Wilson Poulter, Department of Pure Mathematics, University of Waterloo
"Ultraproducts, Hyperreals, and Pseudofinite Graphs"
An ultraproduct is a construction used in mathematical logic to construct limits of first-order structures. Due to a result of Jerzy Los, the first-order properties of these limits is very well understood, allowing one to construct counter-examples as well as structures that are interesting in and of themselves.
In this talk, we will learn what distinguishes this construction from other algebraic constructions i.e. direct sums and direct limits. We will then see briefly how such a construction is obtained, then ourselves construct several interesting examples, these being the hyperreal numbers, and the pseudofinite graphs. Using these, we will prove both Euclid's theorem and Ramsey's theorem in a novel way, revealing a more general method of proof using the ultraproduct.