis. paths. The volumes should be useful to researchers or graduate Contemporary Mathematics, Volume 748 students in mathematics working in these domains and May 2020, 210 pages, Softcover, ISBN: 978-1-4704-4896- to theoretical physicists involved with resurgent functions 7, LC 2019040080, 2010 Mathematics Subject Classification: and alien calculus. 31A10, 33C10, 35G20, 35P20, 35S05, 39B72, 42B35, 46E30, 76D03, 78A05, List US$120, AMS members US$96, This item will also be of interest to those working in discrete MAA members US$108, Order code CONM/748 mathematics and combinatorics. bookstore.ams.org/conm-748 A publication of the European Mathematical Society. Distributed within the Americas by the American Mathematical Society.

IRMA Lectures in Mathematics and Theoretical Physics, Volume 31 March 2020, 354 pages, Hardcover, ISBN: 978-3-03719- 204-7, 2010 Mathematics Subject Classification: 05Exx, 81T15, 81T18, 81Q30, 34C20, 37C10, 18D50, 34M40, 34M60,

746 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 NEW BOOKS

11M32, 30D60, List US$68, AMS members US$54.40, An Excursion into p-Adic Order code EMSILMTP/31 Hodge Theory From Foundations to Recent bookstore.ams.org/emsilmtp-31 Trends Fabrizio Andreatta, Universita Algebraic Combinatorics, degli Studi di Milano, Milano, Resurgence, Moulds Italy, Riccardo Brasca, Institut and Applications de Mathématiques de Jussieu, Uni- (CARMA): Volume 2 versité Paris Diderot, Paris, France, Frédéric Chapoton, Universitë Olivier Brinon, Université de de Strasbourg, France, Frédéric Bordeaux, Talence, France, Xavier Caruso, Université de Bourdeaux, Fauvet, Université de Strasbourg, Talence, France, Bruno Chiarel- France, Claudia Malvenuto, Uni- lotto, University of Padova, Italy, Gerard Freixas i Montplet, versitá di Roma La Sapienza, Italy, Institut de Mathématiques de Jussieu, Université Paris Diderot, and Jean-Yves Thibon, Université Paris, France, Shin Hattori, Tokyo City University, Tokyo, Paris-Est Marne-la-Vallée, France, Japan, Nicola Mazzari, Université de Bourdeaux, Talence, Editors France, Simone Panozzo, Universita degli Studi di Milano, Milano, Italy, Marco Seveso, Universita degli Studi di Milano, This is volume 2 of a 2-volume work comprising a total of Milano, Italy, and Go Yamashita, Research Institute for Math- 14 refereed research articles which stem from the CARMA ematical Sciences, Kyoto University, Kyoto, Japan Conference (Algebraic Combinatorics, Resurgence, Moulds and Applications), held at the Centre International de This volume offers a progressive and comprehensive intro- duction to p-adic Hodge theory. It starts with Tate’s works Rencontres Mathématiques in Luminy, France, from June on p-adic divisible groups and the cohomology of p-adic 26–30, 2017. The conference did notably emphasize the varieties which constitutes the main concrete motivations role of Hopf algebraic techniques and related concepts (e.g. for the development of p-adic Hodge theory. It then moves Rota-Baxter algebras, operads, and Ecalle’s mould calculus) smoothly to the construction of Fontaine’s p-adic period which have lately proved pervasive in combinatorics, but rings and their apparition in several comparison theorems also in many other fields, from multiple zeta values to the between various p-adic cohomologies. Applications and algebraic study of control systems and the theory of rough generalizations of these theorems are subsequently dis- paths. cussed. Finally, Scholze’s modern vision on p-adic Hodge The volumes should be useful to researchers or graduate theory, based on the theory of perfectoids, is presented. students in mathematics working in these domains and to theoretical physicists involved with resurgent functions This item will also be of interest to those working in geometry and topology. and alien calculus.

A publication of the Société Mathématique de France, Marseilles (SMF), This item will also be of interest to those working in discrete distributed by the AMS in the US, Canada, and Mexico. Orders from other mathematics and combinatorics. countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

A publication of the European Mathematical Society. Distributed within Panoramas et Synthèses, Number 54 the Americas by the American Mathematical Society. January 2020, 268 pages, Softcover, ISBN: 978-2-85629- 913-5, 2010 Mathematics Subject Classification: 14F30, IRMA Lectures in Mathematics and Theoretical Physics, 14F40, 11G25, 11F80, List US$75, AMS members US$60, Volume 32 Order code PASY/54 March 2020, 396 pages, Hardcover, ISBN: 978-3-03719- 205-4, 2010 Mathematics Subject Classification: 05Exx, 81T15, bookstore.ams.org/pasy-54 81T18, 81Q30, 34C20, 37C10, 18D50, 34M40, 34M60, 11M32, 30D60, List US$68, AMS members US$54.40, Order code EMSILMTP/32 bookstore.ams.org/emsilmtp-32

