Simons Investigators

Nick Sheridan's work centers around Kontsevich's homological mirror symmetry conjecture, which posits a deep relationship between symplectic topology and algebraic geometry. Working mainly on the symplectic side, he has developed tools for proving the conjecture and applied them to prove the conjecture in a number of cases, most notably the quintic threefold. In cases where the conjecture is established, Sheridan has given applications to enumerative geometry and symplectic topology.

Assaf Naor’s research focuses on analysis and geometry in high dimensions, with emphasis on understanding the structure of metric spaces, including the extent to which they can be realized in “nicer” geometries. He harnesses such insights for a variety of applications in several areas of pure mathematics (analysis, geometry, probability, combinatorics, group theory), and also in order to chart the possibilities and limitations of algorithms. Much of Naor’s work makes progress on the long-standing Ribe program, which is a web of conjectures and analogies between linear and nonlinear geometries that is inspired by a classical rigidity theorem of Ribe.

Alexei Borodin is a professor of mathematics at the Massachusetts Institute of Technology. He studies problems on the interface of representation theory and probability that link to combinatorics, random matrix theory and integrable systems. His most recent work carries over the ideas and techniques of the theory of symmetric functions to solvable lattice models of statistical physics.

Elchanan Mossel’s primary research fields are probability theory, combinatorics, theoretical computer science and statistical inference. Mossel is broad and collaborative in his research. Much of his work spans different areas of mathematics or bridges between mathematics and other sciences. With collaborators, he made fundamental contributions to discrete Fourier analysis and its applications to computational complexity and voting theory. In the area he named ‘combinatorial statistics,’ his collaborative work includes important discoveries on tree broadcast models and associated reconstruction problems, detection of block models, the inference of evolutionary histories and, more recently, deep inference.

Simon Brendle has achieved major breakthroughs in geometry including results on the Yamabe compactness conjecture, the differentiable sphere theorem (joint with R. Schoen), the Lawson conjecture and the Ilmanen conjecture, as well as singularity formation in the mean curvature flow, the Yamabe flow and the Ricci flow.

Vladimir Markovic has made fundamental contributions to the theory of three-dimensional manifolds, resolving several long-standing problems, among them the proof of the Thurston conjecture concerning immersed almost-geodesic surfaces in closed hyperbolic three-manifolds.

James McKernan, in collaboration with Caucher Birkar, Paolo Cascini, and Christopher D. Hacon, has established one of the cornerstones of the Minimal Model Program: the finite generation of canonical rings in all dimensions.

Ian Agol has made major contributions to three-dimensional topology and hyperbolic geometry, completing some of Thurston’s problems elucidating the structure of 3-manifolds. He proved several deep and long-standing conjectures, including the Virtual Haken conjecture, the Marden Tameness conjecture and the Simon conjecture.

Ben Green is an expert in analytic number theory. Among his achievements is the Green–Tao theorem, establishing that primes contain arbitrarily long arithmetic progressions.

Raphaël Rouquier has initiated a new field in mathematics, ‘higher representation theory.’ He constructed novel categories of geometric and representation-theoretic interest and applied these to problems in the theory of finite groups, Lie theory, algebraic geometry and mathematical physics.

Christopher Skinner works in number theory and arithmetic geometry. One of his striking recent results is a proof, in joint work with collaborators, that a positive proportion of elliptic curves defined over the rational numbers satisfy the Birch–Swinnerton-Dyer conjecture.

Alex Eskin is a leading geometer with important contributions to geometric group theory, ergodic theory and number theory. He has applied ideas from dynamical systems to solve counting problems in the theory of Diophantine equations, the theory of the mapping class group and mathematical billiards on rational polygons.

Larry Guth is a geometer with outstanding contributions to Riemannian geometry, symplectic geometry and combinatorial geometry. In Riemannian geometry, he solved a long-standing problem concerning sharp estimates for volumes of k-cycles. In symplectic geometry, he disproved a conjecture concerning higher-dimensional symplectic invariants by constructing ingenious counterexamples. In combinatorial geometry, he adopted a recent proof of the finite field analog of the Kakeya problem to the Euclidean context. He and Bourgain established the best current bounds to the restriction problem. Extending this work, he and Katz essentially solved one of the most well-known problems in incidence geometry, Erdős’s distinct distance problem, which was formulated in the 1940s.

Richard Kenyon’s central mathematical contributions are in statistical mechanics and geometric probability. He established the first rigorous results on the dimer model, opening the door to recent spectacular advances in the Schramm–Loewner evolution theory. In most recent work, he introduced new homotopic invariants of random structures on graphs, establishing an unforeseen connection between probability and representation theory.

Andrei Okounkov’s work spans a wide range of topics at the interface of representation theory, algebraic geometry, combinatorics and mathematical physics. He has made major contributions to enumerative geometry of curves and sheaves, the theory of random surfaces and random matrices. His papers reveal hidden structures and connections between mathematical objects and introduce deep new ideas and techniques of wide applicability.

Ngô’s proof of the fundamental lemma, a deep conjecture of Langlands, inaugurated a new geometric approach to problems in harmonic analysis based on arithmetic geometry. His ideas have already inspired work in many areas, including mathematical physics and geometric representation theory.

Mirzakhani’s work was focused on Teichmüller theory and dynamics of natural geometric flows over the moduli space of Riemann surfaces. One of her major results, in joint work with Eskin and Mohammadi, is a proof that stationary measures for the action of SL2(R) on the space of flat surfaces are invariant, a deep and long-standing conjecture.

