Program

Program:

9am - Breakfast

10am - Chris Laskowski

  • Title: Equivalents of NOTOP

  • Abstract: A countable theory T is classifiable if it is superstable and has both NDOP and NOTOP. For such theories, every model is determined up to isomorphism by an independent tree of countable, elementary substructures. Historically, superstable theories with NOTOP were only studied under the assumption of NDOP, but we prove that countable, superstable theories with NOTOP are extremely well behaved. In particular, models of such theories determined by an independent tree of countable, elementary substructures up to back and forth equivalence over the tree.

    Many equivalents of NOTOP are given, and we prove that for countable, superstable theories, NOTOP implies PMOP, which was previously only known for theories with NDOP. This is joint work with Danielle Ulrich.

11am - Alex Kruckman

  • Title: Some pseudofinite modules

  • Abstract: Recall that a structure is pseudofinite if every sentence satisfied by that structure has a finite model – equivalently, if the structure is elementarily equivalent to an ultraproduct of finite structures. In this talk, we will explore pseudofiniteness for modules over a (unital, but not necessarily commutative) ring R. Most of our results concern the following question: Which rings R have the property that every R-module is pseudofinite? If we restrict attention to commutative reduced rings, these turn out to be exactly the von Neumann regular rings. This is joint work in progress with Alex Van Abel

Noon - Lunch

1:30pm - Graduate student / postdoc presentations

  • Jake Rhody - "Basics of computable model theory"
    • Abstract: Computability theory and model theory share rich connections, including their common interest in definability via first-order formulas. In this talk, we'll give a brief overview of the main tools and ideas used in computability theory, including partial/total computable functions, the "standard enumeration" of such functions, and the notion non-computable sets (in particular, the halting set). We will define a computable model, which is necessarily countable, and adapt the notion of categoricity to a computable context in such a way it remains interesting even among countable structures.
  • Aaron Anderson - "Continuous logic and learning bounds"
    • Abstract: NIP and stability are important dividing lines in model theory, but they also correspond precisely to notions of statistical learnability. In classical logic, this correspondence goes through combinatorial dimensions of families of sets, but these combinatorial dimensions can be generalized to families of real-valued functions. We will briefly survey how these generalized dimensions connect to dividing lines in continuous logic, and how randomizations in continuous logic can be used to deduce bounds for learning algorithms. Joint work with Michael Benedikt.
  • Jeremy Beard - "On the spectrum of limit models"
    • Abstract: The question of whether limit models (a surrogate of saturated models) are isomorphic has proved important when approaching the main test question of AECs, Shelah’s categoricity conjecture.
      In this talk, we present a full characterisation of the spectrum of limit models under reasonable assumptions.
        Theorem. Suppose K is an AEC with monster models, stable in λ, ℵ0-tame, with an independence relation with uniqueness, existence, universal continuity, (≥ κ)-universal local character in some minimal κ < λ+, and non-forking amalgamation. Then if Nl is a (λ, δl)-limit model over M for l = 1, 2 where cf(δ1) < cf(δ2), then N1 ≅M N2 if and only if cf(δ) ≥ κ.
      The theorem also has a 'global' variant. The local and global versions can be used to study limit models in both abstract settings and in natural examples of abstract elementary classes, including any first order complete stable theory.
      Based on recent work with Marcos Mazari-Armida

2:30pm - Rehana Patel

  • Title: The number of ergodic models of an infinitary sentence

  • Abstract: In an ongoing project, Nathanael Ackerman, Cameron Freer and I have been developing a model theory for so-called ergodic structures. These are probability measures on a space of countable structures (in a fixed countable language, with underlying set the natural numbers) that are ergodic and invariant under the logic action. Such measures are of particular interest because they assign probability either one or zero to (the Borel set of models of) any sentence of the infinitary logic Lω1ω, and hence may be considered probabilistic analogues of classical structures.

    In this talk, I will discuss a spectrum question for ergodic structures: Given a sentence of Lω1ω, how many ergodic models does it have? The results combine joint work with Ackerman, Freer, Kruckman and Kwiatkowska

4pm - Henry Towsner

  • Title: An invitation to tame hypergraph regularity

  • Abstract: Tame hypergraph regularity is the project of trying to understand model theoretic and combinatorial dividing lines - especially dividing lines for ternary (and higher arity) relations - by looking at how structures can be approximated by simpler ones. The prototypical results are the proofs that NIP and stable graphs are precisely those which satisfy certain strong forms of Szemeredi's regularity lemma - that is, such graphs are approximately unions of rectangles (with some further control over the approximation for stable graphs).

    For 3-graphs (that is, ternary relations), the picture becomes more complicated - there are variety of ways to approximate such a relation by simpler relations, corresponding to a variety of generalizations of NIP and stable, only some of which are currently understood. We give a (hopefully) not so technical introduction to what is and isn't currently known.

5pm - Aris Papadopoulos

  • Title: Mekler's construction and Murphy's law for 2-nilpotent groups

  • Abstract: A shower thought one may have is the following: we know that given a model-theoretic dividing line admitting a combinatorial definition (think "independence property"), it is relatively easy to construct a purely combinatorial structure (say a graph) which lies precisely on one side of the divide, but can we construct a purely algebraic structure (say a group) with the same model-theoretic behaviour?

    Mekler's construction comes to the rescue, allowing us to build (2-nilpotent) groups from (nice) graphs in a way that preserves the combinatorial complexity of the graph we started with. This is, of course, ancient news.

    In joint work with Boissonneau and Touchard we prove that the fact that Mekler's construction preserves dividing lines such as stability and NIP is no accident. These are special cases of a more general theorem: Mekler's construction preserves all dividing lines admitting a definition through an "indiscernible collapse".