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Home , Demetrios Christodoulou, Robert Langlands, Sylvain Cappell, Yakov Sinai

Spring 2018 • Issue 7 Department of Letter from the CHAIR Congratulations faculty members and 6 postdoctoral Former Faculty Member to the 33 gradu- researchers. ating members of the class of Professors Elliott Lieb and Gang Tian RECEIVES 2017 and the 9 both retired this year. Giants in their Robert P. Langlands has been awarded graduate stu- fields, Prof. Lieb joined our Depart- dents who are ment (joint with ) in 1975 the 2018 Abel Prize “for his visionary now *17. Thank while Prof. Tian has been with us program connecting you for reading since 2003. Speaking of former faculty to number the 2018 edition members, it brings me great pleasure theory;" of Fine Letters! to congratulate Robert Langlands, work done As I write these words, students are professor emeritus at the Institute for while he scrambling to finish their senior and Advanced Study, for winning the Abel was a Ph. D. theses, with graduation a dis- Prize. The Abel committee cited his 30-year-old tant goal, but I am looking forward to famous letter to André Weil (precursor Associate congratulating 34 new math grads, in- to what's now known as the Lang- Professor at cluding a record 10 women, as well as lands program) written as an Associate Princeton. 18 graduate students who are expected Professor in our Department. to complete their final public orals this Previous spring or summer. We were honored to host Umberto Abel Prize Zannier as our fall Minerva Lecturer This year we welcomed Noga Alon and Laure Saint-Raymond, Demetrios recipients include professor emeritus (joint with PACM) as our newest Christodoulou, and Paul Seidel as our , John Nash, and professor professor, as well as 7 new junior Minerva Distinguished Visitors. They . continued on page 2...

INSIDE UPCOMING... the profile of mathematics in mainstream this issue media to ​totally unprecedented levels. —Marcelo Viana 2–5 Faculty Chair, ICM 2018 Organizing Committee 6–7 Honors and Awards Program Committee Chair: János Kollár 8 Distinguished Visitors Princeton Invited Speakers: 9 Faculty Research Alice Chang, Emmy Noether Lecture Assaf Naor, Plenary lecture 10–11 Undergraduate Program August 1–9 John Pardon, invited section lecturer 12 Graduate Program The Biennium of Mathematics 2017- Amit Singer, invited section lecturer 2018, formally proclaimed by the Allan Sly, invited section lecturer 13 Most recent Ph.D.s Alexandr Logunov (starting Sept. 1), Brazilian national parliament, ​encom- invited section lecturer 14 Advisory Council Visits passes a wide range of outreach initia- June Huh (IAS, Visiting Associate tives throughout ​the country, and raised Research Scholar), invited section 15 Fine Hall Update lecturer Faculty Appointments Department Welcomes PROFESSOR NOGA ALON The Department is pleased Professor Alon obtained his Ph.D. in 1983 from the to welcome Noga Alon as our Hebrew University. Following a two-year­ postdoctoral newest professor, joint with position at MIT he joined Tel Aviv University, where he the Program in Applied & remained until coming to Princeton. He is also currently a Computational Mathematics. member of Microsoft Research, Israel. He has supervised A world leader in the field of over 20 doctoral students in mathematics and computer combinatorics, Professor Alon's science, many of whom are now world-leading faculty contributions include solutions members at various institutions. He has received many to longstanding questions on honors and awards, including the Erdös prize (1989), the universal graphs, epsilon-nets, Feher prize (1991), tho Pólya Prize (2000), the Bruno and the structure of large graphs and property-testing, Memorial Award (2001), the Landau Prize (2005), the among many others. Gödel Prize (2005), the Israel Prize (2008), the EMET Prize (2011) and the Dijkstra Prize (2016).

from the Chair ...continued from page 1 all gave beautiful and stimulating lectures. With uniformly acclaimed results, Fine Hall went through its Professors Igor Rodnianski and Allan Sly were named 2017 first major refurbishment since opening in 1969. Our Common Simons Investigators in Mathematics, while professor Amit Room and every office received essentially all new furniture Singer was named an Investigator in Math+X, a program and carpeting, though some of the beautiful and solid furni- designed to encourage novel collaborations between math- ture from the old Fine Hall is still in use. Our administrative ematics and other fields. We are pleased that Andrea Bertozz area was remodeled providing a brighter and more welcoming ’87 *91 was the other Math+X winner. Professor János Kol- reception area. This project required everyone and their lár won the , John Pardon won a Packard Fellow- possessions to move out and back during a three month ship, and Assaf Naor won the Nemmers Prize. Seven of our period. Thanks are due to Professor Igor Rodnianski for Princeton faculty and researchers were named invited speakers spearheading this effort. Detailed oversight, constant for the upcoming International Congress of in attention to big and little things and lots of hand holding by Rio. the staff made this all very bearable.

