Abstract: Motivated by the study of mapping class groups, we discuss the proper homotopy equivalence relation for locally finite graphs. We show that the Borel complexity of this equivalence relation is bireducible with the homeomorphism relation on closed subsets of the Cantor set, so by work of Camerlo and Gao, it is Borel complete. On the other hand, we will see that there is a generic equivalence class. We construct Polish spaces of locally finite graphs in such a way that we can extend these results to the homeomorphism relation for noncompact surfaces with pants decompositions. This is joint work with Jenna Zomback.
Abstract: In continuous logic, there are plenty of examples of interesting stable metric structures. On the other side of the SOP line, there are only a few metric structures where order is relevant, and order appears in a different way in each of them. In this talk, joint work with Diego Bejarano, we present a unified approach to linear and cyclic orders in continuous logic.
We axiomatize these theories, and find generic completions in the ultrametric case, analogous to the complete theory DLO. We study some expansions of these theories, including real closed metric valued fields, from this perspective, and characterize which expansions of metric linear orders should be considered o-minimal.
Abstract: In the classification of first-order theories, many dividing lines are defined by forbidding specific configurations of definable sets (e.g. Stability = No Order Property, Simplicity = No Tree Property, etc). In this talk, we abstract the combinatorial nature of these properties; using the concept of patterns of consistency and inconsistency, we give a general framework for studying this kind of dividing lines. Taking this idea to its limit, we review a notion of maximal complexity (SM) in the context of patterns. Weakening SM, we define new dividing lines (PM and the PM^{(k)} hierarchy), provide examples separating these properties from previously defined dividing lines, and prove various results about them.
Abstract: Based on recent work by Domat-Iyer-Shinko, and independently by Long, we give geometric criteria to determine when the universal minimal flow of a subgroup of the mapping class group of an infinite-type surface is non-metrizable.
Abstract: In this talk, I discuss how the study of Ramsey structures and generalized indiscernibles allows us to define new versions of well-known model-theoretic tree properties. Focusing attention on the colored linear order as an index structure, I prove that the so-called colored tree property gives a characterization of stability in terms of positive formulas, while colored versions of TP1 and TP2 give equivalent versions of TP1 and the Independence Property, respectively.
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