Model Theory Seminar

Fall 2017

CUNY Graduate Center
Room 6417
Fridays 12:30-1:45pm
Organized by Alf Dolich

September 8
Alf Dolich, CUNY
Organization Meeting
We will meet to discuss plans for the semester, in particular if we’d like to have a focused reading on a particular topic.

October 6
John Goodrick, Universidad de los Andes
Hrushovski and Loeser’s model-theoretic study of Berkovich spaces, Part 1
To study the geometry of algebraic varieties over complete algebraically closed valued fields, Berkovich introduced the analytification V^{an} of a variety V, a construction adding “extra points” to V yielding a space which is locally compact and locally path-connected. Hrushovski and Loeser recently gave a new construction of V^{an} as the set of definable stably dominated types in V, and studied these analytifications using tools from model theory (orthogonality, internality, etc.), proving in particular that V^{an} is deformation retractable to a piecewise linear set definable in the value group.

In this talk we will give a broad overview of Hrushovski and Loeser’s approach to Berkovich spaces and begin talking in detail about the first essential concept they use to analyze them, Pro-and Ind-definable sets.

Our reference is Hrushovski and Loeser’s monograph “Non-archimedean tame topology and stably dominated types” (arXiv:1009.0252v5).

October 13
John Goodrick, Universidad de los Andes
Hrushovski and Loeser’s model-theoretic study of Berkovich spaces, Part 2
In this talk (which will not presume knowing the “classical” construction of Berkovich spaces I outlined last week), we will carefully define the Hrushovski-Loeser construction of the “stable completion” V-hat of a variety V defined over a valued field. This completion is the collection of all F-definable stably dominated types concentrating on V over a field of definition F, and it can be presented as a strict pro-F-definable set. We will explain what all of these words mean, why this is important, and how this connects with the classical definition of a Berkovich space.

November 3
Rebecca Coulson, Rutgers University
Twists and Twistability
We introduce the concept of a twist, which is an isomorphism up to a permutation of the structure’s language. We developed this concept in the course of proving results about metrically homogeneous graphs. This concept proved useful for partial classification results as well as finiteness results. The concept of a twist surprisingly is present in other work by Cameron and Tarzi, as well as by Bannai and Ito. We will discuss our results and their connections to other work.

November 10
Philipp Rothmaler, CUNY
Sum-like submodules of direct products
I call a submodule of a direct product of a family of modules sum-like if all of its elements “look the same” as some of its restrictions (projections) to finite supports. I will make this definition precise and discuss properties of the concept—with an eye to chain conditions on endosubmodules and certain definable subgroups. And, though the Baer-Specker group will come up too, this talk will be largely disjoint from the one I gave at MAMLS at Wesleyan three weeks ago.
The concept makes sense also in other categories, for instance, groups, in fact in any concrete category with products and a one-element initial object.

November 17
Brian Wynne, CUNY
Interpreting second-order arithmetic in C(X)
Let C(X) be the ring of all continuous real-valued functions on the topological space X. I will present Cherlin’s proof (from his 1980 paper “Rings of Continuous Functions: Decision Problems”) that second-order arithmetic is interpretable in the ring C(X) whenever X is a non-discrete metric space.

December 1
Jesse Han, McMaster University
Strong conceptual completeness for $\omega$-categorical theories
Suppose we have some process to attach to every model of a first-order theory some (permutation) representation of its automorphism group, compatible with elementary embeddings. How can we tell if this is “definable”, i.e. really just the points in all models of some imaginary sort of our theory?

In the ’80s, Michael Makkai provided the following answer to this question: a functor $Mod(T) \to Set$ is definable if and only if it preserves all ultraproducts and all “formal comparison maps” between them (generalizing e.g. the diagonal embedding into an ultrapower). This is known as strong conceptual completeness; formally, the statement is that the category $Def(T)$ of definable sets can be reconstructed up to bi-interpretability as the category of “ultrafunctors” $Mod(T) \to Set$.

$\omega$-categorical structures are particularly simple: they have few definable sets and are determined up to bi-interpretability by the action of their automorphism groups. Any general framework which reconstructs theories from their categories of models should therefore be considerably simplified for $\omega$-categorical theories.

Indeed, we show:

  1. If $T$ is $\omega$-categorical, then $X : Mod(T) \to Set$ is definable, i.e. isomorphic to $(M \to \psi(M))$ for some formula $\psi \in T$, if and only if $X$ preserves ultraproducts and diagonal embeddings into ultrapowers. This means that all the preservation requirements for ultramorphisms, which a priori get unboundedly complicated, collapse to just diagonal embeddings when $T$ is $\omega$-categorical.
  2. This definability criterion fails if we remove the $\omega$-categoricity assumption. We construct examples of theories and non-definable functors $Mod(T) \to Set$ which exhibit this.

December 8
Alice Medvedev, CUNY