MidWest Model Theory Day

Tuesday, October 26th, 2010 at UIC


Speakers: Andrew Arana, François Loeser, Alex Wilkie

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 me 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!

Abstracts:

Andrew Arana
Title:The preference for "pure" proofs
Abstract: Over the years many mathematicians have voiced a preference for proofs that stay "close" to the theorems being proved, avoiding "foreign", "extraneous", or "remote" considerations. Such proofs have come to be known as "pure". Examples abound, in geometry and number theory for instance, and indeed one can see the Gödel phenomenon as an example as well. In thinking about this preference two main questions arise: how exactly can what is "foreign" to a statement be measured; and what reasons are there for preferring pure proofs of a statement over impure proofs of that same statement. In this talk we address both of these questions, focusing on the first and indicating ways in which model theory bears on its study.

François Loeser
Title: Model Theory and the Fundamental Lemma
Abstract: The Fundamental Lemma is a complicated combinatorial identity between integrals over local fields which plays a central feature in the Langlands program in the theory of automorphic representations. A proof of the Fundamental Lemma was recently completed by Ngo Bau Chau for which he was awarded a Fields Medal in Hyderabad. His proof, which is geometric in nature, works for functions fields over finite fields. It has been proved previously by Waldspurger, using specific representation theory techniques, that the Fundamental Lemma over p-adic fields - which is the case more relevant for applications to number theory - would follow from the function field case, for p large enough. The aim of our talk is to explain how Waldspurger's result - and similar statements for various versions analogues of the Fundamental Lemma which are not covered by Waldspurger's result - follows at once from a general transfer result allowing to transfer identities between integrals depending on parameters from functions fields over finite fields to p-adic fields, for large p, which we obtained in collaboration with Raf Cluckers using our theory of motivic integration in a definable setting. To achieve this, one has to carefully encode all the data appearing in the Fundamental Lemma in a definable way. This is joint work with Raf Cluckers and Tom Hales.

Alex Wilkie
Title:Some model theory for complex analytic functions.
Abstract: After some motivating remarks concerning Zilber's Conjecture on the quasi-minimality of the complex exponential field, I shall discuss the possible role of o-minimality in the study of complex exponentiation. I shall conclude with an observation on the definablility of algebro-logarithmic functions which came to light during this investigation.