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 747 NEW BOOKS

Differential Equations A publication of Wellesley-Cambridge Press. Distributed within the and Linear Algebra Americas by the American Mathematical Society. Gilbert Strang, Massachusetts The Gilbert Strang Series, Volume 2 Institute of Technology May 2016, 573 pages, Hardcover, ISBN: 978-0-9802327- 7-6, 2010 Mathematics Subject Classification: 15–01, List Differential equations and lin- US$95, AMS members US$76, Order code STRANG/2 ear algebra are the two crucial courses in undergraduate math- bookstore.ams.org/strang-2 ematics. This new textbook de- velops those subjects separately K 3 Surfaces and together. The complete book Shigeyuki Kondo, Nagoya Uni- is a year’s course and includes versity, Japan Fourier and Laplace transforms, as well as the Fast Fourier Transform and Singular Value K3 surfaces are a key piece in Decomposition. the classification of complex Undergraduate students in courses covering differential analytic or algebraic surfaces. equations and linear algebra, either separately or together, The term was coined by A. Weil will find this material essential to their understanding. in 1958, a result of the initials Kummer, Kähler, Kodaira, and This item will also be of interest to those working in differential the mountain K2 found in Kara- equations. koram. The most famous exam- ple is the Kummer surface dis- A publication of Wellesley-Cambridge Press. Distributed within the covered in the 19th century. K3 surfaces can be considered Americas by the American Mathematical Society. as a 2-dimensional analogue of an elliptic curve, and the theory of periods—called the Torelli-type theorem for K3 The Gilbert Strang Series, Volume 1 surfaces—was established around 1970. Since then, several January 2014, 502 pages, Hardcover, ISBN: 978-0-9802327- pieces of research on K3 surfaces have been undertaken and 9-0, 2010 Mathematics Subject Classification: 15–01; 34– more recently K3 surfaces have even become of interest in 01, List US$87.50, AMS members US$70, Order code theoretical physics. STRANG/1 The main purpose of this book is an introduction to the Torelli-type theorem for complex analytic K3 surfaces, and bookstore.ams.org/strang-1 its applications. The theory of lattices and their reflection Introduction to groups is necessary to study K3 surfaces, and this book Linear Algebra introduces these notions. The book contains, as well as lattices and reflection groups, the classification of complex Fifth Edition analytic surfaces, the Torelli-type theorem, the subjectivity Gilbert Strang, Massachusetts of the period map, Enriques surfaces, an application to the Institute of Technology moduli space of plane quartics, finite automorphisms of K3 surfaces, Niemeier lattices and the Mathieu group, the This book is designed to help automorphism group of Kummer surfaces and the Leech students understand and solve lattice. the four central problems of The author seeks to demonstrate the interplay between linear algebra that involve: (1) several sorts of mathematics and hopes the book will prove linear systems, (2) least squares, helpful to researchers in algebraic geometry and related (3) eigenvalues, and (4) singular areas, and to graduate students with a basic grounding in values. algebraic geometry. The diagram on the front cover shows the four funda- mental subspaces for the matrix A. Those subspaces lead A publication of the European Mathematical Society (EMS). Distributed to the Fundamental Theorem of Linear Algebra: (1) The within the Americas by the American Mathematical Society. dimensions of the four subspaces, (2) The orthogonality of the two pairs, and (3) The best bases for all four subspaces. EMS Tracts in Mathematics, Volume 32 April 2020, 250 pages, Hardcover, ISBN: 978-3-03719-208- 5, 2010 Mathematics Subject Classification: 14J28, 14C34,

748 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 NEW BOOKS

14J10, 14J15, 14J50, 32G20, List US$88, AMS members US$70.40, Order code EMSTM/32 Differential Equations bookstore.ams.org/emstm-32 Primitive Forms and 83 Related Subjects—Kavli Linear Algebra and Primitive Forms and Related Subjects IPMU 2014 Learning from Data — Kavli IPMU 2014 Kentaro Hori, Kavli Institute for Gilbert Strang, Massachusetts the Physics and Mathematics of the Institute of Technology Universe (IPMU), Changzheng Li, Sun Yat-sen University, Si Li, This is a textbook to help readers Tsinghua University, and Kyoji understand the steps that lead Saito, Kavli Institute for the Physics to deep learning. Linear algebra and Mathematics of the Universe comes first, especially singular (IPMU), Editors values, least squares, and matrix factorizations. Often the goal is a This volume contains the pro- low rank approximation A = CR ceedings of the conference “Primitive Forms and Related (column-row) to a large matrix Subjects”, held at the Kavli Institute for the Physics and of data to see its most important part. This uses the full Mathematics of the Universe (IPMU), the University of array of applied linear algebra, including randomization Tokyo, February 10–14, 2014. for very large matrices. The principal aim of the conference was to discuss Then deep learning creates a large-scale optimization the current status of rapidly developing subjects related problem for the weights solved by gradient descent or better to primitive forms. In particular, Fukaya category, Gro- stochastic gradient descent. Finally, the book develops the mov-Witten and FJRW invariants, mathematical formu- architectures of fully connected neural nets and of Con- lation of Landau-Ginzburg models, and mirror symmetry volutional Neural Nets (CNNs) to find patterns in data. were discussed. The conference had three introductory courses by.experts and 12 lectures on more advanced topics. This item will also be of interest to those working in applications. This volume contains two survey articles and 11 research articles based on the conference presentations. A publication of Wellesley-Cambridge Press. Distributed within the Americas by the American Mathematical Society. This item will also be of interest to those working in geometry and topology. The Gilbert Strang Series, Volume 3 January 2019, 432 pages, Hardcover, ISBN: 978-0-692- 19638-0, 2010 Mathematics Subject Classification: 15–01; Published for the Mathematical Society of Japan by Kinokuniya, Tokyo, and distributed worldwide, except in Japan, by the AMS. 68–01, List US$95, AMS members US$76, Order code STRANG/3 Advanced Studies in Pure Mathematics, Volume 83 December 2019, 415 pages, Hardcover, ISBN: 978-4-86497- bookstore.ams.org/strang-3 085-3, 2010 Mathematics Subject Classification: 53Dxx; 14N35, List US$82, AMS members US$65.60, Order code ASPM/83

