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.

Unless indicated otherwise, the seminars will take place in the Fields Institute, Room 230 from 2 - 3 p.m.
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PAST SEMINARS
Friday February 6, 2009 Room 230 10:30 - 11:30 (note time)	Malgorzata Czapla On the Weak Lipschitzianity and Definable Triangulations with Regularity Conditions
Tuesday February 10, 2009 Room 230	Joris van der Hoeven Transserial Hardy fields In our talk we show that the field of differentially algebraic transseries over the real numbers can be embedded in a Hardy field. The proof relies on the concept of a "transserial Hardy field", which both carries the structure of a Hardy field and of a differential subfield of the field of transseries. We will associate analytic meanings to transseries using a technique of iterated integrals.
Thursday February 12, 2009 Room 230 2-3 p.m.	Andreas Fischer *Recovering o-minimal structures* It is well known that limits of sequences of definable functions are not necessarily definable. We discuss the limits of uniformly convergent sequences of definable continuous functions from n-space space to the reals, where n runs over all natural numbers. We show how to reconstruct the underlying o-minimal structure from the set of all limits.
Thursday February 26, 2009 Room 230 10:30- 11:30 am Room 230 2-3 pm	Kobi Peterzil Applications of o-minimality to arithmetical questions in algebraic geometry This talk gives an overview of the recent work of Pila, Wilkie and Zannier involving rational points of definable sets. Mickael Matusinski Differential Puiseux theorem for generalized power series field of finite rank We consider the field of generalized power series of finite rank (i.e. with a finite number of comparability classes), equipped with a Hardy type derivation. We study differential equations with coefficients in this field. We show that there are elementary transformations such that the exponents of any solution (in the field) can be obtained from those of the equation by a finite number of elementary transformations.
Friday February 27, 2009 Stewart Library 3:30-4:30 pm	Franz-Viktor Kuhlmann Immediate mappings and differential Hensel's Lemmas The concept of "immediate mappings" on ultrametric spaces is a generalization of the notion of immediate extensions of valued fields. A main theorem giving a criterion for the subjectivity of immediate mappings provides a uniform tool to prove all sorts of generalized Hensel's Lemmas, among them differential Hensel's Lemmas for both D-fields in the sense of Scanlon and differential valuations in the sense of Rosenlicht. After presenting a quick introduction to immediate mappings and the main theorem, I will show how these differential Hensel's Lemmas are derived. In the case of D-fields, the Hensel's Lemma we obtain is satisfactory. But for the Rosenlicht case, it is very restricted; I will discuss the problems that occur in this case.
Thursday March 5, 2009 Room 230 2-3 pm	Dinh Si Tiep On the trajectories of horizontal gradient of polynomial functions
Tuesday March 10, 2009 Room 230 2-3 pm	Lorena López Hernanz Parabolic Curves and Separatrices in C2
Thursday March 12, 2009 Room 230 2-3 pm	Dmitry Novikov Non-oscillation of pseudo-Abelian integrals Pseudo-Abelian integrals are integrals of rational one- forms over trajectories of real planar integrable vector fields, generalize usual Abelian integrals and are closely related to the Hilbert 16th problem. I'll try to describe the recent progress and the difficulties appearing in attempts to generalize Vachenko-Khovanskii theorem to pseudo-Abelian integrals.
Friday March 13, 2009 Stewart Library 2-3 pm	Dmitry Novikov Non-oscillation of pseudo-Abelian integrals Part II
Tuesday March 17, 2009 Room 230 11:00 am- 12:00 pm	Andrei Gabrielov Approximation of definable sets by compact sets
Thursday March 19, 2009 Room 230 10:30 - 11:30 am Room 230 2-3 pm Room 230 3:30- 4:30 pm	Alex Rennet Differential equations over polynomially bounded o-minimal structures We show that non-oscillatory solutions of ODEs over a polynomially bounded o-minimal structure are exponentially bounded. This lecture is in preparation for the upcoming course on pfaffian functions. Grisha Kolutsky *On the Hilbert-Smale problem* In 1998 S. Smale suggested to consider a restriction of the second part of the 16th Hilbert problem to Lienard equations. In this talk we will show how to get some explicit upper estimates for the number of limit cycles of (generalized) Lienard equations. There are two different parts of this work. In the first part we deal with some geometrical properties of the phase space. The second part is an application of the theorem of Ilyashenko and Yakovenko (that binds the number of zeros and the growth of a holomorphic function) to a Poincare map. Alex Wilkie Model Theory and Analytic Continuation for Implicitly Defined Functions It is clear that if K is a subfield of the complex field, and if f(z) is an analytic function germ which is algebraic over the field of rational functions K(z), then f may be analytically continued along any path which avoids all those complex numbers which lie in the algebraic closure of K. (We are mainly interested in the case that K is countable, so "most" paths have this property.) I discuss the corresponding situation for function germs f(z) satisfying certain transcendental equations (eg exponential polynomial equations). The methods used are most naturally expressed in the language of nonstandard analysis and the deeper results require a knowledge of o- minimal structures. However, I shall work out the details of a simple case where this background is not explicitly required. I shall then give some applications to definabilty theory for expansions of the complex field by certain analytic functions.