Instructor Hansheng Diao and graduate student Kathleen I close by thanking the Fernholz Foundation and the Class of Emerson received the Excellence in Teaching Awards by the 1971 Endowment for their ongoing support of the Department Undergraduate and Graduate Engineering Student Coun- and thanking the Advisory Council for their valuable advice cils—this is the third year in a row for Hansheng! Hansheng and encouragement. and assistant professor Ana Menezes received this year’s departmental junior faculty teaching award and our graduate I hope that all of those reading this newsletter think back on student teaching awards went to Charles Stibitz and Anibal their time in Fine Hall as one of discovery and growth. I am Velozo. This is the fifth year the Department has awarded always happy to hear from alumni and former members of the these prizes highlighting and encouraging excellence in under- Department, so please do not hesitate to get in touch. graduate instruction. David Gabai *77 *80 Our undergraduate Putnam Competition team had another [email protected] outstanding year, with our team of Murilo Zanarella, Zhuo Qun (Alex) Song, and Xiaoyu Xu coming in third place overall. Page 2 Faculty Appointments

Tristan J. Buckmaster for three years before arriving to Mathematics from Tours University, Assistant Professor France and her BS in Mathematics Analysis Princeton. He completed his PhD and Computer Science at the Uni- Tristan J. Buck- in 2014 at and versity of Science, Vietnam. She is master joins the his MA Mathematics in 2010 and the recipient of the James D. Skinner Department after his BS in Mathematics and Physics graduate fellowship. three years at the in 2009. Courant Institute Yueh-Ju Lin Yunqing Tang of Mathematical Instructor Instructor Sciences, New Geometric Arithmetic York University, where he was an Analysis, Geometry and Instructor and Visiting Assistant Differential Geometry, Number Professor. He completed his PhD and PDEs Theory at the University of Leipzig/Max Yueh-Ju Lin Yunqing Tang Planck Institute for Mathematics joined Princ- received her in the Sciences, Leipzig, Saxony, eton after three PhD in Math- Germany in 2014. He is a recipient years as a Postdoctoral Assistant ematics from Harvard University in of the Leipzig Promotionspreis (PhD Professor, Department of Math- 2016 and holds a BS with honors in Prize) by the Research Academy ematics, University of Michigan. Mathematics from Peking University. Leipzig. She also served as a Postdoctoral She was awarded the AWM Dis- sertation Prize, given for outstanding Gabriele Di Cerbo Fellow at the Mathematical Sci- Assistant ences Research Institute, Berkeley PhD dissertations by female students Professor and as a Research Assistant at the in the US, in 2016. Additionally, Algebraic Institute of Mathematics, Academia she received the New World Math- Geometry Sinica, Taiwan. She received her ematics Award, Gold Medal for her Gabriele Di Cerbo PhD and MS in Mathematics from PhD thesis, awarded for outstanding joins the Depart- University of Notre Dame and her Chinese mathematics students world- ment after serv- BS in Mathematics from National wide, 2016. She received a Merit ing as the Ritt Assistant Professor, Taiwan University. She received the Research Fellowship at the Graduate Department of Mathematics, Colum- Striving for Excellence in Teach- School of Arts and Sciences, Harvard bia University, for four years. He ing Certificate, Kaneb Center for University (2015-2016). earned his PhD in Mathematics from Teaching and Learning, University Princeton University in 2013 and has Joseph Allen Waldron of Notre Dame. a Bachelor of Science degree, 2008, Instructor Birational and Master in Mathematics degree, Oanh Nguyen Geometry Instructor 2009, from the University of Rome, Joseph Allen La Sapienza, Italy. Analysis, Combinator- Waldron was Jonathan Hanselman ics, and Prob- a Postdoctoral Assistant ability Researcher at Professor Oanh Nguyen Ecole Polytech- Low- was previously nique Federale de Lausanne (EPFL), dimensional an Instructor , 2016-2017. He received at Yale Univer- Jonathan Han- his PhD, Mathematics, July 2016 at sity, where she selman was an Queens' College, University of Cam- received her PhD in mathematics. RTG Instructor bridge, UK. She received her MS in Applied at the University of Texas at Austin Page 3 Faculty Retirements Professor Emeritus ELLIOTT LIEB By Michael Aizenman Elliott H. Lieb was born in Boston in lifelong friends and collaborators, but moved to Princeton on leave of absence. 1932, and at the age of five moved to the in 1966 Elliott moved to Northeastern He officially came on board in 1975. Bronx where he was educated in the New University in Boston. It was there that he York City school system, graduating from and F.Y. Wu wrote one of the most cited In Princeton, Elliott addressed important the Bronx High School of Science. papers in condensed matter physics on the problems including stability of matter in magnetic fields, with J-P. Solovej and J. While in high school, Elliott was inter- Yngvason, conditions for the emergence ested in electronics (vacuum tubes then) of ferromagnetism, some with M. Aizen- and amateur radio; one of his proudest man, and an axiomatic presentation of achievements was being able to mas- thermodynamics based on the concept of ter Morse code sufficiently to obtain a entropy, developed jointly with J. Yngva- broadcasting license, W2ZHS. Upon son. entering MIT in 1949 his intention was to acquire an electrical engineering degree, Much of the research was directed toward but thanks to the superb teaching of his understanding the quantum mechanics of first semester physics professor M. Sands atoms and the lowest energy state of the (who collaborated with Feynman and Bose gas. A significant 1987 work with I. Leighton on the famous “Lectures”), Affleck, T. Kennedy and H. Tasaki, car- Elliott switched to the Physics Depart- ried out at Princeton, was the invention ment. and solution of the AKLT quantum spin system. This provides an early example In 1953 he took off on an NSF fellowship of a system exhibiting what is nowadays for Birmingham University in England, Elliott has been guided by referred to as a topological state of matter, for Ph.D. studies in the Department of his drive to understand chal- which is a subject of great current interest. under Profs. R.E. lenging physics phenomena Peierls, S.F. Edwards and G.E. Brown. Elliott’s contributions were recognized, as Birmingham was one of the main cen- in lucid mathematical terms. reflected by many prizes and honors. He ters of theoretical physics in Europe in is a member of five national academies, in- those days. Like many young people then one-dimensional Hubbard model. Elliott cluding the NAS, a foreign member of the and now he steered towards the most also solved the “square ice problem” and UK Royal Society, and has four honorary advanced subject offered, quantum field computed the entropy per atom to be doctorates. He was twice president of the theory. But after graduation in 1956 he (3/2)log (4/3), since known as “Lieb’s International Association of Mathematical tacked towards other directions in physics. ice constant”. Building on the work of Physics. For his work in physics he was H.A. Bethe and C.N. Yang, and trigger- awarded the Boltzmann medal, the Max Throughout the rich scientific career that ing subsequent papers by R. Baxter and Planck medal, the Norwegian Onsager followed, Elliott has been guided by his others, this work played an important prize and the Heineman prize. The Amer- drive to understand challenging physics role by opening up a new area of soluble ican Mathematical Society recognized him phenomena in lucid mathematical terms. models, beyond Onsager’s solution of the with the Birkhoff and Conant prizes, and In the process he would repeatedly break Ising model. the Swedish Royal Academy awarded the new ground in physics and in mathemat- Shock prize. The Austrian government ics. A number of active fields of research In 1968 Elliott returned to MIT, this time awarded him the lifetime Austrian Medal were started or invigorated that way, and as a faculty member. Important scien- of Honor for Science and Art, which can some now enjoy renewed attention. His tific results from that period include the be held by no more than eighteen foreign early work withstands the test of time in foundations of Thomas-Fermi theory and scientists at any time. its clarity and reach. Hartree-Fock theories (with B. Simon), the Lieb-Robinson bound in condensed While Elliott has retired from active pro- Invited by J. Lebowitz, Elliott briefly matter, the Hepp-Lieb maser model, and fessorial duties he continues to do research held a position at the newly created Belfer the thermodynamic limit for Coulomb in mathematics and physics. We wish him Graduate School of Yeshiva Univesity, in systems (with J. Lebowitz). The MIT many years of joy in this endeavor. New York. Elliott and Lebowitz became period lasted until 1974 when Elliott