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MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 749 NEW BOOKS Discrete Mathematics JOIN THE AMS... and Combinatorics or renew your membership today

Séminaire Bourbaki Volume 2017–2018 Exposés 1136-1150 This 70th volume of the Bour- baki Seminar gathers the texts of the 15 survey lectures deliv- ered during the year 2017/2018. Among the topics addressed, the reader will find the Zimmer program, Deligne-Lusztig vari- eties, cocompact-convex groups, MEMBER BENEFITS: Soergel bimodules, the Gan- Gross-Prasad conjectures, expo- • Individual AMS members receive nential sums, Bernoulli convolutions, the Navier-Stokes free standard shipping on orders equations, the combinatorics of matroids, the dynamics delivered to addresses in the of Schrödinger equations, the asymptotic distribution of United States (including Puerto Frobenius eigenvalues, stability conditions in birational Rico) and Canada geometry, the Monge-Ampére equation, harmonic maps in negative curvature, and the Fargues-Fontaine curves. • Discounts on AMS publications including MAA Press books This item will also be of interest to those working in geometry and topology. • Subscriptions to Notices and Bulletin A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the US, Canada, and Mexico. Orders from other • Discounted registration countries should be sent to the SMF. Members of the SMF receive a 30% for world-class meetings discount from list. and conferences

Astérisque, Number 414 • Access to online AMS January 2020, 626 pages, Softcover, ISBN: 978-2-85629- Member Directory 915-9, 2010 Mathematics Subject Classification: 22E40,

37C85, 37C40, 53C24, 20C20, 20C33, 20G05, 20G40, P

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18E30, 37B05, 20H10, 53C35, 51E24, 22F30, 37D40, s

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57S30, 20C08, 20F55, 11F70, 11F67, 11F27, 11T24, 11L05, K a

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y , 82C22, 05A99, 05E99, 14F43, 14F99, 14M25, 14T05, A t l a nt 35P20, 35J10, 35S15, 58J50, 11B05, 14D20, 14E05, 14J28, a C o n v 35J96, 35B65, 35J60, 53C43, 31B05, 31B25, 31B35, 58E20, e n t io n P 14G20, 11F80, 14G22, List US$120, AMS members US$96, h oto gra Order code AST/414 ph y. bookstore.ams.org/ast-414 Learn more and join online at www.ams.org/membership.

750 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 Meetings & Conferences of the AMS May Table of Contents

The Meetings and Conferences section of the Notices gives information on all AMS meetings and conferences ap- proved by press time for this issue. Please refer to the page Meetings in this Issue numbers cited on this page for more detailed information on each event. Paid meeting registration is required to submit an abstract to a sectional meeting. 2020 Invited Speakers and Special Sessions are listed as soon as they are approved by the cognizant program committee; September 12–13 El Paso, Texas p. 752 the codes listed are needed for electronic abstract sub- October 3–4 State College, PA p. 753 mission. For some meetings the list may be incomplete. October 10–11 Chattanooga, Tennessee p. 755 Information in this issue may be dated. October 24–25 Salt Lake City, Utah p. 756 The most up-to-date meeting and conference informa- tion can be found online at www.ams.org/meetings. 2021 Important Information About AMS Meetings: Poten- tial organizers, speakers, and hosts should refer to page January 6–9 Washington, DC p. 757 110 in the January 2020 issue of the Notices for general March 13–14 Atlanta, Georgia p. 757 information regarding participation in AMS meetings and March 20–21 Providence, Rhode Island p. 757 conferences. April 17–18 Cincinnati, Ohio p. 758 Abstracts: Speakers should submit abstracts on the May 1–2 San Francisco, California p. 758 easy-to-use interactive Web form. No knowledge of LATEX is necessary to submit an electronic form, although those July 5–9 Grenoble, France p. 759 who use LATEX may submit abstracts with such coding, and July 19–23 Buenos Aires, Argentina p. 759 all math displays and similarly coded material (such as September 18–19 Buffalo, New York p. 759 accent marks in text) must be typeset in LATEX. Visit www.ams October 9–10 Omaha, Nebraska p. 759 .org/cgi-bin/abstracts/abstract.pl. Questions October 23–24 Albuquerque, NM p. 760 about abstracts may be sent to [email protected]. Close November 20 Mobile, Alabama p. 760 attention should be paid to specified deadlines in this issue. Unfortunately, late abstracts cannot be accommodated. 2022 January 5–8 Seattle, Washington p. 760 Associate Secretaries of the AMS Central Section: Georgia Benkart, University of Wiscon- 2023 sin–Madison, Department of Mathematics, 480 Lincoln January 4–7 Boston, Massachusetts p. 760 Drive, Madison, WI 53706-1388; email: benkart@math .wisc.edu; telephone: 608-263-4283. Eastern Section: Steven H. Weintraub, Department of Mathematics, Lehigh University, Bethlehem, PA 18015- See www.ams.org/meetings for the most up-to-date 3174; email: [email protected]; telephone: information on the meetings and conferences that we offer. 610-758-3717. Southeastern Section: Brian D. Boe, Department of Math- ematics, University of Georgia, 220 D W Brooks Drive, Athens, GA 30602-7403; email: [email protected]; The AMS strives to ensure that participants in its activities telephone: 706-542-2547. enjoy a welcoming environment. Please see our full Policy Western Section: Michel L. Lapidus, Department of Math- on a Welcoming Environment at https://www.ams ematics, University of California, Surge Bldg., Riverside, CA .org/welcoming-environment-policy. 92521-0135; email: [email protected]; telephone: 951-827-5910.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 751 MEETINGS & CONFERENCES

