|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
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|
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.
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.
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.