Page 4 Faculty Retirements Professor Emeritus GANG TIAN By Alice Chang and Paul Yang

Professor Gang Tian was born in faces with positive first Chern class. bility, which is central in the theory Nanjing China. He received his He proved what's now known as the of geometric stability. He initiated B.A. in mathematics from Nanjing Bogomolov–Tian–Todorov theorem the Analytical Minimal Model pro- University in 1982, his M.S. from for Calabi-Yau manifolds. With Ruan, gram through Kähler-Ricci flow, now Peking University in 1984 and his known as Tian-Song MMP theory Ph.D. from Harvard University in in complex geometry. Together with 1988 under the direction of S. T. Yau. John Morgan, he gave an exposition He served Princeton as an Assistant of Perelman's proof of the Poincaré Professor from 1988 to 1990, return- Conjecture and Thurston’s Geometri- ing as full Professor in 2003 after zation Conjecture. More recently, he appointments at Stony Brook, NYU solved the Yau–Tian–Donaldson con- and MIT; he held the chair of Simons jecture, a central problem in Kähler Professor of Mathematics at MIT geometry, which was independently from 1996. He was named a Eugene solved by Chen, Donaldson and Sun. Higgins Professor in 2009 and be- With Jeff Streets, he discovered new came Emeritus September 1, 2017. geometric flows that are now central tools in complex geometry. He currently holds the positions of Vice-President of Peking Univer- Dr. Gang Tian won the Alan T. sity and Professor and Director of Tian made fundamental Waterman Award in 1994 and the the Beijing International Center for contributions to geometric Oswald Veblen Prize in 1996. He Mathematical Research (BICMR). analysis, complex geometry gave an Invited Lecture in the Ge- He is also a member of the Scientific and symplectic geometry. ometry section of the International Council of the Abdus Salam Interna- Congress of Mathematics in 1990, tional Centre for Theoretical Physics and a Plenary Lecture at the Interna- in Italy. he established a theory of quantum tional Congress of Mathematicians in cohomology and Gromov-Witten 2002. He was elected to the National While at Princeton he strengthened invariants on semi-positive symplectic Academy of China in 2001 and the the geometric analysis group. Dur- manifolds, and in particular, sym- American Academy of Arts and Sci- ing his Princeton years, he supervised plectic manifolds of complex dimen- ences in 2004. He has served as editor 33 Ph.D. theses from Princeton and sion-3 as well as Calabi-Yau spaces. for a number of mathematical journals elsewhere, as well as mentoring a This result implies the associativity including the Annals of Mathematics, large number of postdocs. Overall, he of the quantum cohomology ring of and on many mathematics advisory supervised over 40 Ph.D. theses from semi-positive symplectic manifolds. boards. at least 11 different institutions. He constructed the Gromov-Witten invariants for closed symplectic Tian made fundamental contribution manifolds. He developed a compact- to geometric analysis, complex geom- ness theory for high-dimensional So far as the theories of mathematics etry and symplectic geometry. Here Yang-Mills fields and found a deep are about reality, they are not certain; so far as they are certain, they are is a sample of his many results. He connection between high-dimensional not about reality. proved the existence of Kähler-Ein- gauge fields and calibrated geometry. — stein metrics on compact complex sur- Tian introduced the theory of K-sta-