Meetings & Conferences of the AMS

IMPORTANT information regarding meetings programs: AMS Sectional Meeting programs do not appear in the print version of the Notices. However, comprehensive and continually updated meeting and program information with links to the abstract for each talk can be found on the AMS website. See https://www.ams.org/meetings. Final programs for Sectional Meetings will be archived on the AMS website accessible from the stated URL. New: Sectional Meetings Require Registration to Submit Abstracts. In an effort to spread the cost of the sectional meetings more equitably among all who attend and hence help keep registration fees low, starting with the 2020 fall sectional meetings, you must be registered for a sectional meeting in order to submit an abstract for that meeting. You will be prompted to register on the Abstracts Submission Page. In the event that your abstract is not accepted or you have to cancel your participation in the program due to unforeseen circumstances, your registration fee will be reimbursed. El Paso, Texas University of Texas at El Paso

September 12–13, 2020 Program first available on AMS website: July 28, 2020 Saturday – Sunday Program issue of electronic Notices: To be announced Issue of Abstracts: Volume 41, Issue 3 Meeting #1159 Central Section Deadlines Associate secretary: Georgia Benkart For organizers: Expired Announcement issue of Notices: June 2020 For abstracts: July 14, 2020

The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs /sectional.html. Invited Addresses Caroline Klivans, Brown University, Title to be announced. Brisa Sanchez, Drexel University, Title to be announced. Alejandra Sorto, Texas State University, Title to be announced. Special Sessions If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl.

Algebraic, geometric and topological combinatorics (Code: SS 6A), Art Duval, University of Texas at El Paso, Caroline Klivans, Brown University, and Jeremy Martin, University of Kansas. Algebraic structures in topology, logic, and arithmetic (Code: SS 3A), John Harding, New Mexico State University, and Emil Schwab, The University of Texas at El Paso.

752 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 MEETINGS & CONFERENCES

Commutative Algebra (Code: SS 12A), Sara Faridi, Dalhousie University, and Susan Morey, Texas State University. Fixed point theory and its applications (Code: SS 5A), Monther R. Alfuraidan, King Fahd University of Petroleum & Minerals, KSA, Mohamed A. Khamsi, The University of Texas at El Paso, Poom Kumam, King Mongkut’s University of Technology, Thonburi, Thailand, and Osvaldo Mendez, The University of Texas at El Paso. Free Resolutions, Combinatorics, and Geometry (Code: SS 17A), Anton Dochtermann, Texas State University, and Sean Sather-Wagstaff, Clemson University. Geometric Inequalities and Nonlinear Partial Differential Equations (Code: SS 20A), Joshua Flynn, University of Connecti- cut, Jungang Li, Brown University, and Guozhen Lu, University of Connecticut. Geometry of Submanifolds and Integrable Systems (Code: SS 21A), Magdalena Toda and Hung Tran, Texas Tech University. Groups and Their Cohomological Invariants in Arithmetic and Geometry (Code: SS 13A), Stefan Gille and Nikita Karpenko, University of Alberta, and Jan Minac, Western University. High-Frequency data analysis and applications (Code: SS 1A), Maria Christina Mariani and Michael Pokojovy, University of Texas at El Paso, and Ambar Sengupta, University of Connecticut. Leibniz Algebras and related topics (Code: SS 7A), Guy Biyogmam, Georgia College and State University, and Jerry Lod- der, New Mexico State University. Low-dimensional topology and knot theory (Code: SS 4A), Mohamed Ait Nouh and Luis Valdez-Sanchez, University of Texas at El Paso. Methods and applications in data Science (Code: SS 9A), Sangjin Kim, Ming-Ying Leung, Xiaogang Su, and Amy Wagler, The University of Texas at El Paso. Nonlinear analysis and optimization (Code: SS 2A), Behzad Djafari-Rouhani, University of Texas at El Paso, and Akhtar A. Khan, Rochester Institute of Technology. Non-Linear Evolution Equations (Code: SS 19A), Irena Lasiecka and Roberto Triggiani, University of Memphis, and Xiang Wan, George Washington University. Numerical partial differential equations and applications (Code: SS 10A), Son-Young Yi and Xianyi Zeng, The University of Texas at El Paso. Recent advances in scientific computing and applications (Code: SS 11A), Natasha Sharma, University of Texas at El Paso, and Annalisa Quaini, University of Houston. Recent Developments in Commutative Algebra (Code: SS 15A), Louiza Fouli and Jonathan Montaño, New Mexico State University. Statistical methodology and applications (Code: SS 8A), Ori Rosen and Suneel Chatla, University of Texas at El Paso. Stochastic Modeling in Mathematical Biology (Code: SS 16A), Mary Ballyk, New Mexico State University, Si Tang, Lehigh University, and Jianjun Paul Tian, New Mexico State University. Theoretical and Computational Studies of PDEs Related to Fluid Mechanics (Code: SS 14A), Phuong Nguyen, Texas Tech University, Andrei Tarfulea, Louisiana State University, and Kazuo Yamazaki, Texas Tech University. Undergraduate Teaching and Learning of Mathematics (Code: SS 18A), Paul Dawkins and Samuel Obara, Texas State University. State College, Pennsylvania Pennsylvania State University, University Park Campus

