# MidWest Model Theory Day

# Tuesday, October 22, 2013 at UIC

**Speakers**: Vincent Guingona, Bradd Hart, Salma Kuhlmann

**Schedule:**
- 11am: Meet on the first floor of SEO if you're already here. We will leave to lunch at 11:20.
- 11:30am: Lunch at Joy Yee's
- 1pm:
**One talk**
- 2:30:
**Two talk**
- 4pm:
**Three talk**
- 5:30pm: Dinner at Jaks Tap

All talks are about an hour long, 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.

**Let us know (@math.wisc.edu">andrewsmath.wisc.edu) if you are planning to come to lunch and/or dinner so we can make approximately correct reservations!**

**Titles & Abstracts**:

**Vincent Guingona**

**Title:** On VC-minimal fields.

**Abstract:** I discuss recent work on classifying VC-minimal algebraic structures. The work began with J. Flenner and myself when we showed that an ordered group in the pure ordered group language is VC-minimal if and only if it is abelian and divisible. In that same paper, we classified VC-minimal abelian groups and showed that a quasi-VC-minimal Henselian valued field has a divisible value group. In recent work, I have applied our machinery toward classifying VC-minimal fields. I show that a VC-minimal ordered field is real closed and a VC-minimal stable field is algebraically closed. Moreover, I show that we can weaken the condition of VC-minimality to something I call dp-smallness, which sits strictly between VC-minimality and dp-minimality, and still maintain all of these classification results.

**Salma Kuhlmann**

**Title:** Real Closed Fields and Models of Peano Arithmetic

**Abstract:** We say that a real closed field is an **IPA**-real closed field if
it admits an integer part (IP) which is a model of Peano Arithmetic
(PA). In [2] we prove that the value group of an **IPA**-real
closed field must satisfy very restrictive conditions (i.e. must be
an exponential group in the residue field, in the sense of [4]).
Combined with the main result of [1] on recursively saturated real
closed fields, we obtain a valuation theoretic characterization of
countable **IPA**-real closed fields. Expanding on [3], we
conclude the talk by considering recursively saturated o-minimal
expansions of real closed fields and their IPs.

References:

[1] D'Aquino, P. - Kuhlmann, S. - Lange, K.:
*A valuation theoretic characterization of recursively saturated
real closed fields*, arXiv: 1212.6842 (2013)

[2] Carl, M. -
D'Aquino, P. - Kuhlmann, S.$\,$: * Value groups of real closed
fields and fragments of Peano Arithmetic*, arXiv: 1205.2254 (2012)

[3] Conversano, A. - D'Aquino, P. - Kuhlmann, S$\,$: *
$\kappa$-Saturated o-minimal expansions of real closed fields*,
arXiv: 1112.4078 (2012)

[4] Kuhlmann, S.$\,$: *Ordered
Exponential Fields*, The Fields Institute Monograph Series, vol 12.
Amer. Math. Soc. (2000)

**Bradd Hart**

**Title:**Revisiting classification theory from the 1970s

**Abstract:**In joint work with Ilijas Farah, Leonel Robert and Aaron Tikuisis, we have tried to understand the model theory involved in the classification problem of nuclear $C^*$-algebras. One of the original results in this area is a result of Elliott's that shows that any separable inductive limit of finite dimensional $C^*$-algebra $A$ is characterized by its dimension group, $K_0(A)$. I will outline a proof of this theorem in order to highlight the implicit model theory and to give several generalizations.

**Other MWMTDs**:

April 5th, 2016

October 28th, 2014

October 22nd, 2013

April 18th, 2013

October 23rd, 2012

April 26th, 2012

October 11th, 2011

April 7th, 2011

October 26th, 2010