MidWest Model Theory Day

Tuesday, October 22, 2013 at UIC

Speakers: Vincent Guingona, Bradd Hart, Salma Kuhlmann

Schedule: 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.
[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