Thursday March 26, 2009 Room 230 2-3 pm	Vincent Grandjean Conormal spaces, Gauss mappings, limits of tangents (and everything) in a rigid real time setting Whitney regularity conditions are given in terms of limits of tangents and of secants, while Verdier regularity also involves limits of tangents. In the complex analytic world, Verdier regularity for a pair of strata (Y,Z) is equivalent to a condition on the fibres of a certain mapping (Henry-Merle, Teissier, Le-Teissier). From the complex projective point of view, a corollary of Zak's theorem on the tangent and secant varieties states for any irreducible algebraic projective variety X: the dimension of the singular set of X is at least the corank of the projective Gauss mapping of X minus one. In a very informal manner this means that if there is not much limits of tangent spaces at the singular set, then the singular set must be large. I would like to address similar questions in the context of o-minimal geometries in an affine situation. Another way to look at this is in asking the following questions: 1 - Given a connected and enough differentiable and definable submanifold, what is the geometry of its boundary ? More simply how large is a the singular locus ? 2 - For a large class of functions I will prove such a result about the critical locus of a singular level.
Thursday April 2, 2009 Room 230 2-3 pm	Olivier Le Gal A generic condition implying o-minimality for restricted C infinity functions We prove that a transcendance condition (C) on the Tayor series of a restricted smooth function h implies the o-minimality of the expansion of the real field by h. This condition is shown to be generic, in the sense that the set of all functions that verify (C) is residual with respect to the Whitney topology. As corollaries, we re-obtain o-minimal structures that does not admit analytic cell decomposition, and non compatible o-minimal structures. We even obtain o-minimal structures which are not compatible with restricted analytic functions.
Thursday April 9, 2009 Room 230 2-3 pm	Artur Piekosz Grothendieck topology and o-minimality Grothendieck topology is a categorical analogue of usual topology. It originated in algebraic geometry and was already used in the o-minimal context. Also microlocal analysts have already worked with "the subanalytic site". Grothendieck topology allows to define a deeper version of the notion of a topological space. Here several things should be clarified. Then we get an o-minimal version of homotopy theory. As an example, I want to show a Bertini-Lefschetz type theorem about fundamental groups.
Tuesday April 14, 2009 Room 230 2-3 pm	Tamara Servi Pfaffian closure for definably complete Baire structures (joint work with A. Fornasiero) Wilkie (1999) proved that the structure generated by all real Pfaffian functions is o-minimal.Subsequently Speissegger (1999) proved the o-minimality of the Pfaffian closure of an o-minimal structure. We give an alternative proof of this theorem. Moreover our result holds not only over the real numbers but more generally for definably complete Baire structures, which we introduced in 2008 and which form an axiomatizable class.
Thursday April 16, 2009 Room 230 2-3 pm	Armin Rainer *Perturbation of polynomials and normal matrices* Given a smooth family P of complex univariate polynomials, it is natural to study the regularity of its roots. I shall give an overview of the known results and recent developments. In particular, I will show that the roots of a quasianalytic multiparameter family P can be chosen smoothly after applying finitely many local blow-ups and local power substitutions. Using that, I will prove that the roots of P admit a parameterization by functions of bounded variation (not better!), locally. Similar results can be obtained for the eigenvalues and eigenvectors of a quasianalytic multiparameter family of normal matrices.
Thursday April 23, 2009 Room 230 2-3 pm	Yosef Yomdin Moment vanishing, Compositions, and Mathieu conjecture Recently F. Pakovich and M. Muzychuk completely solved the vanishing problem for polynomial moments of the form \int_a^b P^k(x)Q(x) dP(x). This problem can be considered as an infinitesimal version of the Center-Focus problem for Abel differential equation, and the "moment centers" turn out to be pretty accurately described by certain composition relations between P and Q. For Laurent polynomials situation is more complicated. In a very recent work F. Pakovich has achieved a serious progress in understanding vanishing conditions for rational functions and, in particular, for Laurent polynomials. In particular, new relations with the Mathieu conjecture in representations of compact Lie groups have appeared, and (through the recent work of Wenhua Zhao) to certain questions closely related to the Jacobian conjecture.
Thursday May 14, 2009 Stewart Library 11 am	Philipp Hieronymi The real field with two discrete multiplicative subgroups In this talk, I will give the details of the proof that the real field with two discrete multiplicative subgroups defines the integers. The proof presented does not require any previous knowledge of the topic, but it is very computational.
Thursday May 14 , 2009 Stewart Library 2-3 pm	Guillaume Valette Classification of definiable sets from the metric point of view Any definable set may be regarded as a metric space, if endowed with the metric induced by the ambient space. I will present several results related to the classification of metric types of sets definable in an o-minimal structure.