Soundararajan is one of the world’s leaders in analytic number theory and related areas. His work is focused on understanding the zeros and value distribution of L-functions, and on analyzing the behavior of multiplicative functions. In particular, his work (together with co-authors) has led to weak subconvexity bounds for general L-functions and to the proof of the holomorphic quantum unique ergodicity conjecture of Rudnick and Sarnak.

Tataru’s work on nonlinear waves has been deep and influential. He proved difficult well-posedness and regularity results for many new classes of equations. This includes geometric evolutions such as wave and Schrödinger maps, quasilinear wave equations, some of which are related to general relativity, as well as other physically relevant models.

Manjul Bhargava pursues algebraic number theory and the geometry of numbers in the tradition of Gauss and Minkowski. Bhargava has inspired an extraordinary resurgence of this field, with wonderful applications. His overarching goal in this work is to count the basic objects of number theory and to make computational conclusions about their asymptotics. For example, it is conjectured that, in a certain natural sense, the average rank of the group of rational points of an elliptic curve defined over the rationals is 1/2. Bhargava and his student Shankar recently showed that it is less than 1. Previously, it was not even known whether the average rank is finite. In joint work with Dick Gross, Bhargava has also shown that the number of rational points on the majority of hyperelliptic curves is bounded by a certain small number independent of the genus of the curve. This work opens up remarkable new vistas in arithmetic and suggests exciting conjectures.

Alice Guionnet has done very important work on the statistical mechanics of disordered systems (and in particular the dynamics and aging of spin glasses), random matrices (with an emphasis on the combinatorics of maps), and operator algebra/free probability. Her work on large deviations for spectra of random matrices has been very influential. She has extended the large deviation principle to the context of Voiculescu’s free probability theory, and in collaboration with Cabanal-Duvillard, Capitaine, and Biane she proved various large deviation bounds in this more general setting. These bounds enabled her to prove an inequality between the two notions of free entropy given by Voiculescu, settling half of the most important question in the field. With her former students M. Maida and E. Maurel-Segala and more recently with Vaughan Jones and D. Shlyakhtenko, Guionnet has studied statistical mechanics on random graphs through multimatrix models. Their work on the general Potts models on random graphs branches out in promising directions within operator algebra theory. Guionnet resigned her Investigatorship in 2016 to move to the École normale supérieure de Lyon in France.

Christopher Hacon’s works are among the most important contributions to higher-dimensional algebraic geometry since Mori’s in the 1980s. Hacon and his co-authors have solved major problems concerning the birational geometry of algebraic varieties, including the characterization of irregular varieties, boundedness theorems for pluricanonical maps, a proof of the existence of flips, the completion of the minimal model program for varieties of general type, and bounds for the order of automorphism groups of varieties of general type. His work has also led to solutions of other problems, such as the existence of moduli spaces for varieties of general type and the ascending chain condition for log canonical thresholds.

Paul Seidel has done major work in symplectic geometry, in particular on questions inspired by mirror symmetry. His work is distinguished by an understanding of abstract algebraic structures such as derived categories, in sufficiently concrete terms to allow one to derive specific geometric results. On the abstract side, Seidel has made substantial advances towards understanding Kontsevich’s homological mirror symmetry conjecture and has proved several special cases of it. In joint papers with Smith, Abouzaid and Maydanskiy, he has investigated the symplectic geometry of Stein manifolds. In particular, work with Abouzaid constructs infinitely many nonstandard symplectic structures on any Stein manifold of sufficiently high dimension.

Amit Singer works on a broad range of problems in applied mathematics, solving specific applied problems and employing sophisticated theory to allow the solution of general classes of problems. Among the areas to which he has contributed are diffusion maps, cryo-electron microscopy, random graph theory, sensor networks, graph Laplacians, and diffusion processes. His recent work in electron microscopy combines representation theory with a novel network construction to provide reconstructions of structural information on molecules from noisy two-dimensional images of populations of the molecule. He works with a widely varied group of collaborators and graduate students in several disciplines. His work is increasing the range of applicable mathematics. Singer resigned his Math Investigatorship in 2017 to accept the Math+X Investigator award.

Terry Tao is one of the most universal, penetrating and prolific mathematicians in the world. In over 200 publications (in just 15 years) spanning collaborations with nearly 70 mathematicians, he has established himself as a major player in the disparate fields of harmonic analysis, partial differential equations, number theory, random matrices, and more. He has made deep contributions to the development of additive combinatorics through a blend of harmonic analysis, ergodic theory, geometry and number theory, establishing this field as central to the modern study of many mathematical subjects. This work has led to extraordinary breakthroughs in our understanding of the distribution of primes, expanders in groups, and various questions in theoretical computer science. For example, Green, Tao, and Ziegler have proved that any finite set of linear forms over the integers, of which no two are linearly dependent over the rationals, all take on prime values simultaneously infinitely often, provided there are no local obstructions.

Horng-Tzer Yau is one of the world’s leading probabilists and mathematical physicists. He has worked on quantum dynamics of many-body systems, statistical physics, hydrodynamical limits, and interacting particle systems. Yau approached the problems of the quantum dynamics of many-body systems with tools he developed for statistical physics and probability. More recently, he has been the main driving force behind some stunning progress on bulk universality for random matrices. With Laszlo Erdős and others, Yau has proven the universality of the local spectral statistics of random matrices, a problem that was regarded as the main challenge of random matrix theory. This argument applies to all symmetry classes of random matrices. In the special Hermitian case, Terence Tao and Van Vu proved bulk universality concurrently. Yau’s work has been extended in many directions, for instance in his recent results on invariant beta ensembles with Paul Bourgade and Laszlo Erdős.