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.
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