October 3–4, 2020 Program first available on AMS website: August 25, 2020 Saturday – Sunday Program issue of electronic Notices: To be announced Issue of Abstracts: Volume 41, Issue 3 Meeting #1160 Eastern Section Deadlines Associate secretary: Steven H. Weintraub For organizers: Expired Announcement issue of Notices: August 2020 For abstracts: August 11, 2020

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 753 MEETINGS & CONFERENCES

The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs /sectional.html. Invited Addresses Melody Chan, Brown University, Title to be announced. Steven J. Miller, Williams College, Title to be announced. Tadashi Tokieda, Stanford University, Title to be announced. Special Sessions If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl.

Advances in Mathematical Modeling of Infection (Code: SS 21A), Jessica M. Conway, Pennsylvania State University, Troy Day, Queen's University, and Timothy C. Reluga, Pennsylvania State University. Algebraic and Analytic Theory of Elliptic Curves (Code: SS 22A), Alina Cojocaru, University of Illinois at Chicago, Seoyung Kim, Queen's University, Steven J. Miller, Williams College, and Jesse Thorner, University of Florida. Algebraic Singularities in Arbitrary Characteristic (Code: SS 12A), Rankeya Datta, University of Illinois at Chicago, and Takumi Murayama, Princeton University. Analytic Number Theory (Code: SS 11A), Angel V. Kumchev, Towson University, and Siddhi S. Pathak and Robert C. Vaughan, Pennsylvania State University. Automorphic Forms and Galois Representations (Code: SS 3A), Jim Brown, Occidental College, and Krzysztof Klosin, Queens College, CUNY. Cluster Algebras and Plabic Graphs (Code: SS 16A), Chris Fraser, University of Minnesota, and Max Glick, Google Inc. Combinatorics and Computing (Code: SS 17A), Saúl A. Blanco, Indiana University, and Charles Buehrle, Notre Dame of Maryland University. Commutative Algebra and Connections to Algebraic Geometry and Combinatorics (Code: SS 4A), Ayah Almousa, Cornell University, and Kuei-Nuan Lin, Pennsylvania State University, Greater Allegheny. Configuration Spaces across Combinatorics and Topology (Code: SS 18A), Florian Frick and Michael Harrison, Carnegie Mellon University. Conservation Laws and Nonlinear Wave Equations (Code: SS 14A), Alberto Bressan, Pennsylvania State University, Geng Chen, University of Kansas, and Qingtian Zhang, West Virginia University. Drinfeld Modules, Modular Varieties and Arithmetic Applications (Code: SS 10A), Mihran Papikian, Pennsylvania State University, and Dinesh Thakur, University of Rochester. Geometric Dynamics (Code: SS 23A), Mark Levi, Pennsylvania State University, and Sergei Tabachnikov, Pennsylvania State University. Geometry and Arithmetic of Algebraic Varieties (Code: SS 8A), Jack Huizenga, John Kopper, and John Lesieutre, Penn- sylvania State University. Geometry of Groups and 3-manifolds (Code: SS 2A), Abhijit Champanerkar, College of Staten Island and The Graduate Center, CUNY, and Hongbin Sun, Rutgers University. Homological Methods in Algebra (Code: SS 5A), Ela Celikbas and Olgur Celikbas, West Virginia University, and Saeed Nasseh, Georgia Southern University. Legendrian Knots and Surfaces (Code: SS 6A), Honghao Gao, Michigan State University, and Dan Rutherford, Ball State University. Nonlinear Scientific Computing and Applications (Code: SS 1A), Wenrui Hao, Pennsylvania State University. q-Series and Related Areas in Combinatorics and Number Theory (Code: SS 7A), George Andrews, David Little, and Ae Ja Yee, Pennsylvania State University. Recent Developments in Gauge Theory (Code: SS 19A), Siqi He, Stony Brook University, and Ákos Nagy, Duke University. Recent Probabilistic Advances in Mathematical Physics (Code: SS 20A), Alexei Novikiov, Izabella Stuhl, and Yuri Suhov, Pennsylvania State University. Turbulence and Mixing in Fluid Dynamics (Code: SS 15A), Yuanyuan Feng, Anna Mazzucato, and Alexei Novikov, Pennsylvania State University. Variational Aspects of Geometric Analysis (Code: SS 13A), Jeffrey Case, Pennsylvania State University, Casey Kelleher, Princeton University, Chao Li, Princeton University, and Siyi Zhang, University of Notre Dame.

