We will discuss the propagation of certain tree representations in the presence of very large cardinals. These tree representations give generic absoluteness results and have structural consequences in the area of generalized descriptive set theory. In fact these representations give us a method for producing models of strong determinacy axioms.

A subshift is a subset of Cantor space that is both topologically closed and closed under the shift operation. Any element of a subshift can be viewed as a trajectory in a discrete dynamical system. One way a trajectory can be complicated is to have a large effective dimension. We consider the effective dimension spectrum of a subshift X, defined as { dim x : x ∈ X }, where dim is the effective dimension. Most commonly, the dimension spectrum of X is [ 0, h(X) ], where h(X) is the entropy of X. We give examples where the dimension spectrum does not follow this pattern, and discuss partial results and open questions related to the problem of characterizing the dimension spectra of subshifts. No prior knowledge about subshifts will be assumed.

Projective determinacy is the statement that for certain infinite games, where the winning condition is projective, there is always a winning strategy for one of the two players. It has many nice consequences which are not decided by ZFC alone, e.g. that every projective set of reals is Lebesgue measurable. An old so far unpublished result by W. Hugh Woodin is that one can derive specific countable iterable models with Woodin cardinals, $M_n^{\#}$, from this assumption. Work by Itay Neeman shows the converse direction, i.e. projective determinacy is in fact equivalent to the existence of such models. These results connect the areas of inner model theory and descriptive set theory. We will give an overview of the relevant topics in both fields and, if time allows, sketch a proof of the result that for the odd levels of the projective hierarchy boldface $\Pi^1_{2n+1}$-determinacy implies the existence of $M_{2n}^{\#}(x)$ for all reals $x$.

We discuss the Borel complexity of the isomorphism relation (for countable models of a first order theory) as the “right” generalization of the model counting problem. In this light we present recent results of Dave Sahota and the speaker which completely characterize the complexity of isomorphism for o-minimal theories, as well as recent work of Laskowski and Shelah which give a partial answer for omega-stable theories. Along the way, we introduce a few open problems and barriers to generalizing the existing results.

The dynamical and descriptive set theoretic complexity of a countable Borel equivalence relation E can often be understood in terms of the kinds of countable first order structures which are compatible with E in a suitable sense. In this talk I will make this suitable sense precise by discussing the notion of Borel structurability. I will also discuss some recent joint work with Brandon Seward in which we show that the equivalence relation generated by the free part of the translation action of a countable group G on its powerset is structurably-universal among equivalence relations generated by free Borel actions of G.

We will discuss the structure L(V_λ+1) and attempts to generalize facts of descriptive set theory to this structure in the presence of very large cardinals. In particular we will introduce an axiom called Inverse Limit Reflection which we will argue is analogous to the Axiom of Determinacy in this context. The slides for this talk are available here.

I would like to give an overview of recent results in canonical Ramsey theory in the context of descriptive set theory. This is the subject of a recent monograph joint with with Vladimir Kanovei and Jindra Zapletal. The main question we address is the following. Given an analytic equivalence relation on a Polish space, can one find a large subset of the space on which it has a simple form? Canonical Ramsey theory stems from finite combinatorics and is concerned with finding canonical forms of equivalence relations on finite (or countable) sets. We obtain canonization results for analytic and Borel equivalence relations and in cases when canonization is impossible, we prove ergodicity theorems. For a publisher’s book description see:

http://www.youtube.com/watch?v=ZPXBcKeepKk