Southern Wisconsin Logic Colloquium

SWLC Schedule

Refreshments will be served in the 9th floor lounge a half hour before talks. Please check the schedule below. All talks will be in 901 Van Vleck Hall unless stated otherwise.

Date Time Speaker Title Cookies,
dinner, etc.
Tuesday, October 21 4:00 p.m. Gabe Conant, University of Illinois-Chicago Model theory and combinatorics of homogeneous spaces (see abstract) cookies/ beverages at 3:30/
dinner at 6 at
Fugu Asian Fusion
(411 W. Gilman St.)
Wednesday, October 22 4:00 p.m. Andrey Frolov, Kazan Federal University, Russia The categoricity degree of computable structures (see abstract) cookies/ beverages at 3:30
Tuesday, October 28
(Midwest Model Theory Day,
University of Illinois at Chicago)
1:00 p.m. Ward Henson, University of Illinois at Urbana-Champaign Uncountably categorical Banach space structures (see abstract) depart at 8:30 a.m.
from Van Vleck
loading dock/
lunch at 11:30 at Joy Yee's
(1335 S. Halsted)/
dinner at 5:30 TBA
2:30 p.m. Krzysztof Krupiński, University of Wrocław, Poland TBA
4:00 p.m. Ramin Takloo-Bighash, University of Illinois-Chicago Counting orders in number fields and p-adic integrals (see abstract)
Tuesday, November 4 4:00 p.m. Howard Becker, UW TBA cookies/ beverages at 3:30
Tuesday, November 11 4:00 p.m. Reese Johnston, UW TBA (specialty exam) cookies/ beverages at 3:30

Math 873 - Fall 2014 - Recursive Model Theory

Instructor: Uri Andrews

Course Description: This course will serve as a survey of historical and current research in computable model theory. There will be a focus on recursive model theory and the model theory of models of Peano Arithmetic.

Math 975 - Reading Seminar in Logic

Our reading seminar is meeting on Wednesdays at 3:30 in room B231 Van Vleck Hall.

Abstracts of talks

Conant's talk: Model theory and combinatorics of homogeneous spaces

Given a countable subset S of the nonegative reals which contains 0, there is a characterization, due to Laflamme, Delhomme, Pouzet, and Sauer, for the existence of a countable, homogeneous metric space US which has distances only in S and is universal for finite metric spaces with distances in S. We develop a model-theoretic context for the study of US, resulting in characterizations of quantifier elimination, forking independence, stability, and simplicity. Building on previous work of Van Thé, we use tools from model theory to investigate the combinatorial properties of these spaces. In particular, the finitary strong order property provides a concrete combinatorial rank for measuring the complexity of both the theory of US, as well as the algebraic structure inherited by the distance set S. The relationship between this algebraic structure and certain kinds of ordered monoids motivates further conjectures and questions, which are of a similar flavor as some interesting problems in additive and algebraic combinatorics.

Frolov's talk: The categoricity degrees of computable structures

For a computable structure A, the categoricity spectrum is the set of all Turing degrees capable of computing isomorphisms among arbitrary computable copies of A. If the categoricity spectrum of A has a least degree, this degree is called the degree of categoricity of A.

In my talk, I will give a review of results about the categoricity degrees. We focus around rigid structures and some special structures (for instance, fields, linear orderings). In particular, I give examples of rigid structures without degrees of categoricity. Also I show that the categoricity degrees of relatively 0'-categorical linear orderings can be only 0 and 0'. Finally, for any degree b d.c.e. in and above 0'', there exists a computable linear ordering whose the categoricity degree is b.

Henson's talk: Uncountably categorical Banach space structures

The recent progress discussed in this talk concerns new examples of uncountably categorical Banach spaces (of which there have been very few previously known). This is joint work with Yves Raynaud (Univ. of Paris 6).

Model theory is applied to (unit balls of) Banach spaces (and structures based on them, such as Banach algebras or Banach lattices) using the [0,1]-valued continuous version of first order logic. During the talk a sketchy and intuitive description of this logic will be given.

A theory T of such structures is said to be κ-categorical if T has a unique model of density κ. Work of Ben Yaacov and Shelah-Usvyatsov shows that Morley's Theorem holds in this context: If T has a countable signature and is κ-categorical for some uncountable κ, then T is κ-categorical for all uncountable κ.

Known examples of uncountably categorical such structures (including the new ones) are closely related to Hilbert spaces. After the speaker called attention to this phenomenon, Shelah and Usvyatsov investigated it and proved a remarkable result: If M is a nonseparable Banach space structure (with countable signature) whose theory is uncountably categorical, then M is prime over a Morley sequence that is an orthonormal Hilbert basis of length equal to the density of M. There is a wide gap between this result and verified examples of uncountably categorical Banach spaces, which leads to the question: Can a stronger such result be proved, in which the connection to Hilbert space structure is clearly expressed in the geometric language of functional analysis? Here the ultimate goal would be to prove analogues for Banach space structures (or even for general metric structures) of the Baldwin-Lachlan Theorems.

Takloo-Bighash's talk: Counting orders in number fields and p-adic integrals

In this talk I will report on a recent work on the distribution of orders in number fields. In particular, I will sketch the proof of an asymptotic formula for the number of orders of bounded discriminant in a given number field, emphasizing the role played by the model theory of p-adic fields. For this talk, I will not assume any background in algebraic number theory. This is joint work with Nathan Kaplan (Yale) and Jake Marcinek (undergrad, Caltech).

Prepared by Steffen Lempp (">