
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 
December 4, 2014
Stewart Library 
2:003:00pm
Armin Rainer, University of Vienna
On the regularity of roots of smooth polynomials
We show that the roots of a smooth curve of monic polynomials admit
parameterizations that are locally absolutely continuous. More precisely,
any continuous choice of the roots is locally absolutely continuous
with $p$integrable derivatives, uniformly with respect to the coefficients,
where $p>1$ depends only on the degree of the polynomial. This
solves a problem posed by S. Spagnolo over a decade ago in connection
with the solvability of certain systems of partial differential equations.
Joint work with Adam Parusinski.

Past
Seminars 
November 13, 2014
Stewart Library 
2:003:00pm
Gal Binyamini, University of Toronto
Bezouttype theorems for differential fields
We prove analogs of the Bezout and BernsteinKushnirenko theorems
for systems of algebraic differential conditions over differentially
closed fields. Namely, given a system of algebraic conditions on the
first $l$ derivatives of an $n$tuple of functions, which admits finitely
many solutions (in a differentially closed field), we show that the
number of solutions is bounded by an appropriate constant (depending
singly exponentially on $n$ and $l$) times the volume of the Newton
polytope of the set of conditions. I will state the result and try
to present the key geometric ideas for the proof in a simplified setting.
This result sharpens previous results obtained by Hrushovski and
Pillay, and consequently improves estimates for some problems of a
diophantine nature by the same authors. If time permits I'll discuss
some of these application.

3:30 4:40 pm
Patrick Speissegger, McMaster University
Constructing quasianalytic Ilyashenko classes based on logexpanalytic
monomials, part II
This is a continuation of my last talk on quasi analytic Ilyashenko
algebras. I will explain what’s needed to further extend the construction
given last time to arbitrary logexpanalytic monomials; it requires
a detailed understanding of the holomorphic extension properties of
the functions definable in the ominimal expansion of the globally subanalytic
sets by the exponential function. (Joint work with Tobias Kaiser.)

October 23, 2014
Stewart Library 
2:003:00pm
Janusz Adamus, University of Western Ontario
On CRcontinuation of arcanalytic maps
Given a set $E$ in $\C^m$ and a point $p\in E$, there is a unique
smallest complexanalytic germ $X_p$ containing $E_p$, called the
holomorphic closure of $E_p$. We will study the holomorphic closure
of semialgebraic arcsymmetric sets. Our main application concerns
CRcontinuation of semialgebraic arcanalytic mappings: A mapping
$f:M\to\C^n$ on a connected realanalytic CR manifold which is semialgebraic
arcanalytic and CR on a nonempty open subset of $M$ is CR on the
whole $M$.

3:30 4:40 pm
Patrick Speissegger, McMaster University
Constructing quasianalytic Ilyashenko classes based on logexpanalytic
monomials
I will first outline a generalization of Ilyashenko’s
quasianalytic class that includes all Poincaré firstreturn
maps associated to hyperbolic polycycles of planar real analytic vector
fields. I will then explain what’s needed to further extend this
construction to arbitrary logexpanalytic monomials; it requires a
detailed understanding of the holomorphic extension properties of the
functions definable in the ominimal expansion of the globally subanalytic
sets by the exponential function. (Joint work with Tobias Kaiser.)

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.

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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