# Simons Investigators

The Simons Foundation congratulates the 21 mathematicians, theoretical physicists, and theoretical computer scientists who have been selected as Simons Investigators in 2012, the inaugural year of the program.

The Simons Investigators program provides a stable base of support for outstanding scientists, enabling them to undertake long-term study of fundamental questions.

**MATHEMATICS**

**Manjul Bhargava**

*Princeton University*

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**

*Massachusetts Institute of Technology*

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.

**Christopher Derek Hacon**

*The University of Utah*

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**

*Massachusetts Institute of Technology*

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**

*Princeton University*

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.

**Terence Tao**

*University of California, Los Angeles*

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**

*Harvard University*

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.

**COMPUTER SCIENCE**

**Sanjeev Arora**

*Princeton University*

Sanjeev Arora has played a pivotal role in some of the deepest and most influential results in theoretical computer science. He started his career with a major contribution to the proof of the PCP theorem, widely regarded as the most important result in complexity theory in the last 40 years. The PCP theorem states roughly that every proof, of any length, can be efficiently converted into a special format, in which correctness can be verified with high probability by reading small parts of it. The PCP theorem revolutionized our understanding of optimization problems and opened new directions in coding, cryptography and other areas. Arora is also known for his breakthroughs in approximation algorithms, having solved longstanding open problems. Notable examples include his algorithms for the Euclidean traveling salesman problem and for the sparsest cut in a graph. Arora has made important contributions on many other topics, including the unique games conjecture (a conjectured strengthening of the PCP theorem) and the power and limitations of hierarchies of linear and semidefinite programs.

**Shafrira Goldwasser**

*Massachusetts Institute of Technology*

Shafi Goldwasser has had tremendous impact on the development of cryptography and complexity theory. Starting with her thesis on “semantic security”, she laid the foundations of the theory of cryptography. She created rigorous definitions and constructions of well-known primitives such as encryption schemes (both public and private key versions) and digital signatures, and of new ones that she introduced, such as zero-knowledge interactive proof systems invented with Micali and Rackoff. Continuing her work on interactive proofs which allow a probabilistic polynomial time algorithm to verify mathematical proofs via interaction with a powerful prover, Shafi and her co-authors extended the notion of interactive proofs to two-prover systems. The original motivation was cryptographic, but they turned out to be of great significance in complexity theory, paving the way to the equivalent formulation of PCP (probabilistically checkable proofs). The expressive power of two-prover systems is huge (non-deterministic exponential time). furthermore, Shafi and her co-authors showed the connection between a scaled down variant of these systems and the hardness of approximation results for NP-hard problems, which led to the PCP theorem. On the algorithmic front, a problem of great significance is that of recognizing (and generating) prime numbers. Shafi and Kilian designed efficient probabilistic primality provers, which output short proofs of primality, based on the theory of elliptic curves. Together with Goldreich and Ron, Shafi originated the field of combinatorial property testing, devising a class of sub-linear algorithms to test properties in dense graphs.

**Russell Impagliazzo**

*University of California, San Diego*

Russell Impagliazzo has made many deep contributions to cryptography and complexity theory. Russell and collaborators showed that one-way functions exist if and only if pseudorandom generators exist. In other words, one can generate sequences of bits for which it is computationally hard to predict the next bit with accuracy much better than random guessing if and only if there are easy-to-compute functions that are hard to invert on the average. Russell also showed that there are worlds in which certain cryptographic primitives are strictly inequivalent. For example, there are worlds where one-way functions exist but public-key encryption is not possible. One of Russell’s major contributions in complexity theory is the exponential-time hypothesis and its implications. The hypothesis states that there are problems where it is hard to speed up the brute-force solution even by a small amount. Russell helped establish the first complete problem for this class. In joint work with Avi Wigderson, Russell showed that if there are problems in exponential time that require exponential-sized circuits to solve, then any efficient algorithm that uses randomization has an equivalent, efficient one that does not.

**Jon Kleinberg **

*Cornell University*

Jon Kleinberg is noted for his creativity, intellectual ability, research scholarship, diversity of research interests and the impact of his work. He is best known for his contributions in establishing the computational foundations for information retrieval and social networks. His information retrieval work includes the use of link analysis (e.g., hubs and authorities) for ranking, classification and identifying web communities, the web as a graph, and understanding the success of latent semantic analysis. His work in algorithmic social networks (a field that he can be said to have started) includes the understanding of “small worlds” and decentralized search, analysis of bursty streams and influence spread in social networks. Kleinberg has done work in many other fields, including approximation algorithms, communications networks, queuing theory, clustering, computational geometry, bioinformatics, temporal analysis of data streams, algorithmic game theory, online algorithms and distributed computing. His influence is augmented by popular papers in Science and Nature and by two widely used texts, one with Tardos, Algorithm Design, and one with Easley, Networks, Crowds, and Markets: Reasoning about a Highly Connected World.

**Daniel Spielman**

*Yale University*

Daniel Spielman’s work has been important to three distinct research communities: theoretical computer science, applied mathematics, and operations research. His work on smoothed analysis of linear programming provides mathematical justification for why the simplex method to solve problems works well in practice even though worst-case analysis shows that there are instances in which it takes exponential time. A small random perturbation converts any linear programming instance into one that, with high probability is solved efficiently by the simplex algorithm. Similar perturbation results hold for many other problems and provide an alternative to worst-case analysis, which may be too pessimistic. His codes based on expander graphs achieve near-optimal rate and nearly linear time encoding and decoding algorithms. In joint work with Teng, Spielman gave a method of preconditioning a Laplacian matrix A, which yields a near-linear-time algorithm for solving the system Ax = b. This leads to highly efficient algorithms for circuit analysis and network flow computations.

