Courses

Jinyoung Park: Asymptotic enumeration via graph containers and entropy

Abstract: The container methods are powerful tools to bound the number of independent sets of graphs and hypergraphs, and they have been extremely influential in the area of extremal and probabilistic combinatorics. We will focus on more specialized graph container methods due to Sapozhenko (1987) that deal with sets in expander graphs. Entropy, first introduced by Shannon (1948) in the area of information theory, is a measure of the expected amount of information contained in a random variable. Entropy has seen lots of fascinating applications in a wide range of enumeration problems. In this series of lectures, we will discuss some old and new applications of graph containers, entropy methods, and their combinations for various enumeration problems.

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Matija Bucić: Sublinear expander graphs

Abstract: Expander graphs are one of the most widely useful classes of graphs ever considered. In this course, we will explore a (weaker) notion of sublinear expansion due to Komlós and Szemerédi from the early 90's which has found a remarkable number of impressive applications in recent years. We will start with a brief introduction to the theory of expander graphs, introduce the sublinear notion, show the pass to expander and expander decomposition lemmas, establish a number of properties, and illustrate how they were used in some of the aforementioned recent applications.