Spring 2015 Issue 4 Department of Mathematics Princeton University Abel Prize for John Nash *50 From the Chair John Nash *50 received the 2015 Abel Prize This has been quite a year for from the Norwegian Academy of Science and the Mathematics Department; Letters for his work on partial differential however, space allows for mentioning but equations. Nash shares the $800,000 prize a few of the highlights. To start with, we with Louis Nirenberg, a professor emeritus are overjoyed to have Maria Chudnovsky, at NYU's Courant Institute of Mathemati- Fernando Marques and Assaf Naor as new cal Sciences. The prize recognized Nash and members of the senior faculty. Their pres- Nirenberg for “striking and seminal contri- ence has already made a major impact on butions to the theory of nonlinear partial the Department. differential equations and its applications to Manjul Bhargava *01 won the Fields medal geometric analysis.” Nash’s name is attached at last summer’s International Congress of to a range of influential work in mathematics, Mathematicians in Seoul, Korea. Manjul is including the Nash-Moser inverse function not only an extraordinary mathematician theorem, the Nash-De Giorgi theorem (which stemmed from a problem Nash undertook but also an unusually gifted teacher and at the suggestion of Nirenberg), and the Nash embedding theorems, which the academy tabla player. described as “among the most original results in geometric analysis of the twentieth century.” According to David Gabai, the Nash embedding/immersion theorems, which John Nash *50 and Louis Nirenberg from required unusual insight as well as tremendous technical expertise, played an important NYU are this year’s Able Prize winners. role in the development of young mathematicians who came in contact with Nash at While John is famous for winning the 1994 Princeton in the 1970s. Nobel prize in Economics for his Ph. D. the- sis work on non-cooperative games, this is the first major recognition for his extraor- dinary work on isometric embeddings and partial differential equations which most mathematicians consider incomparably deeper work. With Yasha Sinai winning the Abel prize in 2014, this marks the first time Fields Medal for the Abel prize has gone to mathematicians Manjul Bhargava *01 at the same institution in consecutive years. Manjul Bhargava is awarded a Fields Last summer saw the passing of two long Medal for developing powerful new time professors, Harold Kuhn 50* and Ed methods in the geometry of numbers, Nelson. Harold and Ed were original and which he applied to count rings of small influential mathematicians who respectively rank and to bound the average rank of changed the face of mathematical econom- elliptic curves. ics and constructive quantum field theory. A special day was held on April 22, 2015 to from the award citation commemorate Ed’s work. Our finishing graduate students wrote mar- Manjul Bhargava *01, the Brandon Fradd, sity). The prize comes with a monetary velous theses and obtained excellent first Class of 1983, Professor of Mathematics, award of Canadian $15,000 as well. positions. Of the twenty Sloan Foundation was awarded the 2014 Fields Medal on Bhargava’s Ph.D. thesis (written under fellowships in mathematics awarded this August 13, at the opening ceremony of the the direction of Andrew Wiles, the James year, eight went to former Princeton grad- 2014 International Congress of Mathemati- S. McDonnell Distinguished University uate students; Richard Bamler, Boris Bukh, cians in Seoul, Republic of Korea by South Professor of Mathematics, Emeritus), Vivek Shende, Andrew Snowden, Jacob Korean President Park Geun-hye. provided a reformulation of Gauss’s law for Tsimerman, Melanie Wood, and Hau-tieng For 2014, there were four Fields Medalists the composition of two binary quadratic Wu from Math and Lin Lin from PACM. In (Artur Avila of the CNRS in Paris, Martin forms. He showed that the orbits of the addition, assistant professors Vlad Vicol and Hairer of the University of Warwick in group SL(2,Z)3 on the tensor product Tasho Kaletha were also winners. the UK, and former Princeton professor of three copies of the standard integral We have many excellent teachers among Maryam Mirzakhani of Stanford Univer- continued on next page continued on next page ....continued Fields Medal for Manjul Bhargava (continued) Department Chair’s letter (continued) representation correspond to quadratic Persi Diaconis, whose research incorpo- our faculty and students. Peter Sarnak won rings (i.e., rings of rank 2 over Z) together rates decks of cards and coin-flipping with the University’s Phi Beta Kappa teaching with three ideal classes whose product is other random problems. Bhargava thought award and grad student Thomas Beck won trivial. This recovers Gauss’s composition they would be a good way to introduce stu- a teaching award from the Graduate School. law in an original and computationally ef- dents to a field