
SCIENTIFIC PROGRAMS AND ACTIVTIES 

May 30, 2024  
Geometry and Model Theory Seminar 200809

Past Seminars 200405 
Past Seminars 200506 
Past Seminars 200607 
Past Seminars
200708 
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.
Unless indicated otherwise, the seminars will take place in the
Fields Institute, Room 230 from 2  3 p.m.
Please subscribe to the Fields mail
list to be informed of upcoming seminars.
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 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 Room 230

Andreas Fischer Recovering ominimal 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 nspace space to the reals, where n runs over all natural numbers. We show how to reconstruct the underlying ominimal structure from the set of all limits. 
Thursday Room 230

Mickael Matusinski 
Friday Stewart Library 
FranzViktor 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 Dfields 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 Dfields, 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 Room 230 
Dinh Si Tiep On the trajectories of horizontal gradient of polynomial functions 
Tuesday Room 230 
Lorena López Hernanz Parabolic Curves and Separatrices in C2 
Thursday Room 230 
Dmitry Novikov 
Friday March 13, 2009 Stewart Library 23 pm 
Dmitry Novikov Nonoscillation of pseudoAbelian integrals Part II 
Tuesday Room 230 
Andrei Gabrielov 
Thursday
Room 230
Room 230 
Alex Rennet Grisha Kolutsky Alex Wilkie 
Thursday
Room 230

I would like to address similar questions in the context of ominimal 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 
Olivier Le Gal A generic condition implying ominimality for restricted C infinity functions We prove that a transcendance condition (C) on the Tayor series of a restricted smooth function h implies the ominimality 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 reobtain ominimal structures that does not admit analytic cell decomposition, and non compatible ominimal structures. We even obtain ominimal structures which are not compatible with restricted analytic functions. 
Thursday 
Artur Piekosz Grothendieck topology and ominimality Grothendieck topology is a categorical analogue of usual topology. It originated in algebraic geometry and was already used in the ominimal 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 ominimal version of homotopy theory. As an example, I want to show a BertiniLefschetz type theorem about fundamental groups. 
Tuesday April 14, 2009 Room 230 23 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 ominimal.Subsequently Speissegger (1999) proved the ominimality of the Pfaffian closure of an ominimal 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 
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 blowups 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

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 CenterFocus 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 Stewart Library 
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 23 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 ominimal structure. 
Thursday Room 230 
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 byproduct, we also prove that it has ominimal open core, which is to say that all the open sets definable in (R,E) are already definable in the real field. 
Tuesday Stewart Library 
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 Stewart Library 
Serge Randriambololona 
Thursday Room 230 Room 230 
David Trotman 
Tuesday Room 230 (two talks) 
Adam Parusinski

Wednesday Room 230 
Margaret Thomas OMinimal Structures Without Mild Parameterization We consider parameterization in ominimal structures and look at how the ominimal 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 ominimal structures mild parameterization might be found and, using work of Le Gal, show that the analogous reparameterization theorem does not hold for ominimal structures in general. 
Thursday Room 230
Room 230 
Juan Diego Caycedo 
Thematic Program on Ominimal Structures and Real Analytic Geometry JanuaryJune 2009