Summer semester 2025
Time: Tuesday 11:15-12:45 (A 314) and Thursday 09:15-10:45 ( Felix_Klein Hs) . First lecture is on 8 April.
Seminar: Tuesday 13:15-14:45 (SG 3-10). First seminar is on 15 April.
Lecturer: Alexey Bufetov (alexey.bufetov AT gmail.com)
Prerequisites: Basic Probability (e.g., what is a random variable ?). The rest will be defined and explained.
The goal of this course is to give an introduction to Integrable Probability, modern and rapidly developing part of Probability Theory.
Consider a uniformly random permutation of size N, written as a word with letters from 1 to N. What is the length of the longest increasing subsequence in this word?
Such a naive question turns out to be connected with a variety of topics -- not only combinatorics and probability, but also representation theory of symmetric groups, Fredholm determinants, and partial differential equations, among others. We will show that the length of a typical longest increasing sequence is of order 2 N^{1/2}, and that it has fluctuations of order N^{1/6}. These results are in no way simple: The journey towards them will be quite long, but exciting, and with a few of unexpected turns. We will also discuss several other models in which similar phenomena occur.
The first part of the course will follow the book ``The surprising mathematics of longest increasing subsequences" by Dan Romik. It can be downloaded on the author's webpage . We will cover Chapter 1 and parts of Chapters 3 and 4.
The subsequently covered topics can be partially influenced by tastes of listeners.