I work primarily in geometric functional analysis, exploring connections among functional analysis, convex geometry and probability theory. Some of my work extends to probability theory, statistics, geometric combinatorics, convex and discrete geometry, harmonic analysis, theoretical computer science and numerical analysis.
Geometric functional analysis strives to understand and use high dimensional structures in mathematics. High dimensions often have srong regularization effect and help us to see the overall picture. While counter-intuitive, this is very similar to the methodology of probability theory. Taking more independent observations, we increase the dimension of the (product) probability space. As the dimension grows to infinity, the classical limit theorems such as the Central Limit Theorem begin to manifest themselves. Thus the picture becomes simpler in higher than in lower dimensions.
Techniques of geometric functional analysis are useful to explore, build, or use various high-dimensional structures such as Banach spaces, convex sets, matrices, signals and other massive data sets.
On the technical level, I have been interested in the following topics:
Geometric functional analysis: concentration of measure, sections of convex bodies, John's decompositions, geometry of Banach spaces
Random matrices: non-asymptotic analysis of invertibility and extreme singular values
Statistics: estimation of covariance matrices, statistical learning theory
Operator theory: norms, invertibility, coordinate restrictions, approximations
Geometric combinatorics: Vapnik-Chervonenkis dimension, integer points in convex sets
Numerical analysis: linear programming, solvers of linear systems, Monte-Carlo methods for large matrices, low rank approximations