754 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 MEETINGS & CONFERENCES Chattanooga, Tennessee University of Tennessee at Chattanooga

October 10–11, 2020 Program first available on AMS website: September 1, 2020 Saturday – Sunday Program issue of electronic Notices: To be announced Issue of Abstracts: Volume 41, Issue 4 Meeting #1161 Southeastern Section Deadlines Associate secretary: Brian D. Boe For organizers: Expired Announcement issue of Notices: August 2020 For abstracts: August 18, 2020

The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs /sectional.html. Invited Addresses Giulia Saccà, , Title to be announced. Chad Topaz, Williams College, Title to be announced. Xingxing Yu, Georgia Institute of Technology, Title to be announced. Special Sessions If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl.

Active Learning Methods and Pedagogical Approaches in Teaching College Level Mathematics (Code: SS 20A), Hashim A. Saber, University of North Georgia. Advances in Graph Theory (Code: SS 5A), Xiaofeng Gu, University of West Georgia, and Dong Ye, Middle Tennessee State University. Advances in Image Reconstruction Algorithms for Inverse Tomography Problems (Code: SS 22A), Sanwar Uddin Ahmad, Colorado State University, and Taufiquar R Khan, Clemson University. Advances in the Modeling and Computation of Fluid Flows and Fluid-Structure Interactions (Code: SS 11A), Jin Wang and Eleni Panagiotou, University of Tennessee at Chattanooga. Applicable Analysis of PDE Systems which Govern Fluid Flows and Flow-Structure Interactions (Code: SS 12A), Pelin Guven Geredeli, Iowa State University, and George Avalos, University of Nebraska-Lincoln. Applied Knot Theory (Code: SS 4A), Jason Cantarella, University of Georgia, Eleni Panagiotou, University of Tennessee at Chattanooga, and Eric Rawdon, University of St Thomas. Boundary Value Problems for Differential, Difference, and Fractional Equations (Code: SS 2A), John R Graef and Lingju Kong, University of Tennessee at Chattanooga, and Min Wang, Kennesaw State University. Coding Theory, Cryptography, and Number Theory (Code: SS 16A), Ryann Cartor, Shuhong Gao, Kevin James, and Felice Manganiello, Clemson University. Commutative Algebra (Code: SS 1A), Simplice Tchamna, Georgia College, and Lokendra Paudel, University of South Carolina, Salkehatchie. Convexity and Probability in High Dimensions (Code: SS 21A), Steven Hoehner, Longwood University, Stanislaw Szarek, Case Western Reserve University, and Elisabeth Werner, Case Western Reserve University. Geometric and Topological Generalization of Groups (Code: SS 19A), Bikash C Das, University of North Georgia. Homological Commutative Algebra (Code: SS 9A), Hugh Geller, James Gossell, and Sean Sather-Wagstaff, Clemson University. Interactions Between Algebra, Geometry and Topology in Low Dimensions (Code: SS 6A), Alex Casella and Lorenzo Ruffoni, Florida State University at Tallahassee, and Michelle Chu, University of Illinois at Chicago. Mathematics in Industry and National Laboratories (Code: SS 15A), Samantha Erwin and John Gounley, Oak Ridge National Laboratory. Modern Applied Analysis (Code: SS 8A), Boris Belinskiy, University of Tennessee at Chattanooga. Nonstandard Elliptic and Parabolic Regularity Theory with Applications (Code: SS 10A), Hongjie Dong, Brown University, and Tuoc Phan, University of Tennessee, Knoxville.

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 755 MEETINGS & CONFERENCES

Polynomials, Approximation Theory, and Potential Theory (Code: SS 13A), Aaron Yeager, College of Coastal Georgia, and Erik Lundberg, Florida Atlantic University. Probability and Statistical Models with Applications (Code: SS 7A), Sher Chhetri, University of South Carolina, Sumter, and Cory Ball, Florida Atlantic University. Quantitative Approaches to Social Justice (Code: SS 17A), Chad Topaz, Williams College. Structural and Extremal Graph Theory (Code: SS 3A), Hao Huang, Emory University, and Xingxing Yu, Georgia Institute of Technology. Title to be Announced (Code: SS 18A), Giulia Saccà, Columbia University. Topological Data Analysis and Artificial Intelligence (Code: SS 14A), Vasilios Maroulas, University of Tennessee, Knox- ville, Farzana Nasrin, University of Tennessee, Knoxville, Eleni Panagiotou, University of Tennessee at Chattanooga, and Theodore Papamarkou, Oak Ridge National Laboratory. Salt Lake City, Utah University of Utah

October 24–25, 2020 Program first available on AMS website: September 17, Saturday – Sunday 2020 Program issue of electronic Notices: To be announced Meeting #1162 Issue of Abstracts: Volume 41, Issue 4 Western Section Deadlines Associate secretary: Michel L. Lapidus For organizers: Expired Announcement issue of Notices: August 2020 For abstracts: September 1, 2020

The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs /sectional.html. Invited Addresses Bhargav Bhatt, University of Michigan, Ann Arbor, Title to be announced. Jonathan Brundan, University of Oregon, Eugene, Title to be announced. Andrei Okounkov, Columbia University, Title to be announced (Erdo ˝s Memorial Lecture). Mariel Vazquez, University of California, Davis, Title to be announced. Special Sessions If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl.

