In this first of two talks, I will explain the notion of definability in continuous logic and connect it with the notion of spectral gap in the theory of unitary representations and in ergodic theory.

In  Ramsey  Theory, ultrafilters often play an instrumental role.
By using nonstandard models of the integers, one can replace those
third-order objects (ultrafilters are families of subsets) by simple
points.

In this talk we present a nonstandard technique that is grounded
on the above observation, and show its use in proving some new results
in Ramsey Theory of Diophantine equations.
 

In this talk, we will continue with basics of measurable cardinals and their relationship to non-trivial elementary embeddings.  We proceed with basic facts about the constructible universe, L.  After laying this groundwork, we show L cannot have a measurable cardinal.  Time permitting, we will discuss the dichotomy introduced by Jensen's covering lemma: either L is a good approximation to V, or there is a non-trivial elementary embedding from L to L.

A major project in set theory aims to explore the connection between large cardinals and so-called generic absoluteness principles, which assert that forcing notions from a certain class cannot change the truth value of (projective, for instance) statements about the real numbers. For example, in the 80s Kunen showed that absoluteness to ccc forcing extensions is equiconsistent with a weakly compact cardinal. More recently, Schindler showed that absoluteness to proper forcing extensions is equiconsistent with a remarkable cardinal. (Remarkable cardinals will be defined in the talk.) Schindler's proof does not resemble Kunen's, however, using almost-disjoint coding instead of Kunen's innovative method of coding along branchless trees. We show how to reconcile these two proofs, giving a new proof of Schindler's theorem that generalizes Kunen's methods and suggests further investigation of non-thin trees.

Forcing and elementary embeddings are central topics in set theory. Most of what set theorists have focused on are the study of forcing and elementary embeddings over models of ZFC. In this talk, we focus on forcing and elementary embeddings over models of the Axiom of Determinacy (AD). In particular, we focus on answering the following questions: work in V which models AD. Let P be a forcing poset and g ⊆ P be V -generic.

1) Does V [g] model AD?

2) Is there an elementary embedding from V to V [g]?

Regarding question 1, we want to classify what forcings preserve AD. We show that forcings that add Cohen reals, random reals, and many other well-known forcings do not preserve AD. Regarding question 2, an analogous statement to the famous Kunen’s theorem for models of ZFC, can be shown: suppose V = L(X) for some set X and V models AD, then there is no elementary embedding from V to itself. We conjecture that there are no elementary embeddings from V to itself. We present some of the results discussed above. There is still much work to do to completely answer questions 1 and 2. This is an ongoing joint work with D. Ikegami.