[[Syllabus]]

MATH 7370 Probability Theory II MoWe 3:30pm - 4:45pm

The course discusses probabilistic models of growth, transport, and related phenomena, which are modeled by stochastic particle systems. Starting from basic examples, we will examine limit shapes of growing interfaces (governed by probabilistic hydrodynamics), and explore finer asymptotic questions (governed by random matrix type laws and beyond). Most modern developments in studying stochastic particle systems is done at the interplay of algebraic / combinatorial, probabilistic, and analytic tools. I will introduce all of these tools in suitable detail, and use them to answer natural questions arising when looking at concrete particle systems. In particular, we will explore TASEP (totally asymmetric simple exclusion process) and its deformations such as ASEP and q-TASEP; limiting continuous PDEs and stochastic PDEs (like the famous Kardar-Parisi-Zhang equation); longest increasing subsequences of random permutations; and stochastic vertex models (solvable by tools of quantum integrability like Yang-Baxter equation and Bethe Ansatz).

Graduate real analysis generally should suffice for understanding the material. We will also use algebraic / combinatorial tools such as partitions, determinants, multivariable symmetric polynomials, and symmetric functions. Therefore, you should have seen some algebra at graduate or advanced undergraduate level (but exact tools will be introduced along the way). Finally, as probabilistic concepts are very natural in the context of particle systems, no probability prerequisite is needed.

There is an [[Ru/Аннотация курса |accompanying course in Russian]], to be taught in Spring 2021 at the Independent University of Moscow