Page 5 Honors and Awards

Michael Aizenman which has led to breakthroughs in Sir Andrew Wiles was presented the theory of algorithms". Sir Andrew with an honorary Peter Ozsváth Wiles, Princ- degree by Tech- eton's James S. was elected to nion University at McDonnell its June 2018 the Nathional Academy of Distinguished ceremony. Science in rec- University ognition of his Professor of distinguished and continuing Mathematics, Javier Gómez-Serrano achievement in Emeritus and a Royal Society Re- was one of seven original research. search Professor at the University of young mathemati- Oxford, was awarded the 2017 Copley cians to receive the John Pardon Prize by the Royal Society, which third annual was named a included a gift of £25,000. First Vicent Caselles 2017 Packard awarded in 1731, the Copley Medal is Mathematical Fellow. The the Royal Society’s oldest and most Research Award of David and Lucile prestigious award. the Royal Spanish Packard Founda- Mathematical Society. The award recog- tion named 18 nizes "creativity, originality and achieve- 2017 Simons Investigators fellows this year, ment in the early years of the scientific Igor Rodnianski and Allan Sly were each of whom profession," and is awarded to Spanish named 2017 Simons Investigators in will receive $875,000 over five years mathematicians around 30 years old. mathematics by the Simons Founda- to pursue their research. tion as leaders in their fields. Amit János Kollár Singer was named a Simon Investiga- has been awarded received an hon- tor in Math +X as one of the leaders the 2017 Shaw in the mathematical analysis of noisy Prize in math- orary doctorate data provided by cryo-EM. The Si- ematics. Professor from King's Col- mons Investigators program provides a Kollár shares this lege London in stable base of support for outstanding year's prize with October 2017. A , mini-conference scientists, enabling them to undertake professor at the to celebrate the long-term study of fundamental ques- Collège de France, for their "remarkable award to Professor Sarnak was held tions. results in many central areas of algebraic at Goodenough College. Fellows of the American geometry, which have transformed the Mathematical Society field and led to the solution of long- Allan Sly Alex Ionescu, , standing problems." was awarded an Fernando Codá Marques, John Par- NSF CAREER don, and Peter Sarnak were elected to Assaf Naor grant. With this the 2018 class of Fellows of the Ameri- has been named five-year grant can Mathematical Society. The Fellows the 2018 Frederic Professor Sly of the American Mathematical So- Esser Nemmers will study ciety program recognizes members Mathematics Prize "stochastic who have made outstanding contribu- Recipient "for his processes and embeddings on tions to the creation, exposition, profound work networks." on the geometry advancement, communication, and of metric spaces, utilization of mathematics. Page 6 Honors and Awards DIAO and EMERSON Honored with Excellence in Teaching Award Mathematics De- student-run, and University’s partment instructor any professor or top honor for Hansheng Diao and AI instructing graduate graduate student an engineering, students, along Kathleen Emerson mathematics, or with Chantal were awarded this physics course is Berman, Cole year's Excellence in eligible. This is Bunzel and Teaching Award by the third consecu- Matthew Ed- the Undergradu- tive year that Diao wards. Georgios Moschidis ate and Graduate has received this Engineering Stu- honor. The fellowships support the final dent Councils. This Kathleen Emerson and Hansheng Diao year of study at Princeton and are student-nominated awarded to one Ph.D. student in each award is for a professor or TA who MOSCHIDIS RECEIVES of the four divisions (humanities, "was especially dedicated, taught the Top Graduate Honor social sciences, natural sciences and material clearly, and simply de- engineering) whose work has exhib- Georgios Moschidis is one of four serves to be recognized for the hard ited the highest scholarly excellence. winners of this year’s Porter Ogden work he or she put into the course." Jacobus Fellowship, Princeton These teaching awards are entirely

