Model Theory Seminar

A one-dimensional asymptotic class, as introduced by Macpherson and Steinhorn in 2008, is a collection of finite structures whose definable subsets in a single variable grow approximately linearly with respect to the size of the structure, in a definable and well-behaved fashion. The motivating example is the collection of finite fields, as proved by Chatzidakis, van den Dries and Macintyre in 1992. In this talk, we survey Steinhorn and Macpherson's foundational 2008 paper. We give examples and nonexamples of one-dimensional asymptotic classes, as well as more general notions such as N-dimensional and multidimensional classes. We show how infinite ultra-products of one-dimensional asymptotic classes are model-theoretically nice, with particular emphasis on the existence of a well-behaved dimension and measure on definable subsets and the consequences of such.

The logic $L_{\omega_1 \omega}$ is obtained by closing finitary first-order logic under countable disjunction and conjunction. There is a kind of normal form for such sentences. For any structure $\mathcal{A}$ there is a sentence of $L_{\omega_1 \omega}$, known as its Scott sentence, which describes $\mathcal{A}$ up to isomorphism among countable structures. Given a countable scattered linear order $L$ of Hausdorff rank $\alpha < \omega_1$, we show that it has a $d$-$\Sigma_{2 \alpha+1}$ Scott sentence. From Ash's calculation of the back and forth relations for all countable well-orders, we obtain that this upper bound is tight, i.e., for every $\alpha < \omega_1$ there is a linear order whose optimal Scott sentence has this complexity.

I will survey what is known about model companions of theories of (ordered) valued differential fields and discuss my ongoing work towards isolating a model companion for a certain theory of ordered valued differential fields, including positive results at the level of the value group.

This talk is a sequel to my talk in the Kolchin seminar. Let $p$ be a prime number and $e\geq 1$ a fixed natural number. We will consider the theory of separably closed non-trivially valued fields of characteristic $p$ and degree of imperfection $e$, either in a language where a $p$-basis is named or with $e$ commuting stacks of Hasse derivations. Denote the latter by $SCVH_{p,e}$.

We will first sketch a proof of the classification of imaginaries in $SCVH_{p,e}$ by the geometric sorts of Haskell-Hrushovski-Macpherson, using prolongations. We will then explain how these may be used to reduce more phenomena of geometric model theory in $SCVH_{p,e}$ to the algebraically closed case, e.g., a description of the stable part and the stably dominated types, yielding metastastability of $SCVH_{p,e}$. This is joint work with Moshe Kamensky and Silvain Rideau.

I will present results by Kaplan, Levi, and Simon (2017) showing that groups with two model-theoretic properties – omega-categorical and dp-minimal – are nilpotent-by-finite, i.e., they have normal nilpotent subgroups of finite index. Nilpotent-by-finite is a strong group-theoretic property, which can inform the use of groups in exploring various theories, including the theories of structures other than groups. This talk will define these properties (nilpotent-by-finite, omega-categorical, dp-minimal, as well as NIP), give examples of structures with and without these properties, and explain the group-theoretic consequences of the model-theoretic properties that are used in the paper.

In this talk, I will discuss fields with a strong form of model completeness (called almost quantifier elimination) and sketch a proof that PAC fields with almost QE are simple. Time permitting, I will discuss a version of the theorem for differential fields.

In their 2004 seminal paper, 'Fraisse Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups,' Kechris, Pestov, and Todorcevic, tied together the fields of model theory, Ramsey theory, descriptive set theory, and topological dynamics, via the concept of homogeneity. A key tool used is a combinatorial concept called finite constraint. We will show that a class of graphs called metrically homogeneous graphs, of interest to model theorists and combinatorialists, is finitely constrained, and we show how this is used to derive a whole host of topological and topological dynamical consequences.