# SCIENTIFIC PROGRAMS AND ACTIVITIES

September 30, 2014
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
 Set Theory Seminar Series 2014-15 Fields Institute, Room 210 Friday 1:30 pm Organizing Committee: Miguel Angel Mota, Ilijas Farah, Juris Steprans, Paul Szeptycki
 List of previous talks Seminars held during 2013-14 York U Set Theory Page
 2014 Fridays Upcoming Seminars at 1:30 p.m. in the Fields Institute, Room 210 October 3 Ilijas Farah Omitting types in logic of metric structures is hard One of the important tools for building models with prescribed second-order properties is the omitting types theorem. In logic of metric structures omitting types is much harder than in classical first-order logic (it is Pi-1-1 hard). Although the motivation for this work comes from C*-algebras, the talk will mostly be on descriptive set theory. The intended takeaway from the talk is "logic of metric structures blends with descriptive set theory beautifully." This is joint work with Menachem Magidor. 2014-15 Past Seminars Speaker and Talk Title September 26 No seminar September 19 Daniel Soukup Trees, ladders and graphs The chromatic number of a graph $G$ is the least (cardinal) number $\kappa$ such that the vertices of $G$ can be covered by $\kappa$ many independent sets. A fundamental problem of graph theory asks how large chromatic number affects structural properties of a graph and in particular, is it true that a graph with large chromatic number has certain obligatory subgraphs? The aim of this talk is to introduce a new and rather flexible method to construct uncountably chromatic graphs from non special trees and ladder systems. Answering a question of P. Erdos and A. Hajnal, we construct graphs of chromatic number $\omega_1$ without uncountable infinitely connected subgraphs. September 12 Konstantinos Tyros A disjoint union theorem for trees In this talk we will present an infinitary disjoint union theorem for level products of trees. An easy consequence of the dual Ramsey theorem due to T.J. Carlson and S.G. Simpson is that for every Suslin measurable finite coloring of the power set of the natural numbers, there exists a sequence $(X_n)_{n\in\mathbb{N}}$ of disjoint non-empty subsets of $\mathbb{N}$ such that the set $\Big\{\bigcup_{n\in Y}X_n:\; Y\;\text{non-empty subset of }\mathbb{N}\Big\}$ is monochromatic. The result that we will present is of this sort, where the underline structure is the level product of a finite sequence of uniquely rooted and finitely branching trees with no maximal nodes of height $\omega$ instead of the natural numbers. As it is required by the proof of the above result, we develop an analogue of the infinite dimensional version of the Hales--Jewett Theorem for maps defined on a level product of trees, which we will also present, if time permits. September 5 Diana Ojeda Finite forms of Gowers' Theorem on the oscillation stability of $c_0$ We give a constructive proof of the finite version of Gowers' $FIN_k$ Theorem and analyze the corresponding upper bounds. The $FIN_k$ Theorem is closely related to the oscillation stability of $c_0$. The stabilization of Lipschitz functions on arbitrary finite dimensional Banach spaces was proved well before by V. Milman. We compare the finite $FIN_k$ Theorem with the Finite Stabilization Principle found by Milman in the case of spaces of the form $\ell_{\infty}^n$, $n\in N$, and establish a much slower growing upper bound for the finite stabilization principle in this particular case. August 29 Seminar cancelled August 22 Mike Pawliuk Various types of products of Fraisse Classes, various types of amenability and various types of preservation results. This is joint work with Miodrag Sokic. In a recent paper of Jakub Jasinski, Claude Laflamme, Lionel Nguyen Van Thé and Robert Woodrow (Arxiv: 1310.6466) it was shown that certain Fraisse Classes are actually Ramsey classes. For many of those cases we have determined whether their automorphism groups are extremely amenable or not. Some of these spaces turn out to actually be a special type of product of Fraisse classes. We were able to prove that unique ergodicity (a type of amenability) is preserved under this type of product. August 1 Martino Lupini Functorial complexity of Polish and analytic groupoids I will explain how one can generalize the theory of Borel complexity from analytic equivalence relations to groupoids by means of the notion of Borel classifying functor. This framework allows one to capture the complexity of classifying the objects of a category in a functorial way. I will then present the first results relating the functorial complexity of a groupoid and the complexity of its associated orbit equivalence relation, focusing on the case of Polish groupoids: For Polish groupoids with essentially treeable equivalence relations any Borel reduction between the orbit equivalence relations extends to a Borel classifying functor. On the other hand for any countable non-treeable equivalence relation E there are Polish groupoids of different functorial complexity both having E as associated orbit equivalence relation. The proof of these results involves a generalization of some fundamental results on the descriptive set theory of actions of Polish groups --such as the Becker-Kechris theorem on Polishability of Borel G-spaces-- to actions of Polish groupoids. July 25 Saeed Ghasemi An analogue of Feferman-Vaught theorem for reduced products of metric structures I will give a metric version of the Feferman-Vaught theorem for reduced products of discrete spaces. We will use this to show that, under the continuum hypothesis, the reduced powers of any metric structure over atomless layered ideals are isomorphic. As another application, I will give an example of two reduced products of sequences of matrix algebras over Fin, which are elementarily equivalent, therefore isomorphic under the CH, with no trivial isomorphisms between them. June 27 Christopher Eagle Model theory of abelian real rank zero C*-algebras We consider algebras of the form $C(X)$, where $X$ is a $0$-dimensional compact Hausdorff space, from the point of view of continuous model theory. We characterize these algebras up to elementary equivalence in terms of invariants of the Boolean algebra $CL(X)$ of clopen subsets of $X$. We also describe several saturation properties that $C(X)$ may have, and relate these to topological properties of $X$ and saturation of $CL(X)$. We will discuss some consequences of saturation when we view $C(X)$ as a $C^*$-algebra. All the necessary background on continuous logic will be provided. This is joint work with Alessandro Vignati.