DEPARTMENT TEACHING AWARDS Graduate Student and Junior Faculty not hesitate to put forth the effort for This is the fifth year the Department described as inspirational, as someone students who are struggling. Hansh- of Mathematics Graduate and Junior who is always willing to dedicate extra eng’s lectures were described as excep- Faculty Teaching Awards have been time for students when needed. Artem tionally clear, well-organized, inspir- given. The awards were created to makes sure the entire class compre- ing, informative, and challenging, and recognize excellence in teaching at hends complicated concepts and does one student described the undergraduate level. him as “literally the This year’s graduate re- best teacher I’ve had cipients are Alexandros here at Princeton.” Eskenazis and Artem Ana has been an inspi- Kotelskiy, who are cur- ration to her students rent students. The Junior with an engaging and Faculty Teaching Award approachable teaching was given to Instruc- style that one student tor Hansheng Diao and described as a “little Assistant Professor Ana lightbulb going off in Menezes to recognize ex- my head.” cellence in teaching at the Selection for both undergraduate level. awards was made by Alexandros’ enthusiasm a committee of faculty for mathematics was Left to right: Artem Kotelskiy, Alexandros Eskenazis, Hansheng Diao. members. Not pictured: Ana Menezes Page 7 Distinguished Visitors The Minerva Program: DEMETRIOS CHRISTODOULU, LAURE SAINT-RAYMOND, PAUL SEIDEL, AND UMBERTO ZANNIER It was another productive year for the Department's Minerva Program. Umberto Zannier, this year's Minerva Laure Saint-Raymond, who specializes in differential Distinguished Lecturer, joined us from the Scuola Normale equations, delivered a mini-course with talks on Taylor Superiore-Pisa while , Laure Proudman columns: "rotation stabilizes the fluid around Saint-Raymond, and Paul Seidel came as Minerva Distin- the 2D flow..."; Ekman pumping: "rotation stabilizes the guished Visitors. boundary layer in the inviscid limit..."; : "when inertial waves meet topography..."; and Quasi-resonant forc- Umberto Zannier, the author of Lecture Notes on Diophan- ing: "how generalized eigenfunctions can be observed in a tine Analysis and the editor of Diophantine Geometry, bowl..." delivered a series of three Minerva Lectures on games and the Betti map: "The games of Steiner and Poncelet and Christodoulou, a professor emeritus at ETH Zürich who re- algebraic group schemes"; "Torsion values for sections in ceived his PhD from Princeton in 1971 and was a professor abelian schemes and the Betti map"; and "Ambients for the here from 1992—2001, visited the department in the Fall to Betti map and the question of its rank." These talks illus- collaborate with colleagues at Princeton. trated how one can vary the data of the games, obtaining families of elliptic curves and sections on elliptic schemes, Paul Seidel, professor at MIT, is visiting this Spring. While for which torsion values can be sought and related to the here, Seidel has lectured on the Fukaya categories of Calabi- Betti map. Yau hypersurfaces.

Special Colloquium for WEST POINT CADETS In November of 2017, Diana Thomas, a mathematics from West Point Military Academy who had a brief tour of professor at West Point Military Academy, inquired about the Princeton campus prior to the colloquium. the possibility of having a number of cadets attend the departmental colloquium. After consultations with the In the colloquium, Peter Sarnak spoke about the local-to- chair, Paul Yang organized a program of short talks about global principle in , with the interesting lead mathematical topics. question about sums of squares and cubes. Dave Gabai gave a talk about the role of in the topology of On February 17, a special mathematics colloquium was 3-manifolds: he illustrated the topics with a number of gad- held at Princeton for West Point cadets. On the program gets, including his own flock of nearly impossible to man- there were three short talks given by members of our de- age hair. Evita Nestoridi spoke about the optimal strategy partment on number theory, topology and probability. In to shuffle a deck of cards and the mathematical idea behind attendance were 15 cadets and five members of the faculty mixing time of two distinct card shuffling models. Faculty Research Mathematicians Find WRINKLE IN FAMED FLUID EQUATIONS By Kevin Hartnett, Quanta Magazine cian John Nash and was im- The Navier-Stokes equations ported more recently into the capture in a few succinct study of fluids by De Lellis terms one of the most ubiq- and Székelyhidi. uitous features of the physi- cal world: the flow of fluids. Using this approach, Buck- The equations, which date to master and Vicol prove that the 1820s, are today used to these very weak solutions to model everything from ocean the Navier-Stokes equations currents to turbulence in the are nonunique. They demon- wake of an airplane to the flow strate, for example, that if you of blood in the heart. Tristan Buckmaster Vlad Vicol start with a completely calm fluid, like a glass of water sit- While consider the equa- ting still by your bedside, two scenar- tions to be as reliable as a hammer, “We’re figuring out some of the inher- ios are possible. The first scenario is mathematicians eye them warily. To a ent issues with these equations and the obvious one: The water starts still , it means little that the why it’s quite possible [that] people and remains still forever. The second is equations appear to work. They want have to rethink them,” said Buckmas- fantastical but mathematically permis- proof that the equations are unfailing: ter. sible: The water starts still, erupts in that no matter the fluid, and no matter the middle of the night, then returns how far into the future you forecast its Buckmaster and Vicol’s work shows to stillness. flow, the mathematics of the equations that when you allow solutions to the will still hold. Such a guarantee has Navier-Stokes equations to be very Buckmaster and Vicol prove the proved elusive. The first person (or rough (like a sketch rather than a pho- existence of many nonunique weak team) to prove that the Navier-Stokes tograph), the equations start to output solutions (not just the two described equations will always work — or to nonsense: They say that the same above) to the Navier-Stokes equations. provide an example where they don’t fluid, from the same starting condi- The significance of this remains to be — stands to win one of seven Millen- tions, could end up in two (or more) seen. nium Prize Problems endowed by the very different states. It could flow one Clay Mathematics Institute, along with way or a completely different way. If Buckmaster and Vicol are also think- the associated $1 million reward. that were the case, then the equations ing in terms of layers, and they have don’t reliably reflect the physical world their sights set on Leray solutions Mathematicians have developed many they were designed to describe... — proving that those, too, allow for a ways of trying to solve the problem. multitrack physics in which the same New work posted online in Septem- In their new paper, Buckmaster and fluid from the same position can take ber raises serious questions about Vicol consider solutions that are even on more than one future form. whether one of the main approaches weaker than Leray solutions — solu- pursued over the years will succeed. tions that involve the same averag- “Tristan and I think Leray solutions The paper, by Tristan Buckmaster ing principle as Leray solutions but are not unique. We don’t have that yet, and Vlad Vicol also relax one additional requirement but our work is laying the foundation of Princeton University, is the first re- (known as the “energy inequality”). for how you’d attack the problem,” said sult to find that under certain assump- Vicol. tions, the Navier-Stokes equations They use a method called “convex provide inconsistent descriptions of integration,” which has its origins in This article excerpt was reprinted from the physical world. work in geometry by the mathemati- www.quantamagazine.org. Page 9 Undergraduate Program Congratulations to the GRADUATING WOMEN

