We will focus on probability theory in high dimensions with applications in data science. This course is intended for senior undergraduate students and graduate students who are interested in mathematical tools used in data science and high-dimensional statistics. Examples of high-dimensional probabilistic problems include random matrices, estimation for high-dimensional data, randomized algorithms, optimization in a disordered system etc.

This course will be divided into two parts. In the first part, we will focus on R. Vershynin's book "High-Dimensional Probability". We will talk about concentration inequalities for random variables, random vectors in high dimensions and some basic random matrix theory. If time allows, we will also cover some other interesting (and important) topics in this book.

In the second part, we will more or less follow the book "Information, Physics, and Computation" by M. Mezard and A. Montanari, and discuss how methods in information theory and statistical physics can be applied to random optimization problems. This book is a physics book, but we will try to make things rigorous. Tentative topics include the Random Energy Model, the random code ensemble, number partitioning, factor graphs and graph ensembles.