I am working in geometric functional analysis. This area studies high-dimensional sets and linear operators, combining ideas and methods from convex geometry, functional analysis and probability. While the complexity of a set may increase with the dimension, it is also possible that passing to a high-dimensional setting may reveal properties of an object, which are obscure in low dimensions. For example, the average of a few random variables may exhibit a peculiar behavior, while the average of many will be close to a constant with high probability.
Probabilistic considerations play a prominent role in geometric problems. In particular, to prove the existence of a section of a convex body having a certain property, one can show that a random section possesses this property with positive probability. This powerful method allows to prove results in situations, where deterministic constructions are unknown, or unavailable.
In recent years I have been also studying spectral properties of random matrices. Here the connection between the areas runs in the opposite direction: the origins of the problems are purely probabilistic, while the methods draw from functional analysis and convexity.
