## Geometry and Model Theory Seminar at the Fields Institute

### Overview

The idea of the seminar is to bring together people from the group
in geometry and singularities at the University of Toronto (including
Ed Bierstone, Askold Khovanskii, Grisha Mihalkin and Pierre Milman)
and the model theory group at McMaster University (Bradd Hart, Deirdre
Haskell, Patrick Speissegger and Matt Valeriote).

As we discovered during the programs in Algebraic
Model Theory Program and the Singularity
Theory and Geometry Program at the Fields Institute in 1996-97,
geometers and model theorists have many common interests. The goal of
this seminar is to further explore interactions between the areas.

Seminars will take place in the Fields Institute Library Thursdays
from 2 - 3 p.m.

### Upcoming Seminars

March 03, 2005

**Patrick Ahern**, University of Wisconsin, Madison

*On the method of Ecalle and Voronin*

Abstract: We discuss the method of Ecalle and Voronin and give some

applications of it.

### Past Seminars

January 13, 2005

**Wieslaw Pawlucki**, Uniwersytet Jagiellonski

*A linear extension operator for Lipschitz functions on o-minimal
sets*

For any subset E of R^n definable in an o-minimal structure S on the
field of real numbers R, a construction of a continuous linear extension
operator for Lipschitz functions on E which preserves definability
in S will be presented.

November 4, 2004

**Sergei Starchenko**, Notre Dame University

Complex-Analytic subsets in Analytic-Geometric Categories

We consider a subanalytic subset A of a complex analytic manifold
M (when M is viewed as a real manifold) and formulate conditions under
which A is a complex analytic subset of M.

October 28, 2004

**Thomas Scanlon**, University of California, Berkeley

*Regular types in partial differential fields*

In superstable theories, every types may be analyzed in terms of regular
types. So, to understand general types its is necessary to know the
regular types. The theory of differentially closed fields with $n$
commuting derivations is totally transcendental, and, in particular,
superstable. In the case of $n = 1$, every regular types is either
locally modular or non-orthogonal to the generic type of a definable
field. We extend this result to the case of $n \geq 1$ showing that
a non-locally modular regular type is nonorthogonal to a regular generic
type of some definable additive group. We discuss some properties
of additive groups with non-locally modular regular generic types.

[This is a report on joint work with Rahim Moosa and Anand Pillay.]

October 21, 2004

** Thierry Zell**, Georgia Tech

*Topology of Hausdorff limits in o-minimal structures*

Let's fix an o-minimal expansion of the real field and consider
a definable family of compact definable sets. It is well-known that
any set that occurs as a Hausdorff limit in this family has to be
definable too, which implies in particular that the sum of the Betti
numbers of such a limit is always finite. In this talk, I will give
a quantitative version of that result: an upper-bound on the Betti
numbers of a Hausdorff limit, expressed in terms of the Betti numbers
of very simple sets defined from the fibers. One immediate application
of this result is to new upper-bounds in the semialgebraic case. It
also allowed to answer completely the problem of giving effective
estimates for the Betti numbers of relative closures of semi-Pfaffian
sets.

October 7, 2004

**Rahim Moosa**, University of Waterloo

*Nonstandard meromorphic groups.*

A. Pillay and T. Scanlon proved the Chevellay structure theorem
for meromorphic groups (these are "compactifiable" complex
Lie groups) which describes every meromorphic group as an extension
of a complex torus by a linear algebraic group. In this talk I will
discuss current work with

T. Scanlon and M. Aschenbrenner toward a relative (or uniform) version
of this theorem. Equivalently, we aim to extend the structure theorem
to groups definable in nonstandard models of the theory of compact
complex spaces. I will focus on the strongly minimal case.

September 23, 2004

**Tobias Kaiser**, Universität Regensburg

*Definability results for the Poisson equation*

Let U be a simply connected, bounded and semianalytic domain
in the real plane. We study the Poisson equation Lu = f in U, u =
g on the boundary of U, where L is the Laplace operator and f and
g are bounded, semianalytic functions. We ask for several classes
of domains whether the solution u lives in an o-minimal structure.
For example, we show for analytically smooth domains that the solution
is definable in the expansion of the real exponential field by all
restricted analytic functions

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