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
I study number theory: the arithmetic of whole numbers. My research is concerned with mathematical objects, called L-functions, which encode the behavior of prime numbers. The study of prime numbers has been a central concern of mathematics since ancient times, and the idea that their properties can be studied via L-functions arose in the nineteenth … Continued
My research interests include the mathematical theory of waves (for instance, the waves in incompressible fluids), and the analysis of patterns in sets of number-theoretic interest, such as the prime numbers.
My research develops applied algebraic geometry within a broad mathematical context, addressing fundamental problems in algebra, geometry and combinatorics that are relevant for nonlinear models. This involves algebraic geometry (especially over the real numbers), commutative algebra, combinatorics, polyhedral geometry, and more. On the applications side, I am interested in statistics, optimization, computer vision, and the … Continued
Many statistical models are fundamentally algebraic in that they can be described either parametrically or implicitly via polynomial. My research focuses on studying the mathematical and statistical properties of these algebraic statistical models. Major challenges are to elucidate how theoretical algebraic aspects of the models interact with applications. Currently, most problems I work on in … Continued
My research involves the study Low Dimensional Topology: smooth four dimensional manifolds, three dimensional manifolds, and embedded circles (knots) inside three-manifolds. There are various mathematical methods used in this research: gauge theory, symplectic geometry, Floer homology and ideas from topological quantum field theory. Some of my more recent work on studying knots and surgery problems … Continued
My current research focuses on the area of mathematics called number theory. Solutions of problems in this area often draw from all other areas of mathematics. I am particulary interested in establishing connections between mathematical objects of number-theoretic interests coming from seemingly disperate sources, say analysis on one hand and algebra on the other. For … Continued
I study number theory and algebraic geometry. I am interested in solving polynomial equations with the requirement that the coordinates of the solutions be either integers (like -37) or rational numbers (like -3/5). Such problems have been studied for their intrinsic interest since the time of the ancient Greeks. Starting in the 20th century, they … Continued
I work in modern mathematical physics and am particularly interested in geometric and representation-theoretic problems originating in supersymmetric gauge and string theories.