Some prominent conjectures in the theory of C*-algebras ask whether or not every C*-algebra of a particular form embeds into an ultrapower of a particular C*-algebra.  For example, the Kirchberg Embedding Problem asks whether every C*-algebra embeds into an ultrapower of the Cuntz algebra O_2.  In this series of lectures, we show how techniques from model theory, most notably model-theoretic forcing, can be used to give nontrivial reformulations of these conjectures.  We will start from scratch, assuming no knowledge of C*-algebras nor model theory.

Recent years have seen an increasing number of applications of descriptive set theory in ergodic theory and dynamical systems. We present some set theoretic background and survey some of the applications.

Slides for this series of talks can be found here:

https://www.dropbox.com/sh/om8efuv6ez10ysb/AADOA4SPbdjXKoDajEftFb2pa?dl=0

I will discuss recent investigations of various reducibility notions between Pi^1_2 principles of second-order arithmetic, the most familiar of which is implication over the subsystem RCA_0. In many cases, such an implication is actually due to a considerably stronger reduction holding, such as a uniform (a.k.a. Weihrauch) reduction. (Here, we say a principle P is uniformly reducible to a principle Q if there are fixed reduction procedures Phi and Gamma such that for every instance A of P, Phi(A) is an instance of Q, and for every solution S to Phi(A), Gamma(A + S) is a solution to A.) As an example, nearly all the implications between principles lying below Ramsey's theorem for pairs are uniform reductions. In general, the study of when such stronger implications hold and when they do not gives a finer way of calibrating the relative strength of mathematical propositions, and has led to the development of a number of new forcing techniques for constructing models of second-order arithmetic with prescribed combinatorial properties. In addition, this analysis sheds light on several open questions from reverse mathematics, including that of whether the stable form of Ramsey's theorem for pairs (SRT^2_2) implies the cohesive principle (COH) in \omega (standard) models of RCA_0.

The pursuit of better understanding the universe of set theory V motivated an extensive study of definable inner models M whose goal is to serve as good approximations to V. A common property of these inner models is that they are contained in HOD, the universe of hereditarily ordinal definable sets. Motivated by the question of how ``close" HOD is to V, we consider various related forcing methods and survey known and new results. This is a joint work with Spencer Unger.

We complete the proof of the main theorem by showing that if X^3 is isomorphic to X, then X^{\omega} has a parity-reversing automorphism. By our previous results this implies X^2 is isomorphic to X as well. The proof generalizes to show that for any n > 1, if X^n is isomorphic to X, then X^2 is isomorphic to X. Time permitting we will discuss related results, including the existence of an A and X such that A^2X is isomorphic to X, while AX is not.