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 THANK YOU 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 field by supporting a prize, or honor a mentor or give in memory of an esteemed colleague or great mathematician. Each mathematics 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. FEATURES 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