|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||Generalized Bohr compactification and model-theoretic connected components (see abstract)|
|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.
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.
More precisely, assume G is a group definable in an arbitrary structure M. Extending a classical definition from topological dynamics, we introduce the notion of the generalized externally definable Bohr compactification of G. We prove that there is a natural continuous surjection from the generalized externally definable Bohr compactification of G to G*/G*000M, where G* is the interpretation of G in the monster model and G*000M is the smallest M-invariant subgroup of G* of bounded index. Using this, we show that that for any parameter set A the quotient G*00A/G*000A is isomorphic to the quotient of a compact Hausdorff group by a dense subgroup. Assuming that all types in SG(M) are definable, we also conclude that if G is definably strongly amenable (e.g. G is nilpotent), then G*00M=G*000M. (Recall that in a joint paper with J. Gismatullin, we have proved that if M=G is a non-abelian free group expanded by predicates for all subsets, then G*00A≠G*000A.)
All of this applies to the case when all types in SG(M) are definable (then, instead of externally definable objects one can just work with the definable ones), in particular, to classical topological dynamics by taking as M the group G expanded by predicates for all subsets.
We also show that in the NIP context [externally] definable amenability coincides with externally definable strong amenability.