Lecture Series: What can infinity tell us about the finite?

Speaker: Maryanthe Malliaris (Chicago) 

Abstract: It turns out that model theoretic theorems about infinite objects can have strong consequences for finite objects.  This is not only, as might be supposed, because of classical compactness, or because of pseudofiniteness in the sense of taking a single infinite limit object to represent an increasing family of finite objects.  Rather it appears that in some cases a productive correspondence arises between the structural changes visible as a natural number n varies and the structural changes visible as an infinite cardinal lambda varies. These lectures will discuss some illustrative examples involving Ramsey theory, Erdos-Hajnal, Szemeredi regularity, quasirandomness, and differential privacy, as well as related questions involving cardinal invariants, ultrapowers, and the countable; and will present the developing work in core model theory which supports these advances. There will be three lectures:  Part 1: Model theory and set theory.  Part 2: Finite and infinite combinatorics. Part 3: Regularity and algorithms.