SCIENTIFIC PROGRAMS AND ACTIVTIES

2:00-3:00pm

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.

2:00-3:00pm

Gal Binyamini, University of Toronto
Bezout-type theorems for differential fields

We prove analogs of the Bezout and Bernstein-Kushnirenko 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.

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3:30- 4:40 pm

Patrick Speissegger, McMaster University
Constructing quasianalytic Ilyashenko classes based on log-exp-analytic 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 log-exp-analytic monomials; it requires a detailed understanding of the holomorphic extension properties of the functions definable in the o-minimal expansion of the globally subanalytic sets by the exponential function. (Joint work with Tobias Kaiser.)

2:00-3:00pm

Janusz Adamus, University of Western Ontario
On CR-continuation of arc-analytic maps

Given a set $E$ in $\C^m$ and a point $p\in E$, there is a unique smallest complex-analytic germ $X_p$ containing $E_p$, called the holomorphic closure of $E_p$. We will study the holomorphic closure of semialgebraic arc-symmetric sets. Our main application concerns CR-continuation of semialgebraic arc-analytic mappings: A mapping $f:M\to\C^n$ on a connected real-analytic CR manifold which is semialgebraic arc-analytic and CR on a non-empty open subset of $M$ is CR on the whole $M$.

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3:30- 4:40 pm

Patrick Speissegger, McMaster University
Constructing quasianalytic Ilyashenko classes based on log-exp-analytic monomials

I will first outline a generalization of Ilyashenko’s quasi-analytic class that includes all Poincaré first-return 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 log-exp-analytic monomials; it requires a detailed understanding of the holomorphic extension properties of the functions definable in the o-minimal expansion of the globally subanalytic sets by the exponential function. (Joint work with Tobias Kaiser.)

2:00-3:00pm

Ethan Jaffe, University of Toronto
Pathological phenomena in Denjoy-Carleman classes

We provide explicit constructions of three functions in the theory of Denjoy-Carleman classes. First, generalizing a classical result and a result of Rolin, Speissegger, and Wilkie, we construct a function in any given Denjoy-Carleman class which is nowhere in any smaller one. Second, we also construct a function which is formally of a given Denjoy-Carleman class at every point, but is not actually in the class. Third, we construct a smooth example of function quasianalytic of a given Denjoy-Carleman class on every curve in that class, but which fails to actually be in the class.

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3:30- 4:40 pm

André Belotto, University of Toronto
Local monomialization of a system of First Integrals

We consider a non-singular 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 blowings-up 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.

2:00-3:00 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 well-ordered 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 o-minimal expansions of the real field, such as quasianalytic Denjoy-Carleman classes, Gevrey multisummable series and a class containing some Dulac Transition Maps of real analytic planar vector fields.

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3:30- 4:40 pm

Jean-Philippe 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).