- 11am: Meet on the first floor of SEO if you're already here. We will leave to lunch at 11:20.
- 11:30am: Lunch at Joy Yee's
- 1pm:
**One talk** - 2:30:
**Two talk** - 4pm:
**Three talk** - 5:30pm: Dinner at The Parthenon

It is probably easiest to park in the university parking lot on Morgan St. between Roosevelt and Taylor.

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 $\kappa$

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.

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.

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