
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

Geometry
and Model Theory Seminar 201415
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 199697,
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 Ominimal
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,
Upcoming
Seminars 
October 2, 2014
Stewart Library 
2:003:00pm
Ethan Jaffe, University of Toronto
Pathological phenomena in DenjoyCarleman classes
We provide explicit constructions of three functions in the theory
of DenjoyCarleman classes. First, generalizing a classical result
and a result of Rolin, Speissegger, and Wilkie, we construct a function
in any given DenjoyCarleman class which is nowhere in any smaller
one. Second, we also construct a function which is formally of a given
DenjoyCarleman class at every point, but is not actually in the class.
Third, we construct a smooth example of function quasianalytic of
a given DenjoyCarleman class on every curve in that class, but which
fails to actually be in the class.

3:30 4:40 pm
André Belotto, University of Toronto
Local monomialization of a system of First Integrals
We consider a nonsingular analytic manifold M and a singular
foliation F with N analytic globally defined first integrals (f_1,...
f_N). We present a monomialization of these first integrals. More precisely,
for each point P in M, there exists a finite collection of local blowingsup
G_i:(M_i,F_i) >(M,F) covering P, such that the singular foliation
F_i has N monomial first integrals at every point, i.e. at each point
Q_i of M_i, there exists a coordinate system u=(u_1,...,u_m) and N monomial
first integrals (u^{a1}, ..., u^{aN}) of F_i such that the exponents
a1,..., aN are linearly independent.

Past
Seminars 
July 31, 2014
Stewart Library 
2:003:00 pm
Tamara Servi, Centro de Matemática e Aplicações
Fundamentais
Multivariable Puiseux Theorem for convergent generalised
power series
The classical Puiseux Theorem says that the solutions y=g(x)
of a real analytic equation f(x,y)=0 in a neighbourhood
of the origin are convergent Puiseux series. The aim of
my talk is to extend this result, and its multivariable
version, to the class of convergent generalised power series.
A generalised power series (in several variables) is a series
with real nonnegative exponents whose support is contained
in a cartesian product of wellordered subsets of the real
line. Let A be the collection of all convergent generalised
power series. I will show that, if f(x_1,...,x_n,y) is in
A, then the solutions y=g(x_1,...,x_n) of the equation f=0
can be expressed as terms of the language which has a symbol
for every function in A and a symbol for division. This
result extends to other classes of functions definable in
polynomially bounded ominimal expansions of the real field,
such as quasianalytic DenjoyCarleman classes, Gevrey multisummable
series and a class containing some Dulac Transition Maps
of real analytic planar vector fields.

3:30 4:40 pm
JeanPhilippe Rolin, Université de Bourgogne
Formal embeddings of transseries into flows
This work is inspired by some results about the fractal
analysis of the orbits of a diffeomorphism in one variable.
In order to perform a similar analysis for an extended class,
we prove a normal form result and the embedding into a flow
for a diffeomorphism given by a transseries (joint work
with P. Mardesic, M. Resman and V. Zupanovic).


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