Algebraic combinatorics and applications in harmonic analysis (Code: SS 4A), Joseph Iverson and Sung Y. Song, Iowa State University, and Bangteng Xu, Eastern Kentucky University. Building bridges between commutative algebra and nearby areas (Code: SS 6A), Benjamin Briggs and Josh Pollitz, Uni- versity of Utah. Commutative Algebra (MSC 13) (Code: SS 5A), Adam Boocher, University of San Diego, Eloísa Grifo, University of California, Riverside, and Jennifer Kenkel, University of Michigan. Extremal Graph Theory (Code: SS 1A), József Balogh, University of Illinois, and Bernard Lidick y ´, Iowa State University. Geometry and Representation Theory of Quantum Algebras and Related Topics (Code: SS 7A), Mee Seong Im, United States Military Academy, West Point, Bach Nguyen, Temple University, and Arik Wilbert, University of Georgia. Monoidal Categories in Representation Theory (associated with the Invited Address by Jon Brundan) (Code: SS 2A), Jonathan Brundan, Ben Elias, and Victor Ostrik, University of Oregon. PDEs, data, and inverse problems (Code: SS 3A), Vira Babenko, Drake University, and Akil Narayan, University of Utah.

756 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 MEETINGS & CONFERENCES Washington, District of Columbia Walter E. Washington Convention Center

January 6–9, 2021 Associate secretary: Brian D. Boe Wednesday – Saturday Announcement issue of Notices: October 2020 Meeting #1163 Program first available on AMS website: November 1, 2020 Joint Mathematics Meetings, including the 127th Annual Program issue of electronic Notices: To be announced Meeting of the AMS, 104th Annual Meeting of the Mathe- matical Association of America (MAA), annual meetings of Issue of Abstracts: To be announced the Association for Women in Mathematics (AWM) and the National Association of Mathematicians (NAM), and the win- Deadlines ter meeting of the Association of Symbolic Logic (ASL), with sessions contributed by the Society for Industrial and Applied For organizers: April 16, 2020 Mathematics (SIAM). For abstracts: To be announced Atlanta, Georgia Georgia Institute of Technology

March 13–14, 2021 Program first available on AMS website: To be announced Saturday – Sunday Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Meeting #1164 Southeastern Section Deadlines Associate secretary: Brian D. Boe For organizers: To be announced Announcement issue of Notices: To be announced For abstracts: To be announced

The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs /sectional.html. Special Sessions If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl.

Differential Graded Methods in Commutative Algebra (Code: SS 1A), Saeed Nasseh, Georgia Southern University, and Adela Vraciu, University of South Carolina, Columbia. Providence, Rhode Island Brown University

March 20–21, 2021 Program first available on AMS website: To be announced Saturday – Sunday Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Meeting #1165 Eastern Section Deadlines Associate secretary: Steven H. Weintraub For organizers: To be announced Announcement issue of Notices: To be announced For abstracts: To be announced

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 757 MEETINGS & CONFERENCES

The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs /sectional.html. Special Sessions If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl.

Commutative Algebra, Laura Ghezzi, Department of Mathematics, New York City College of Technology-CUNY, Saeed Nasseh, Georgia Southern University, and Oana Veliche, Northeastern University. Recent Advances in Schubert Calculus and Related Topics (Code: SS 2A), Cristian Lenart and Changlong Zhong, State University of New York at Albany. Cincinnati, Ohio University of Cincinnati

April 17–18, 2021 Program first available on AMS website: To be announced Saturday – Sunday Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Meeting #1166 Central Section Deadlines Associate secretary: Georgia Benkart For organizers: To be announced Announcement issue of Notices: To be announced For abstracts: To be announced San Francisco, California San Francisco State University

May 1–2, 2021 Program first available on AMS website: To be announced Saturday – Sunday Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Meeting #1167 Western Section Deadlines Associate secretary: Michel L. Lapidus For organizers: To be announced Announcement issue of Notices: To be announced For abstracts: To be announced

The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs /sectional.html. Special Sessions If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl.

Differential Geometry and Geometric PDE, Alfonso Agnew, Nicholas Brubaker, Thomas Murphy, Shoo Seto, and Bog- dan Suceava ˘, California State University, Fullerton.

