In this seminar we will discuss the following question: What is the length of the longest increasing subsequence of a random permutation?
This question, which appears innocent at first sight, has occupied mathematicians for the last 50 years and has reached its resolution only in the last decade. The question is taken from the field of combinatorics, or discrete probability, but it turns out to have connections with various other fields in mathematics such as algebra (representation theory of the symmetric group), analysis (special functions, variational problems) and differential equations (Painlevé equations). In this seminar we will study the problem and the results obtained for it during the years, while learning of its connections to the other fields.

The seminar is based on the first two chapters of the book of Dan Romik:
The surprising mathematics of longest increasing subsequences.
The book is freely available from the author's website at:
https://www.math.ucdavis.edu/~romik/lisbook/

The seminar is intended for third year undergraduate students. Recommended background is one of the courses Probability for the sciences or Probability for mathematicians. Required background is Introduction to probability, linear algebra 2 and hedva 3 (hedva 3 may be taken in parallel).

During the seminar each student will lecture on a topic or two from the book. Before the lecture the student will meet with the instructor to receive comments. If there are few participants some students may be required to give two talks. The first lecture will be given by the instructor.