A record number of ten women will graduate with BA in mathematics this June. The ten seniors are:

Boriana Gjura: Her thesis advised by Mark Zhandry of the Computer Science Department is on Obfuscating Compute- and-Compare Programs. Boriana will pursue a PhD in Theoretical Computer Science at Harvard.

Abby Hickok: Her thesis is on Khova- nov holomogy and advised by Zoltán

Szabó. Abby will start graduate school Front row from left to right: Abby Hickok, Gloria Yin, Luya Wang, Iden Kalemaj, Jessica Shi, Andreea Magalie. Back row standing left to right: Daphne Yang, Boriana Gjura, Rachel Spady. in mathematics at UC Berkeley. Inset photo: Casandra Monroe

Iden Kalemaj: She is working with Chun-Hung Liu on Jessica Shi: Her thesis, advised by Maria Chudnovsky, is Jones’ Conjecture in graph theory relating the number of on Dominating Sets in Graphs with No Long Induced Paths. minimal vertices needed to cover all cycles in a planar graph, Jessica will start a PhD program in Computer Science at and the maximal number of vertex-disjoint cycles that can be MIT. packed in a planar graph. Iden will be working as an Ana- lyst at Analysis Group in Boston next fall; she plans apply to Rachel Spady: She is working with Faruk Gul of the Eco- graduate school in combinatorics in a couple of years. nomics Department on market bargaining processes in the presence of intermediaries. Andreea Magalie: She is working with Simon Levin of the Ecology and Evolutionary Biology Department on the effect Luya Wang: She is working with John Pardon on the of network structure on stable social groups, in particular contact invariant in Heegaard Floer homology and some ap- how homophilous and targeted interactions affect the equi- plications of the open book decompositions. Luya will start librium points in a heterogenous population. Andreea will a math PhD program at UC Berkeley. start a PhD program in mathematical biology at Georgia Tech. Daphne Yang: Her thesis on Finite Orbits of Polynomial Automorphisms in Affine Three-Space with Applications is Casandra Monroe: Casandra works with John Pardon for advised by Peter Sarnak. After graduation, Daphne is going her thesis on the Volume Conjecture in knot theory. Casa- to work at Barclays Capital as a Trading Analyst on the US ndra is a Mellon Mays Undergraduate Fellow mentored Treasuries desk. by Jon Fickenscher. The Fellowship’s central mission is to increase diversity in the faculty ranks of higher education. Gloria Yin: Her thesis, advised by Fernando Codá As an initial step toward the Fellowship’s ultimate goal, Marques, is on Curvature on Discrete Surfaces. Gloria will Casandra has won an NSF Graduate Fellowship to pursue take a year off before starting math graduate school at either doctoral studies in mathematics at the University of Caltech or the University of Chicago. Illinois at Urbana-Champaign.

Page 10 Undergraduate Program

MATH CLUB IMPACTS by Matthew Tyler, Math Club President

This past year, the Math Club has continued to host social events, provide undergraduates with new opportunities to learn mathematics, and help spread mathematics and the joy of learning it to the broader community.

