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:
- Noon: Lunch in SEO 636.
- 1pm: Talk #1: Caroline Terry, A
stable arithmetic regularity lemma in finite-dimensional vector spaces
over fields of prime order
- 2:30: Talk #2: John Goodrick, Model theory of groups of finite dp-rank and finite burden
- 4pm: Talk #3: Ronnie Nagloo, Towards Strong Minimality and the Fuchsian Triangle Groups
- 5:30pm: Dinner at Bracket Room.
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.