SCIENTIFIC PROGRAMS AND ACTIVTIES

October 31, 2014
THE FIELDS INSTITUTE
FOR RESEARCH IN MATHEMATICAL SCIENCES

Geometry and Model Theory Seminar 2011-12
at the Fields Institute

Organizers: Ed Bierstone, Patrick Speissegger

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. It served as the main seminar for the program on O-minimal structures and real analytic geometry, which focussed on such interactions arising around Hilbert's 16th problem.

The seminar meets once a month at the Fields Institute, Room 230, on a Thursday announced below, for one talk 2-3pm and a second talk 3:30-4:30pm. Please subscribe to the Fields mail list to be informed of upcoming seminars.

PAST SEMINARS 2011-12

Thursday March 22, 2012
Stewart Library

2:00-3:00pm

Pantelis E. Eleftheriou, University of Waterloo
Definable quotients of locally definable groups
A locally definable group in an o-minimal structure is a group whose domain is a countable union of definable sets U_i and whose multiplication is definable when restricted to each U_i x U_j. An important example is the universal cover of a definable group. In recent work with Y. Peterzil, we examined the following converse:
Conjecture. Let U be a connected abelian locally definable group which is generated by a definable set. Then U is the cover of some definable group.

We will report progress on this conjecture, mentioning joint work with Y. Peterzil, as well as work by other authors.
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3:30-4:30pm

Martin Bays, McMaster University
Abelian integrals and categoricity
An integral of the form $\Int_{P_0}^P f(x,y)dx$, where $f$ is a rational function and $x$ and $y$ satisfy a polynomial dependence $p(x,y)=0$, is known as an Abelian integral. Fixing one endpoint $P_0$ and allowing the other to vary on the Riemann surface $p(x,y)=0$, we obtain a multifunction whose value depends on the (homology class of) the path along which we integrate.

We consider the model-theoretic status of such multifunctions, and in particular the problem of giving categorical elementary descriptions of structures incorporating them and their interactions with the complex field. Following work by Zilber and Gavrilovich, we will find that the classical theory of Abelian varieties, along with Faltings' work and some model theoretic ideas due to Shelah, allow us to give partially satisfactory answers in some special cases.

August 29, 2011

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Martin Bays, McMaster
Some definability results in abstract Kummer theory
Classically, Kummer theory for a commutative algebraic group reduces Galois theoretic properties of points of the group to "linear", group theoretic properties. We observe that a geometric part of this theory goes through in arbitrary commutative groups of finite Morley Rank, and deduce from the proof a useful definability property in the original case of algebraic groups. (This is joint work with Misha Gavrilovich and Martin Hils.)

Anand Pillay, University of Leeds
Nash groups
The category of Nash manifolds lies in between the real algebraic and real analytic categories. I revisit the category of Nash groups, giving a correct account of the virtual algebraicity of affine Nash groups, a generalization to real closed fields as well as some new examples of nonaffine Nash groups. (Joint with E. Hrushovski.)

June 30, 2011

Chris Miller
Oscillatory trajectories, Minkowski dimension and definability.
In recent joint work with A. Fornasiero and P. Hieronymi, we showed that an expansion of the real field (in the sense of model theory) avoids defining the set of all integers if and only if every compact definable subset of the line has either nonempty interior or Minkowski dimension zero. I will give an outline of the proof, and discuss consequences for expansions of the real field by oscillatory trajectories of planar vector fields.

 

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