Roughly every two weeks, we organize a Board Game Night in the Fine Hall 3rd Floor Common Room. These gath- erings are often preceded by a panel or advising session about a topic relevant to Math Club members’ lives, such as The Math Club after board game night applying to graduate school or choos- ing math courses for next semester. offer the Mentoring Moebius peer Princeton This May, we are also hosting our usual mentorship program, which brings PUTNAM TEAM end-of-year banquet in the Professors’ together underclassmen, upperclass- WINS THIRD PLACE Lounge. men, and graduate students to meet in small groups over coffee or dinner At the 78th annual William Lowell Additionally, we host a colloquium each month to talk about math, life, or Putnam Mathematical Competition, each week, during which a professor anything in between. organized by the Mathematical As- gives a talk for undergraduates about sociation of America, includes the a topic related to mathematics. Recent This past year, PUMaC, the math competition we run for high school United States and Canada. In 2017, talks have been on subjects as varied 4,638 students from 575 institutions as minimal surfaces, modular forms, students, had another successful run. Now in its 12th year, PUMaC offers participated and spent two 3-hour and natural algorithms, and these talks sessions solving 12 problems. have been a great way to both gain high schoolers from around the coun- exposure to subjects not often discussed try and around the globe the chance to come to Princeton and spend the Princeton University placed third in the classroom, and to dive deeper in the team competition receiving into a topic of interest. Moreover, we day taking various mathematics tests and having fun. an award of $15,000 and each team member (Murilo Corato Zanarella, RYAN CHEN in careers in mathematics, the natural Zhuo Qun Song, and Xiaoyu Xu) will named sciences and engineering. receive $600. Additionally, Zhuo Qun (Alex) Song and Xiaoyu Xu Goldwater Scholar Chen, a mathematics major from were recognized as part of the next By Liz Fuller-Wright, Office of Lawrenceville, Georgia, is planning ten highest ranking individuals after Communictions to pursue a Ph.D. in mathematics the Putnam Fellows, with an award Prineton with the goal of entering academia. of $1,000 each. Rodrigo Angelo, Mu- junior “I am currently interested in study- rilo Corato Zanarella, Andrei Graur, Ryan Chen ing number theory, and have been and Eric Neyman received honorable pursuing this interest in my junior has been mention. independent work and summer awarded research,” Chen said. One- and two- a one-year year Goldwater Scholarships cover Goldwater The book of nature is tuition, fees, room and board up to a Scholarship, maximum of $7,500 per year. Chen written in the language an annual is one of 211 scholarship winners se- of Mathematics. award for lected from a field of 1,280 students —Galileo outstanding undergraduates interested nationwide.

Page 11 Graduate Program

Graduate Profile From the GEORGIOS MOSCHIDIS DIRECTOR OF GRADUATE STUDIES: ZOLTÁN SZABÓ I work in the field of , Being a Director of Graduate Studies (DGS) is a rewarding job. Together with which lies at the intersection of analy- my fellow DGS, Javi Gómez-Serrano, and with the help of our excellent grad- sis, geometry and physics. Of central uate administrator, Jill LeClair, we aim to provide guidance and support for our importance in this field is the ques- very talented graduate students. It is great to see them learning exciting new tion of stability of certain spacetime results and methods in Mathematics, and progressing from first year students to solutions of the Einstein equations outstanding researchers. under perturbations of their initial state. During my years as a graduate One of the more challenging aspects of the job is graduate admissions. This is student at Princeton, my research has a multi-step process, starting with a committee of seven faculty members who been focused on the stability proper- meet around the end of December and read all the applications. Later it involves ties of spacetimes modelling isolated, the full faculty and also at various points the Graduate School as well. This year stationary self-gravitating systems, as we had the annual Open House in the beginning of March. Almost all of the well as on instability phenomena as- newly admitted students visited Princeton to learn more about the Ph.D. pro- sociated to the presence of a negative gram and talk with students and professors. This was a one-day event, with lots cosmological constant. of interesting lectures by our own graduate students and faculty, who all helped recruit this remarkable group of students. At the end it was a great success. We Growing up in Greece, I finished my are excited to welcome a large and diverse class of 16 new graduate students, undergraduate studies at the Na- who will join us in Fall 2018. tional Technical University of Athens. There, I had the chance to attend a talk of Mihalis Dafermos on black holes, exposing me for the first time to a subject which instantly captivated my imagination. My admission to Princeton gave me the opportunity to join the vibrant relativity community here. The close interaction between students, postdocs and professors within this community never ceased to provide me with both ideas and motivation thoughout my graduate years. I also consider myself lucky to have entered the field at the right time: Despite the century-long history of general relativity, the necessary analytical tools for addressing many prominent questions set forth already Professor Szabó, Director of Graduate Studies in the 60s-70s, during the so-called newcomers in the field. that I met here, as well as all the people golden age of the theory, were only who made everyday life in Fine Hall recently developed (mostly by people Looking back at my years in Princeton, flow unobstructed. I will certainly miss in Fine Hall); as a result, a plethora of I am deeply grateful for all the friends all of them in the forthcoming years! interesting problems are accessible to

Page 12 Most recent Ph .D.s Name/ Field Undergrad Advisor Thesis Title Original Placement

Gregory Gauthier The structure of graphs with no Princeton University (Math)/ Structural Graph Georgia Tech Seymour cycles of length 0 (mod 3) Lecturer Theory Institute of Mathematical Sci- University of North Matthew Mechanisms of Lagrangian analyti- ences (ICMAT), Carolina, Chapel Fefferman Hernandez city in fluids Madrid/ Postdoctoral Hill PDEs Researcher Equidistribution in shrinking sets Australian University College London/ Peter Humphries Sarnak and L4-norm bounds for automor- National University Research Associate Number Theory phic forms

Dynamics of the incompressible Korea Institute for In-Jee Jeong Brown University Sinai Euler equations at critical Advanced Study/ PDEs regularity Postdoctoral Researcher

