A central goal of my research is to mathematically understand how intricate structures behave as their “complexities” tend to infinity. I have studied this question under various guises, including analyzing statistical mechanical systems (crystal growth and interacting particle systems) with many particles; random matrices of high dimension; and surfaces of large genus. My work develops … Continued
The main topic of my research is nodal geometry. In the beginning of the 19-th century Napoleon Bonaparte set a prize for the best mathematical explanation of Chladni’s resonance experiments. Nodal sets observed in these experiments fascinated scientists for years, but our mathematical understanding of zeros of solutions of elliptic differential equations, such as nodal … Continued
I explore problems in geometric topology and related fields, including symplectic geometry. Topology is the study of properties of shapes which are preserved under continuous deformations (as if things are made of a clay-like material which can be deformed arbitrarily but which cannot be glued to itself or cut). Topological questions are by definition insensitive … Continued
My research interests lie at the intersection of two fields of mathematics: algebraic geometry (which studies solutions to systems of polynomial equations in many variables) and number theory (which studies properties and relationship of numbers). The interaction between these fields is often mediated through a third field: topology (which studies the qualitative features of shapes). … Continued
My research tackles some of the oldest unsolved mysteries of mathematics. In ancient Greece, Diophantus asked how many integral solutions equations have (for example, one integral solution to y^2=x^3+3x+5 is x=4, y=9). My research group works on such questions by exploiting a connection between the number of solutions to equations and the geometry of the … Continued
My research group works to unify algebraic structures within mathematics, build bridges between these structures and domains of physics, and discover universal phenomena within these domains. We have uncovered universal distributions (modern day parallels of the bell curve) in models of interface growth, traffic flow, mass transport, turbulence, and shock-fronts.
I am a mathematician whose research is in combinatorics and related fields of mathematics and computer science. In particular, my research includes extremal combinatorics, algebraic and probabilistic methods in combinatorics, Ramsey theory, graph theory, additive combinatorics, combinatorial geometry, and applications of combinatorics to theoretical computer science. My research on regularity methods and their applications has … Continued
I work at the crossroads of algebraic geometry, representation theory, and number theory. I’m interested in applying methods from geometry to solve problems in regarding numbers and symmetries, especially those related to the Langlands program.
My general research interests include: Partial differential equations, non-equilibrium statistical physics, Interacting particle systems, Probability theory, Quantum dynamics of many-body systems. My recent interests include: Gross-Pitaevskii equation, dynamics of Bose-Einstein condensation, energy of Bose gas, diffusion of random Schrodinger equations and local Wigner semicircle law for random matrices.
Bounds for exponential sums are fundamental throughout much of (analytic) number theory, and key to the robustness of applications in theoretical computer science, cryptography, and so on. They are the primary tool for testing equidistribution (apparent “randomness”) of number theoretic sequences. For a century until late 2010, exponential sum estimates of higher degree, while serviceable, … Continued