758 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 MEETINGS & CONFERENCES Grenoble, France Université de Grenoble-Alpes

July 5–9, 2021 Program first available on AMS website: Not applicable Monday – Friday Program issue of electronic Notices: Not applicable Issue of Abstracts: Not applicable Meeting #1168 Associate secretary: Michel L. Lapidus Deadlines Announcement issue of Notices: To be announced For organizers: September 16, 2020 For abstracts: To be announced Buenos Aires, Argentina The University of Buenos Aires

July 19–23, 2021 Program first available on AMS website: Not applicable Monday – Friday Program issue of electronic Notices: Not applicable Issue of Abstracts: Not applicable Meeting #1169 Associate secretary: Steven H. Weintraub Deadlines Announcement issue of Notices: To be announced For organizers: To be announced For abstracts: To be announced Buffalo, New York University at Buffalo (SUNY)

September 18–19, 2021 Program issue of electronic Notices: To be announced Saturday – Sunday Issue of Abstracts: To be announced Eastern Section Associate secretary: Steven H. Weintraub Deadlines Announcement issue of Notices: To be announced For organizers: To be announced Program first available on AMS website: To be announced For abstracts: To be announced Omaha, Nebraska Creighton University

October 9–10, 2021 Program issue of electronic Notices: To be announced Saturday – Sunday Issue of Abstracts: To be announced Central Section Associate secretary: Georgia Benkart Deadlines Announcement issue of Notices: To be announced For organizers: To be announced Program first available on AMS website: To be announced For abstracts: To be announced

MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 759 MEETINGS & CONFERENCES Albuquerque, New Mexico University of New Mexico

October 23–24, 2021 Program issue of electronic Notices: To be announced Saturday – Sunday Issue of Abstracts: To be announced Western Section Associate secretary: Michel L. Lapidus Deadlines Announcement issue of Notices: To be announced For organizers: To be announced Program first available on AMS website: To be announced For abstracts: To be announced

The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs /sectional.html. Special Sessions If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl.

Inverse Problems, Hanna Makaruk, Los Alamos National Laboratory, and Robert Owczarek, University of New Mexico. Mobile, Alabama University of South Alabama

November 20, 2021 Program issue of electronic Notices: To be announced Saturday Issue of Abstracts: To be announced Southeastern Section Associate secretary: Brian D. Boe Deadlines Announcement issue of Notices: To be announced For organizers: To be announced Program first available on AMS website: To be announced For abstracts: To be announced Seattle, Washington Washington State Convention Center and the Sheraton Seattle Hotel

January 5–8, 2022 Program issue of electronic Notices: To be announced Wednesday – Saturday Issue of Abstracts: To be announced Associate secretary: Georgia Benkart Announcement issue of Notices: October 2021 Deadlines Program first available on AMS website: To be announced For organizers: To be announced For abstracts: To be announced Boston, Massachusetts John B. Hynes Veterans Memorial Convention Center, Boston Marriott Hotel, and Boston Sheraton Hotel

January 4–7, 2023 Program issue of electronic Notices: To be announced Wednesday – Saturday Issue of Abstracts: To be announced Associate secretary: Steven H. Weintraub Announcement issue of Notices: October 2022 Deadlines Program first available on AMS website: To be announced For organizers: To be announced For abstracts: To be announced

760 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 POSITION AVAILABLE ASSOCIATE EXECUTIVE DIRECTOR for Meetings and Professional Services AMERICAN MATHEMATICAL SOCIETY

Applications are invited for the position of Associate Executive Director for Meetings and Professional Services. Come be part of an extraordinary collection of talent who support the Society’s extensive activities. This is an exciting opportunity to have a direct impact on the ways we advance research and create connections in the mathematics community. The Associate Executive Director heads the Meetings and Professional Services division and is part of the executive leadership team. Departments in the division support a variety of AMS meetings, programs, and activities that engage our members and the entire mathematical community. This robust range of activities includes meetings such as the Joint Mathematics Meetings, projects such as the Annual Survey and Mathjobs.org, membership activities such as the AMS Graduate Student Chapters, and activities such as our AMS Mathematics Research Communities and the AMS Fellows program, as well as a number of education initiatives, various travel support programs, and several outreach activities. Responsibilities of the Associate Executive Director include: • Developing and implementing long-range plans for all parts of the division • Overseeing departments in the division, including budgetary planning and control • Leadership and vision to ensure existing AMS programs optimize their impact, as well as in creating, planning, and implementing new programs • Collaborating with other mathematical organizations • Representing the division with AMS governance and the mathematical community • Working closely with senior executive staff, as well as department directors across the organization, to ensure excellence and professionalism Candidates should have an earned doctorate in one of the mathematical sciences as well as adminis- trative experience. A strong interest in professional programs and services is essential, as is experience with grant writing. This position reports to the AMS Executive Director and also interacts with the AMS governance on the Council and Board of Trustees. This position is full time, located in our Providence, RI headquarters. Salary is negotiable and will be commensurate with experience. Inquiries about the position are encouraged. Please contact exdir@ ams.org. This position is open until filled. Please submit letter of interest, CV, and three professional references to be considered for the position.

The American Mathematical Society is committed to creating a diverse environment and is proud to be an equal opportunity employer. All quali ed applicants will receive consideration for employment without regard to race, color, religion, gender, gender identity or expression, sexual orientation, national origin, genetics, disability, age, veteran status, or immigration status. American Mathematical Society Distribution Center

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