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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 from 2 - 4:30 p.m.
Upcoming Seminars
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2005
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2006
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November 3, 2005 - 2:10 at the University of Toronto -
Bahen Building - Room BA 3116
Guillaume Rond, University of Toronto
Artin approximation in rings of power series
Askold Khovanskii, University of Toronto
Insolvability of equations in finite terms
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November 17, 2005
December 15, 2005
October 20, 2005
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September 22, 2005
Edward Bierstone, Toronto
Uniform bounds in analytic rings
Patrick Speissegger, Dept. of Mathematics & Statistics,
McMaster
The theorem of the complement for nested
sub-Pfaffian sets
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January 19, 2006
Febuary 16, 2006
March 16, 2006
Krzysztof Kurdyka, Université de Savoie, France
An analogue of Lojasiawicz's gradient inequality
for maps
April 20, 2006
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September 22, 2005 -- 2:00 p.m.
Edward Bierstone, University of Toronto
Uniform bounds in analytic rings
I will discuss questions of uniformity or linearity of bounds
for invariants that play important roles in the solution of equations
(involving analytic functions) of various forms; e.g., f(y) =g(p(y)),
f(x)=A(x)g(x), f(x,g(x))=0 (where, in each case, g(x) is the unknown).
The invariants corresponding to the these classes of equations are
the Chevalley function, Artin-Rees exponent, Artin function (respectively).
Uniform or linear bounds for the Chevalley function, for example,
characterize regularity or "tameness" properties of analytic
mappings and their images (subanalytic sets).
The new results are joint with Janusz Adamus and Pierre Milman.
September 22, 2005 --3:30 p.m.
Patrick Speissegger, Department of Mathematics & Statistics,
McMaster University
The theorem of the complement for nested sub-Pfaffian sets
We show that the Pfaffian closure of an o-minimal structure with analytic
cell decomposition is model complete. This is achieved by proving
a theorem of the complement for nested sub-Pfaffian sets over the
o-minimal structure in question. (Joint work with Jean-Marie Lion)
November 3, 2005 - 3:10 p.m.
BA 3116
Guillaume Rond, University of Toronto
Artin approximation in rings of power series
We will talk about the Artin function of a system of equations with
coefficients in the ring of power series. The existence of such of
a function is a generalisation of the implicit function theorem. We
will give the basic properties of these functions. We will show how
one can connect these functions with a result of diophantine approximation
in field of power series. Then we will give a counter-example to the
conjecture that all Artin function are bounded by affine functions.
November 3, 2005, 3:10 p.m.
BA 3116
Askold Khovanskii, University of Toronto
Insolvability of equations in finite terms
Abel, Galois, Liouville, Picard, Vessiot, Kolchin and others found
a lot of results about solvability and insolvability of equations
in finite terms. According to them, algebraic equations are usually
not solvable by means of radicals. Ordinary linear differential equations
and holonomic systems of linear differential equations in partial
derivatives are not usually solvable by quadratures. Galois theory
belongs to algebra. In fact results about insolvability of differential
equations belongs to differential algebra and are also purely algebraic.
About 30 years ago I constructed a topological version of Galois
theory for functions in one complex variable. According to it, there
are topological restrictions on the way the Riemann surface of a function
representable by quadratures covers the complex plane. If the function
does not satisfy these restrictions, then it is not representable
by quadratures. Beside its geometric clarity the topological results
on nonrepresentability of functions by quadratures are stronger than
the algebraic results. By now I have constructed a multi-dimensional
topological version of Galois theory.
No preliminary knowledge is required.
March 16, 2006 - 2:00 p.m.
Fields Institute
Krzysztof Kurdyka, Université
de Savoie, France
An analogue of Lojasiawicz's gradient inequality for maps
We state and prove a result analogous to Lojasiewicz's Gradient Inequality
for analytic (even o-minimal) mappings with values in k- dimensional
Euclidean space, k>1. This inequality gives a uniform bound on
the measure of submanifolds that are transverse to the fibres of the
mapping. We shall also discuss a possible relation to the characterization
of "blowup-free" maps.
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