MidWest Model Theory Day 13:
Groups and Logic
Tuesday, April 23, 2019 at UIC
Spring 2019 MWMT is April 23.
Speakers: Nir Avni (Northwestern), Rizos Sklinos (Stevens), Turbo Ho (Purdue)
Schedule:
- Noon: Lunch in SEO 636.
- 1pm: Talk #1: Nir Avni
- 2:30: Talk #2: Rizos Sklinos
- 4pm: Talk #3: Turbo Ho
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 (from university lots), and
we can validate them. This is easiest if you bring your actual ticket with you.
The Poster
Abstracts:
Avni: Rigidity and bi-interpretability with Z for higher rank lattices.
Abstract: A lattice in a Lie
group is a discrete subgroup with finite co-volume. In many contexts,
there is a dichotomy between lattices in Lie groups of rank one and
lattices in Lie groups of higher rank, where the two classes behave in
qualitatively different ways. I will talk about this dichotomy in the
context of Model Theory.
Sklinos: Fraisse constructions in the free group
Abstract: In an influential
paper Fraisse obtained the ordered rationals as a limit of finite
linear orders through amalgamations. Furthermore his construction
implied the (ultra)-homogeneity, the countability and universality of
the limit structure. Since then various adaptations of Fraisse's method
had given very interesting examples in many mathematical disciplines.
The random graph in graph theory and Philip Hall's universally locally
finite group in group theory to name a few.
In joint work with Kharlampovich and Myasnikov we look into the
possibility of applying Fraisse constructions in classes of groups that
played a central role in answering Tarski's question on nonabelian free
groups. In particular, we modify Fraisse's method to prove that
nonabelian limit groups form a $\forall$-Fraisse class and finitely
generated elementary free groups form an elementary-Fraisse class.
Ho: Scott sentence of finitely-generated groups
Abstract: Scott showed that for
every countable structure A, there is a L_{\omega_1,\omega} sentence,
called the Scott sentence, whose countable models are the isomorphic
copies of A. The quantifier complexity of a Scott sentence can be
thought of as a measure of the complexity of the structure. Knight et
al. have studied the Scott sentences of many structures. In particular,
Knight and Saraph showed that a finitely-generated structure always has
a \Sigma_3 Scott sentence. In this talk, we will focus on
finitely-generated groups. On the one hand, most "natural"
finitely-generated groups have a d-\Sigma_2 Scott sentence. On the
other hand, we give a characterization of finitely-generated structures
where the \Sigma_3 Scott sentence is optimal. We then give a
construction of a finitely-generated group where the \Sigma_3 Scott
sentence is optimal.
This is joint work with Matthew Harrison-Trainor.
Midwest Model Theory Day is supported by NSF award number 1700095.