MidWest Model Theory Day

Tuesday, October 3, 2017 at UIC

Fall 2017 MWMT is October 3.

Speakers: Joel Nagloo (CUNY), John Goodrick (Los Andes), Caroline Terry (Maryland)

Schedule:
All talks are about an hour long including questions, in SEO 636. There will also be coffee & cookies in 636.
It is probably easiest to park in the university parking lot on Morgan St. between Roosevelt and Taylor; please bring in parking tickets, and we can validate them.





Abstracts:

A stable arithmetic regularity lemma in finite-dimensional vector spaces over fields of prime order

In this talk we present a stable version of the arithmetic regularity lemma for vector spaces over fields of prime order. The arithmetic regularity lemma for $\mathbb{F}_p^n$ (first proved by Green in 2005) states that given $A \subseteq \mathbb F_p^n$, there exists $H \leq \mathbb F_p^n$ of bounded index such that $A$ is Fourier-uniform with respect to almost all cosets of $H$. In general, the growth of the index of $H$ is required to be of tower type depending on the degree of uniformity, and must also allow for a small number of non-uniform elements. Our main result is that, under a natural stability theoretic assumption, the bad bounds and non-uniform elements are not necessary. Specifically, we present an arithmetic regularity lemma for $k$-stable sets $A\subseteq \mathbb{F}_p^n$, where the bound on the index of the subspace is only polynomial in the degree of uniformity, and where there are no non-uniform elements. This result is a natural extension to the arithmetic setting of the work on stable graph regularity lemmas initiated by Malliaris and Shelah.


Model theory of groups of finite dp-rank and finite burden

We will present several recent results concerning the model theory of groups whose theories are of finite rank, where "rank" means either dp-rank or burden (inp-rank). First, we review some results (joint with A. Dolich) about ordered Abelian groups of finite burden, where "burden" or "inp-rank" is a generalization of weight which is useful in unstable theories; in this context, we can show that unary definable sets satisfy various desirable properties. Next, we present more recent results (joint with J. Dobrowolski) showing that inp-minimal groups with an ordering invariant under left translations are Abelian, and also showing that finite weight stable groups cannot be too far from being Abelian. Finally, we will present some open questions and possible future directions for research.

Towards Strong Minimality and the Fuchsian Triangle Groups
 
From the work of Freitag and Scanlon, we have that the ODEs satisfied by the Hauptmoduls of arithmetic subgroups of $SL_2(\mathbb{Z})$ are strongly minimal and geometrically trivial. A challenge is to now show that same is true of ODEs satisfied by the Hauptmoduls of all (remaining) Fuchsian triangle groups. The aim of this talk is to both explain why this an interesting/important problem and also to discuss some of the progress made so far.
 


Midwest Model Theory Day is supported by NSF award number 1700095.