Model Theory Seminar

This first meeting will be partially devoted to organizing for the semester. But, I will also begin talking about Jamshid Derakhshan' survey paper on the model theory of the adeles entitled 'Model Theory of the Adeles and Number Theory'.

We will discuss the paper of Cluckers, Derakhshan, Leeknegt and Macintyre on uniformly defining valuation rings in Henselian valued fields with finite or pseudofinite residue fields.

I will continue talking about Derakhsan's survey article 'Model Theory of Adeles and Number Theory'.

We prove that a random group, in Gromov's density model with $d\lt 1/2$, satisfies an AE sentence (in the language of groups) if and only if this sentence is true in a nonabelian free group. This is a joint work with R. Sklinos.

A few years ago, a problem arose in some of my work that I wasn’t able to solve, forcing me to add a technical hypothesis to a theorem - this has bothered me ever since. The issue has to do with the relationship between independence in a stable (or simple or NSOP_1) theory and independence in a stable reduct. In this rather informal talk, I will describe the problem and some partial results. The audience is welcome to provide proofs or counterexamples.

In this talk, I explore the connections between Statistical Learning Theory and Model Theory. This includes the connections between PAC-learning and NIP and the connections between differentially private PAC-learning and stability. Finally, I examine the work that my colleagues and I have started on improving the sample complexity of differentially private PAC-learning algorithms using techniques from stability theory. This work is joint with Alexei Kolesnikov, Miriam Parnes, and Natalie Piltoyan.

Associated to certain valued differential fields like transseries and Hardy fields are so-called asymptotic couples, which were introduced by M. Rosenlicht. These are ordered abelian groups equipped with a map induced by the derivation of the valued differential field. I will describe ongoing work with A. Gehret and E. Kaplan on the asymptotic couple of the field of logarithmic transseries, in which we show various notions of smallness coincide. For example, the structure is d-minimal in the sense that every unary definable set with empty interior is a finite union of discrete sets. This enables us to classify all dimension functions on the structure.

In the classification of complete first-order theories, many dividing lines have been defined in order to understand the complexity and the behavior of some classes of theories. In this talk, using the concept of patterns of consistency and inconsistency, we describe a general framework to study combinatorially defined dividing lines and we introduce a notion of maximal complexity by requesting the presence of all the exhibitable patterns of definable sets. Weakening this notion, we define new properties (Positive Maximality and the $PM^{(k)}$ hierarchy) and prove some results about them.