MidWest Model Theory Day

Tuesday, October 28, 2014 at UIC

Speakers: C. Ward Henson, Krzysztof Krupiński, Ramin Takloo-Bighash,

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:

C. Ward Henson
Title: Uncountably categorical Banach space structures
Abstract: 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 $\kappa$-categorical if $T$ has a unique model of density $\kappa$. Work of Ben Yaacov and Shelah-Usvyatsov shows that Morley's Theorem holds in this context: if $T$ has a countable signature and is $\kappa$-categorical for some uncountable $\kappa$, then $T$ is $\kappa$-categorical for all uncountable $\kappa$.

Known examples of uncountably categorical such structures (including the new ones) are closely related to Hilbert space. 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.

Ramin Takloo-Bighash
Title: Counting orders in number fields and p-adic integrals.
Abstract: 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).

Krzysztof Krupiński
Title: Generalized Bohr compactification and model-theoretic connected components
Abstract: Newelski proposed and developed a new, based on topological dynamics, approach to study groups from the model-theoretic perspective. Going further in this direction, for a given group we relate its generalized Bohr compactification with its model-theoretic connected components, and using this, we obtain new information about these components.

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 $G^*/{G^*}^{000}_M$, where $G^*$ is the interpretation of $G$ in the monster model and ${G^*}^{000}_M$ is the smallest $M$-invariant subgroup of $G^*$ of bounded index. Using this, we show that that for any parameter set $A$ the quotient ${G^*}^{00}_A/{G^*}^{000}_A$ is isomorphic to the quotient of a compact Hausdorff group by a dense subgroup. Assuming that all types in $S_G(M)$ are definable, we also conclude that if $G$ is definably strongly amenable (e.g. $G$ is nilpotent), then ${G^*}^{000}_M={G^*}^{00}_M$. (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 ${G^*}^{000}_M \ne {G^*}^{00}_M$.)

All of this applies to the case when all types in $S_G(M)$ 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.

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