Advisor: Atilla Yilmaz. Research interests: large deviations, optimal control.

Advisor: Atilla Yilmaz. Research interests: homogenization, Hamilton-Jacobi equations, differential games.

Webpage. Research interests: statistical physics, random matrices and log/Riesz gases.

Advisor: Wei-Shih Yang. Research interests: random fields.

Research interests: random matrix theory, mathematical physics.

Advisor: Atilla Yilmaz. Research interests: mean-field games.

Webpage. Research interests: quantum computing, quantum random walks, statistical mechanics.

In this course, we introduce the theory of Ito calculus and stochastic differential equations based on Brownian motion and Gaussian processes (avoiding the subtleties of measure and integration whenever possible), illustrate the concepts and results with concrete examples and numerical projects, and present some of the fundamental applications of the theory to option pricing in finance.

This course provides a self-contained introduction to the area of high-dimensional probability and statistics from a non-asymptotic perspective, aimed at students across the mathematical sciences. It includes a focus on core methodology and theory (tail bounds, concentration of measure, random matrices, random graphs and networks) as well as in-depth exploration of various applications (to statistical learning theory, sparse linear and graphical models, community detection, as examples).

In this course, we introduce the axioms and fundamental notions of probability based on measure and integration, develop the theory with the accompanying probabilistic intuition which is equally important, formulate and prove the strong law of large numbers for independent random variables, define and characterize weak convergence of probability measures, and give a rigorous treatment of the central limit theorem.