**PHYSICS**

**Igor Aleiner**

*Columbia University*

Igor Aleiner is an influential leader of condensed matter theory research, renowned both for his fundamental contributions to our understanding of the quantum mechanical interplay of electron-electron interactions and disorder in condensed matter systems (in particular many-body localization) and for the theoretical power displayed in his tour de force calculations. He has used a variety of quantum field theoretic and random matrix methods to obtain profound results in the theory of quantum chaos, the study of mesoscopic fluctuation effects in interacting electron systems, the theory of transport in interacting disordered systems, and the properties of graphene.

**Michael Brenner**

*Harvard University*

Michael Brenner is a versatile theoretical physicist whose diverse contributions involve collaborations with biologists, physicists, and engineers from a variety of subfields. His work seamlessly integrates analytical and computational approaches to solve problems ranging from fundamental issues in fluid mechanics to engineering design to the evolution of protein functionality and from the aerodynamics of whale flippers to the ejection of fungal spores. He is known for generating creative and original questions and answers. Particularly noteworthy are his achievements in understanding the singularities and nonlinearities that control how droplets, jets and sheets of fluid change shape and break up. His work in this area has potential impact for optimizing devices ranging from inkjet printers to cell sorters. His research has also led to the development of general methods for simplifying the dynamical models of many coupled oscillators that arise in contexts such as atmospheric chemistry.

**Sharon Glotzer**

*University of Michigan*

Sharon Glotzer is a leader in the use of computer simulations to understand how to manipulate matter at the nano- and meso-scales. Her work in the late 1990s demonstrating the nature and importance of spatially heterogeneous dynamics is regarded as a breakthrough. Her ambitious program of computational studies has revealed much about the organizing principles controlling the creation of predetermined structures from nanoscale building blocks, while her development of a conceptual framework for classifying particle shape and interaction anisotropy (patchiness) and their relation to the ultimate structures the particles form has had a major impact on the new field of “self-assembly’’. Glotzer recently showed that hard tetrahedra self-assemble into a quasicrystal exhibiting a remarkable twelve-fold symmetry with an unexpectedly rich structure of logs formed by stacks of twelve-member rings capped by pentagonal dipyramids.

**Matthew Hastings**

*Duke University*

Matthew Hastings’ work combines physical insight and mathematical power to make profound contributions to a range of topics in physics and related fields. His Ph.D. thesis produced breakthrough insights into the multifractal nature of diffusion-limited aggregation, a problem that had stymied statistical physicists for more than a decade. Hastings’ recent work has focused on proving rigorous results on fundamental questions of quantum theory, including the stability of topological quantum order under local perturbations. His results on area laws and quantum entanglement and his proof of a remarkable extension of the Lieb-Schulz-Mattis theorem to dimensions greater than one have provided foundational mathematical insights into topological quantum computing and quantum mechanics more generally.

**Chris Hirata**

*California Institute of Technology*

Chris Hirata is an outstanding young cosmologist and astrophysicist whose research ranges from purely theoretical investigations to original data analysis. He is known for his sophisticated analysis of radiative transfer through the epoch of reionization. He has also shown that primordial dark matter fluctuations can impact contemporary observations. His work with experimental and observational groups on systematizing the extraction of cosmological data from cross correlation of different extragalactic surveys is having an important impact on precision cosmology.

**Charles Kane**

*University of Pennsylvania*

Charles Kane and co-workers showed, extending earlier work by Thouless and collaborators, that the electronic band structures of all crystals could be classified in terms of the momentum space topology of the electronic states, and that as a consequence there exist protected states at interfaces between topologically nontrivial crystals and topologically trivial crystals. Along with related work by Shou-Cheng Zhang and others, Kane’s results have created a large and vibrant research field focused on the search for and measurement of topologically nontrivial materials, including materials that are topologically nontrivial as a result of broken symmetries. Kane’s recent work has turned towards applications for example the use of interfaces between topological insulators and ordinary superconductors to achieve a solid-state realization of Majorana fermions and the exploration of possible applications of these excitations in topological quantum computing.

**Hirosi Ooguri**

*California Institute of Technology*

Hirosi Ooguri is a mathematical physicist and string theorist of exceptional creativity and breadth. His work on Calabi-Yau manifolds has yielded important new insights into the D-brane structures crucial to string theory, while his work on the relationship of supersymmetric gauge theories to string theory and to gravity has fostered the rapid development of the AdS/CFT correspondence, which relates quantum properties of gauge theories to solutions of higher-dimensional classical field equations in the presence of black holes and curved space-time. He is perhaps best known for his innovations in the use of topological string theory to compute Feynman diagrams in superstring models.

**Frans Pretorius**

*Princeton University*

Frans Pretorius has made seminal contributions to the numerical solution of the equations of general relativity, in particular inventing a new computational scheme based on harmonic decomposition of the Ricci tensor, which is now a textbook method in the field. Thanks in large part to Pretorius’ innovations, accurate computer simulations of such general relativistic phenomena as the merger of two black holes have become possible for the first time after several decades of effort. These results enable the calculation of expected gravitational-wave signals that may be detected by present or planned gravitational wave observatories. Pretorius has also contributed to mathematical issues in general relativity such as the no-hair theorem in higher dimensions and the Gregory-Laflame instability of black strings.

**Eliot Quataert**

*University of California, Berkeley*

Eliot Quataert is an outstanding theoretical astrophysicist whose research combines many areas of physics, including gas dynamics, plasma physics, radiative transfer and nuclear physics. He is also known as a particularly effective mentor of students and postdocs. He has made fundamental contributions to the theory of astrophysical turbulence and transport properties in hot plasmas, as well as to stellar and black-hole astrophysics.