Kähler-Einstein metrics and nor- Yuchen Liu /Gibbs Peking University Kollár malized volumes of Algebraic Assistant Professor valuations Geometry Community detection in the sto- Princeton University Colin Sandon MIT Abbe chastic block model: (PACM)/Postdoctoral Re- Random Graphs fundamental limits search Associate

Nonlinear wave dynamics in black Radix Trading, Quantitative John Stogin Princeton University Klainerman hole spacetimes Technologist Hyperbolic PDEs

Mehdi Yazdi Sharif University of On Thurston’s Euler class one University of Oxford/ Gabai Topology Technology conjecture Glasstone Fellow

Ruixiang Zhang Perturbed Brascamp-Lieb inequali- Harmonic Analysis Peking University Sarnak ties and applications to Parsell- IAS Member and Number Theory Vinogradov systems

“I feel like I won the grad school lottery. I can't imagine a better place to concentrate on math. I have made friends who came to Fine Hall from all over the world. I had the opportunity to do an internship I probably wouldn't have heard of if it weren't for my advisor, Peter Sarnak, and also got to supervise undergraduate research. The Minerva program hosting long-term visitors gave me a chance to meet my future postdoc boss, Maryna Viazovska. It's been a great experience, and I owe a big thank you to a lot of the people reading this!” — Matthew de Courcy-Ireland

Page 13 Advisory Council ADVISORY COUNCIL and strengthen its status as one of the We are grateful to each of the council Visits world's preeminent institutions for members for taking time out of their mathematical research and instruction. busy schedules to come to Princeton and provide this valuable outside review. On April 19 and 20th this year the De- partment's Advisory Council convened at Fine Hall. The council, composed of distinguished alumni and other math- ematicians with close ties to the depart- ment, met with faculty, graduate and undergraduate students, staff, and the Dean of the Faculty over the two-day meeting.

The council meets every few years, last in 2014, to evaluate the state of the Department and to provide rec- ommendations on how it can retain From left: János Kollár (Dept. Assoc. Chair), John Morgan, Sylvain Cappell, Jean Taylor, Luisa Fernholz (Council Chair), Flavio Bartmann, Elliott Stein, Henry Laufer, David Gabai (Dept. Chair) TWIN Pictures By Evan O'Dorney (G2)

It is exciting when a simple formula turns out to yield some dazzling artwork. It is even more exciting when two simple and apparently unrelated formulas give subtly connected artistic results. On the left, we have an image of the projective plane , made by plotting the points (a/d,b/d) where a, b, and d are integers in a bounded range. This image is relevant to the simultaneous Diophantine approximation of irrational numbers and the design of musical scales. On the right, we have colored each integer-coordinate point black or white according to whether its distance from the origin, , is above or below its closest integer. • The striking dark bands on the right occur at points that nearly form a 3-4-5 right triangle, corresponding to the circle x2 + y2 = 1 on the left passing through the points (3/5,4/5) and (4/5,3/5) (red); • Between those bands is a grating of elliptical arcs that can be explained by the circle's passing near the point 2/3,2/3 (blue).

Page 14 Fine Hall Update NEW LOOK for Fine Hall

The summer of 2017 was a busy time in Fine Hall; but rather than halls filled with summer programs and conferences, the building was emptied and taken over by construction crews making improvements both seen and unseen.

Beginning the day after Commence- ment, every office was boxed up and emptied of the old furniture—much of it original to the building's debut in 1969. With the building emptied of a much-needed update. When the ensure a comfortable environment for persons and things, a crew of over 50 original Fine Hall—now Jones Hall— quiet study, friendly meetings, and, of began installation of a fire sprinkler was built, Oswald Veblen insisted that course, afternoon tea. system throughout the building. the faculty have quarters more like private studies than traditional offices. Many thanks are due to Professor Igor The third floor administrative area was We're glad to be able to keep part of Rodnianski who spearheaded this ef- completely renovated, including a glass this tradition alive, with all faculty fort, the staff who helped coordinate it, wall between the reception desk and offices having full sofas, creating an and the University's central adminis- the new mail room, bringing natural atmosphere conducive to individual tration for funding this project. light to the Department's front door. research, collaboration, and meeting with students. As the major construction work was Department completed, all offices were outfitted ADMINISTRATION The Common Room, the Department's with brand new carpet and furniture— gathering place, was also refreshed to David Gabai Chair János Kollár Associate Chair Zoltán Szabó and Javier Gómez-Serrano Directors of Graduate Study János Kollár Department Representative Jennifer Johnson Assoc. Department Representative John Pardon Senior Advisor Mark McConnell Junior Advisor Vlad Vicol Placement Officer Page 15 Department of Mathematics Fine Hall — Washington Rd. Princeton, NJ 08544 math.princeton.edu [email protected]

Alumni , faculty, students, friends,Annual Alumni connect with us, write to us at OPEN HOUSE Friday, June 1, 2018 • 2:00-3:30pm 3rd Floor Common Room, Fine Hall Incoming Graduate Students China • India • Israel • Korea • Spain • Caltech • Cornell Univeristy Harvard University • MIT • NYU • UC Berkeley • University of Chicago

Undergraduate Class of 2017 11 Graduates in Graduate School 19 Employed: Education • Finance and Insurance • Health Care and Social Assistance • Information, Professional Scientific, and Technical Services