Mathematics People
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Mathematics People half of the most important question in the field. With 2012 Simons Investigators her former students M. Maida and E. Maurel-Segala and Named more recently with Vaughan Jones and D. Shlyakhtenko, Guionnet has studied statistical mechanics on random The Simons Foundation has named twenty-one mathema- graphs through multimatrix models. Their work on the ticians, theoretical physicists, and theoretical computer general Potts models on random graphs branches out in scientists as Simons Investigators for 2012, the inaugural promising directions within operator algebra theory. year of the program. The Simons Investigators program Christopher D. Hacon of the University of Utah has provides a stable base of support for outstanding sci- produced some of the most important contributions to entists, enabling them to undertake long-term study of higher-dimensional algebraic geometry since Mori’s in fundamental questions. The names and institutions of the the 1980s. Hacon and his coauthors have solved major awardees whose work involves the mathematical sciences problems concerning the birational geometry of algebraic and brief excerpts from the prize citations follow. varieties, including the characterization of irregular va- Manjul Bhargava of Princeton University pursues rieties, boundedness theorems for pluricanonical maps, algebraic number theory and the geometry of numbers a proof of the existence of flips, the completion of the in the tradition of Gauss and Minkowski. His overarching minimal model program for varieties of general type, and goal in this work is to count the basic objects of number bounds for the order of automorphism groups of variet- theory and to make computational conclusions about ies of general type. His work has also led to solutions of their asymptotics. For example, it is conjectured that, in other problems, such as the existence of moduli spaces for a certain natural sense, the average rank of the group of varieties of general type and the ascending chain condition rational points of an elliptic curve defined over the ratio- for log canonical thresholds. nals is 1/2. Bhargava and his student Shankar recently Paul Seidel of the Massachusetts Institute of Tech- showed that it is less than 1. Previously, it was not even nology has done major work in symplectic geometry, in known whether the average rank is finite. In joint work particular on questions inspired by mirror symmetry. His with Dick Gross, Bhargava has also shown that the number work is distinguished by an understanding of abstract of rational points on the majority of hyperelliptic curves algebraic structures, such as derived categories, in suf- is bounded by a certain small number independent of the ficiently concrete terms to allow one to derive specific genus of the curve. geometric results. On the abstract side, Seidel has made Alice Guionnet of the Massachusetts Institute of substantial advances toward understanding Kontsevich’s Technology has done very important work on the statis- homological mirror symmetry conjecture and has proved tical mechanics of disordered systems (and in particular several special cases of it. In joint papers with Smith, the dynamics and aging of spin glasses), random matrices Abouzaid, and Maydanskiy, he has investigated the sym- (with an emphasis on the combinatorics of maps), and plectic geometry of Stein manifolds. In particular, work operator algebra/free probability. She has extended the with Abouzaid constructs infinitely many nonstandard large deviation principle to the context of Voiculescu’s symplectic structures on any Stein manifold of sufficiently free probability theory, and, in collaboration with Cabanal- high dimension. Duvillard, Capitaine, and Biane, she proved various large Amit Singer of Princeton University works on a broad deviation bounds in this more general setting. These range of problems in applied mathematics, solving specific bounds enabled her to prove an inequality between the applied problems and employing sophisticated theory to two notions of free entropy given by Voiculescu, settling allow the solution of general classes of problems. Among 1452 NOTICES OF THE AMS VOLUME 59, NUMBER 10 Mathematics People the areas to which he has contributed are diffusion maps, ing of the PCP theorem) and the power and limitations of cryoelectron microscopy, random graph theory, sensor hierarchies of linear and semidefinite programs. networks, graph Laplacians, and diffusion processes. His Shafrira Goldwasser of the Massachusetts Institute recent work in electron microscopy combines representa- of Technology has had tremendous impact on the devel- tion theory with a novel network construction to provide opment of cryptography and complexity theory. Start- reconstructions of structural information on molecules ing with her thesis on “semantic security”, she laid the from noisy two-dimensional images of populations of foundations of the theory of cryptography. She created the molecule. rigorous definitions and constructions of well-known Terence Tao of the University of California Los An- primitives, such as encryption schemes (both public and geles has produced more than 200 publications in just private key versions) and digital signatures, and of new fifteen years, spanning collaborations with nearly seventy ones that she introduced, such as zero-knowledge inter- mathematicians and establishing himself as a major player active proof systems invented with Micali and Rackoff. in the disparate fields of harmonic analysis, partial dif- Continuing her work on interactive proofs which allow a ferential equations, number theory, random matrices, and probabilistic polynomial time algorithm to verify math- more. He has made deep contributions to the development ematical proofs via interaction with a powerful prover, of additive combinatorics through a blend of harmonic she and her coauthors extended the notion of interactive analysis, ergodic theory, geometry, and number theory, proofs to two-prover systems. They turned out to be of establishing this field as central to the modern study of great significance in complexity theory, paving the way to many mathematical subjects. This work has led to ex- the equivalent formulation of PCP. The expressive power traordinary breakthroughs in our understanding of the of two-prover systems is huge (nondeterministic expo- distribution of primes, expanders in groups, and various nential time). Furthermore, she and her coauthors showed questions in theoretical computer science. For example, the connection between a scaled down variant of these Green, Tao, and Ziegler have proved that any finite set systems and the hardness of approximation results for of linear forms over the integers, of which no two are NP-hard problems, which led to the PCP theorem. On the algorithmic front, a problem of great significance is that linearly dependent over the rationals, all take on prime of recognizing (and generating) prime numbers. Goldwas- values simultaneously infinitely often, provided there are ser and Kilian designed efficient probabilistic primality no local obstructions. provers, which output short proofs of primality, based Horng-Tzer Yau of Harvard University is one of the on the theory of elliptic curves. Together with Goldreich world’s leading probabilists and mathematical physicists. and Ron, Goldwasser originated the field of combinatorial He has worked on quantum dynamics of many-body property testing, devising a class of sublinear algorithms systems, statistical physics, hydrodynamical limits, and to test properties in dense graphs. interacting particle systems. Yau approached the problems Russell Impagliazzo of the University of California of the quantum dynamics of many-body systems with San Diego has made many deep contributions to cryp- tools he developed for statistical physics and probability. tography and complexity theory. He and his collabora- More recently, he has been the main driving force behind tors showed that one-way functions exist if and only if some stunning progress on bulk universality for random pseudorandom generators exist. In other words, one can matrices. With Laszlo Erdo˝s and others, Yau has proved generate sequences of bits for which it is computationally the universality of the local spectral statistics of random hard to predict the next bit with accuracy much better than matrices, a problem that was regarded as the main chal- random guessing if and only if there are easy-to-compute lenge of random matrix theory. functions that are hard to invert on the average. He also Sanjeev Arora of Princeton University has played a showed that there are worlds in which certain crypto- pivotal role in some of the deepest and most influential graphic primitives are strictly inequivalent. For example, results in theoretical computer science. He started his there are worlds in which one-way functions exist but career with a major contribution to the proof of the PCP public-key encryption is not possible. One of his major (probabilistically checkable proofs) theorem, widely re- contributions in complexity theory is the exponential-time garded as the most important result in complexity theory hypothesis and its implications. The hypothesis states that in the last forty years. The PCP theorem states roughly that there are problems for which it is hard to speed up the every proof, of any length, can be efficiently converted brute-force solution even by a small amount. Impagliazzo into a special format in which correctness can be verified