considered dry and daunt- Departmental teaching awards went to in- fective manner. He then studied orbits in ing to many. “The Mathematics of Magic structors Jeffrey Case and Mihai Fulger more complicated integral representations, Tricks and Games” was designed to show and graduate students Nathan Dowlin and which correspond to cubic, quartic, and the artistic and creative side of mathemat- Thomas Beck. quintic rings, and counted the number of ics, which Bhargava asserted is how math- Departmental staff play the crucial behind rings with bounded discriminant. ematicians approach their research. He the scenes role that enables us to focus on Bhargava next turned to the study of repre- wanted to offer this seminar to freshmen our research, teaching and studies. Eszter sentations with a polynomial ring of invari- to capture their enthusiasm early in their Rudy joined the Department as Grants and ants. The simplest such representation is college careers while introducing them Business Manager, replacing Alberta Molnar given by the action of PGL(2,Z) on the to the ‘correct’ and ‘fun’ side of math- who got promoted to a managerial position space of binary quartic forms. This has two ematics. For this initiative, Bhargava was in the Sponsored Research office, and is off independent invariants, which are related awarded several grants from the University, to a great start. to the moduli of elliptic curves. Together including the 250th Anniversary Fund for Innovation in Undergraduate Education Our Department Manager Kathy Apple- with his graduate student, Arul Shankar, gate received the Presidential Achievement Bhargava used delicate estimates on the and a grant from the University’s Council on Science and Technology. Award. Kathy has done tremendous work number of integral orbits of bounded for the Department and so we are delighted height to bound the average rank of elliptic Students appreciated his enthusiasm and that she has been recognized at the highest curves. Generalizing these methods to outstanding commitment to his classes. university level. curves of higher genus, he recently showed His availability to the students and dedica- that most hyperelliptic curves of genus at tion to providing interesting topics and Coming full circle, Scott Kenney has an- least two have no rational points. visiting lecturers was unsurpassed in the nounced his pending retirement. To many Department. of us Scott was the embodiment of the Math Bhargava’s work is based both on a deep Department. During his 25 years as De- understanding of the representations of Sharing his love of mathematics extends be- partment Manager, chairs came and went, arithmetic groups and a unique blend of yond the university. He recently returned but from day visitors to long-time faculty he algebraic and analytic expertise. from a tour of India where he traveled to was there as a constant, welcoming and ap- Bhargava joined the Mathematics Depart- a large number of high schools to discuss proachable soul always ready to tackle prob- ment as a full professor in 2003. In addi- the beauty of mathematics with students. lems from the complex to the mundane. He is also head of a committee devoted to tion to the Fields Medal, he has received While members of our faculty, students and numerous other awards including the 2012 bringing scientists from all over the world to visit schools throughout India. staff have received high profile recognitions Infosys Prize, the 2011 Fermat Prize (pre- for their research, teaching and service, sented by the Toulouse Mathematics Insti- Bhargava was born in Ontario, Canada but their awards reflect but a small fraction of tute in France) and the Packard Foundation raised in New York on Long Island with the tremendous productivity, energy, and Fellowship in Science and Engineering in frequent trips to Jaipur, India, to visit fam- enthusiasm that permeates Fine Hall. 2004. He was elected to the U.S. National ily. His interests include Sanskrit poetry Academy of Sciences in 2013. and Tabla drumming. Considered a tabla Finally, it is my pleasure to thank Bob and expert, he has given numerous free tabla Luisa Fernholz, Wei-Tong Shu and the Class Bhargava is an outstanding teacher on both of 1971 Fund for their very generous and the graduate and undergraduate levels. In performances at student cultural programs, international fairs, and charity benefits on ongoing support of the activities of our De- recent years, he developed a very popular partment. freshman seminar called “The Mathemat- both the Harvard and Princeton campuses. ics of Magic Tricks and Games,” which had Thank you for reading this newsletter and the maximal enrollment of 15 students reconnecting with the Department. I’d love in Spring 2013 and Spring 2015. He also to hear your thoughts and views. taught “The Magic of Numbers” in the Amazing. Everyone was engaged and it was always so much fun. I don’t know how this course could be improved.