SCIENTIFIC PROGRAMS AND ACTIVTIES

December 21, 2014
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
 Geometry and Model Theory Seminar 2014-15 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,

 Upcoming Seminars Past Seminars December 4, 2014 Stewart Library 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. November 13, 2014 Stewart Library 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. --------------------------------- 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.) October 23, 2014 Stewart Library 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 $M$. --------------------------------- 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.) October 2, 2014 Stewart Library 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. --------------------------------- 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. July 31, 2014 Stewart Library 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. --------------------------------- 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).

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