Thursday May 28, 2009 Room 230 2-3 pm	Ayhan Gunaydin The real field with the rational points of an elliptic curve (joint work with P. Hieronymi) We consider the model theoretic structure (R,E), where R is the real field and E is the group of rational points of an ellitic curve. We axiomatize this structure and show that it eliminates quantifiers up to existential formulas. As a by-product, we also prove that it has o-minimal open core, which is to say that all the open sets definable in (R,E) are already definable in the real field.
Tuesday June 2, 2009 Stewart Library 10:30 - 11:30 am	Anna Valette Asymptotic variety of polynomials mappings The asymptotic variety of a polynomial map is the locus of points at which this map fails to be proper. We will discuss some nice geometrical properties of the asymptotic variety and we will see how it can be determined effectively in some special cases.
Tuesday June 2, 2009 Stewart Library 2-3 pm	Serge Randriambololona Some (non-)elimination results for curves Geometric structures generalise o-minimal structures, strongly minimal structures (such as the field of complex numbers) or even the field of p-adic numbers. I will present some positive and negative results concerning quantifier-elimination for definable curves in this setting. (Joint work with S. Starchenko)
Thursday June 11, 2009 Room 230 10:30 - 11:30 am Room 230 2-3 pm	Bradd Hart Conceptual completeness for continuous logic David Trotman An index theorem for generic vector fields on closure orderable finite partitions by definable manifolds (e.g. definably Whitney stratified definable sets) (with Henry C. King) The usual Poincare-Hopf theorem for manifolds-with-boundary M expresses the Euler-Poincare characteristic of M as a sum of the local indices of the zeros of any vector field v exiting the boundary. Marston Morse (Amer. J. of Math. 51, 1929) and Charles Pugh (Topology 7, 1968) proved an extension allowing tangencies of v to the boundary. We generalise further to stratified-sets-with-boundary, and to even more general ‘radial manifold complexes’, including all finite partitions by definable submanifolds (in some o-minimal structure) inducing a filtration by closed subsets. The stratified vector fields need no longer be continuous. We introduce a notion of virtual zero and virtual index to treat generic vector fields, such as gradients of generic Morse functions. I will also discuss problems related to the stratifying of definable sets and maps.
Tuesday June 16, 2009 Room 230 (two talks) 10:30 - 11:30 am 2 - 3 pm	Adam Parusinski Gradient conjecture in o-minimal structures Let x(t) be a trajectory of the gradient vector field of a C^1 function defined on an open subset of R^n and definable in an o-minimal extension of the real field. We show that this trajectory admits a tangent line at its limit point. The proof is based on the main ideas of [Kurdyka, Mostowski, Parusinski, 2000] and [Kurdyka, Parusinski, 2006], and a geometric analysis of the asymptotic of the spherical part of gradient.
Wednesday June 17, 2009 Room 230 2-3 pm	Margaret Thomas O-Minimal Structures Without Mild Parameterization We consider parameterization in o-minimal structures and look at how the o-minimal Reparameterization Theorem of Pila and Wilkie might be enhanced. In particular, we are interested in whether or not we can have any greater control over the bounds on the derivatives of the parameterizing functions. Work by Pila shows that a certain choice of bounds, namely `mild' bounds, could improve the original corollary to the Reparameterization Theorem (a result about the bound on the number of rational points of bounded height lying on definable sets), at least in the particular case of Pfaff curves. We consider in which o-minimal structures mild parameterization might be found and, using work of Le Gal, show that the analogous reparameterization theorem does not hold for o-minimal structures in general.
Thursday June 18, 2009 Room 230 10:30 - 11:30 am Room 230 2-3 pm	Zbigniew Jelonek How to construct new non-trivial but stably trivial vector bundles? I will show that any complex algebraic variety X of dimension > 6 has a birational modification X' such that on X' there is a stably trivial but not- trivial algebraic vector bundles. I give some application to affine algebraic geometry. Juan Diego Caycedo Complex elliptic curves with a distinguished dense subgroup Let E be a complex elliptic curve and let exp be its exponential map. The map exp is a homomorphism from the additive group of the complex numbers onto E. Consider a subgroup G of E obtained as the image under exp of a real line through the origin having trivial intersection with the kernel of exp. Then G is dense in E in the Euclidean topology. Consider the structure on E with a predicate for each of the Zariski closed subsets of its cartesian powers and a predicate for G. We show that if E does not have complex multiplication, is invariant under complex conjugation and a version of the Schanuel Conjecture holds for exp, then the theory of this structure is omega-stable and has quantifier elimination after adding predicates for the existentially definable sets. To do this we write axioms for the theory by means of a "predimension function", the proof that these axioms hold in the structure uses facts from the theory of analytic sets and o-minimality. Although conditionally, this provides new examples of stable expansions of the complex field.

Thematic Program on O-minimal Structures and Real Analytic Geometry January-June 2009

SCIENTIFIC PROGRAMS AND ACTIVTIES

Geometry and Model Theory Seminar 2008-09 at the Fields Institute

Overview

PAST SEMINARS

Geometry and Model Theory Seminar 2008-09
at the Fields Institute