A common motif in high dimensional probability and geometry is that the behavior of objects of interest is often dictated by their marginals onto a fixed number of directions. This is manifested in the fact that several classical functional inequalities are {\it dimension free} (hence, have no explicit dependence on the dimension), the extremizers of those inequalities being functions that only depend on a fixed number of variables. Another related example comes from statistical mechanics, where Gibbs measures can often be decomposed into a small number of "pure states" which exhibit a simple structure.

In this talk, we present an analytic approach that helps reveal phenomenona of this nature. The approach is based on pathwise analysis: We construct stochastic processes, driven by Brownian motion, associated with the high-dimensional object which allow us to make the object more tractable, for example, through differentiation with respect to time.

I will try to explain how this approach works and will briefly discuss several results that stem from it, including functional inequalities in Gaussian space, high dimensional convexity as well results related to decomposition Gibbs measures into pure states.