SCIENTIFIC PROGRAMS AND ACTIVTIES

April 21, 2014
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
Geometry and Model Theory Seminar 2013-14
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
  TBA
Past Seminars
Wednesday,
Nov. 13

Olivier Le Gal, Université de Savoie
Trajectories associated to formal invariant curves

If X is an analytic vector field near the origin of R^n which admits a formal invariant curve c, we prove that there exists a real trajectory c' of X having c as asymptotic expansion at 0. From this we obtain the o-minimality of all isolated trajectories of analytic vector fields without using the Borel-Laplace summation process. (Joint work with Felipe Cano and Fernando Sanz)

André Belotto, University of Toronto
A problem of little return time with an approach by resolution of singularities

Let M be a real analytic manifold, N an analytic smooth submanifold and X an analytic vector field such that, for each point p in the submanifold N, the condition (called geometric quasi-transversality)
dimX(p) + dim T_pN = dim(X(p)+T_pN) holds. Does there exist a neighborhood U of p (in the ambient space M) and a positive real number d>0 such that, for every point q in the intersection of U and N, the trajectory \gamma_q(t), with time |t| < d, intersects N only at q?

We will give a partial answer to this question, and we will discussthe difficulties involved with a generalized version of the problem. This question was inspired by a question of J.-F. Mattei about the germs of singular holomorphic 1-forms.


Please note the first talk will start at 1:00 p.m. and the second talk will start at 2:00 p.m.

Wednesday
Sept. 25

Jacob Tsimerman, Harvard University
Transcendence, Arithmetic, and o-minimality

o-minimality has recently emerged as a powerful technique in certain number-theoretic questions. We will explain how certain classical geometric transcendence theorems can be tackled by combining tools from o-minimality with a counting theorem of Pila-Wilkie. In particular, we give an alternative proof of the classical Ax-Lindemann theorem and discuss its generalizations to other functions arising from Shimura varities, such as elliptic functions.

Chris Miller, Ohio State University
Coincidence of dimensions in expansions of the real field

For expansions of the field of real numbers, nondefinability of the set of natural numbers is equivalent to equality of the upper Minkowski and euclidean dimensions on images of closed definable sets under definable continuous maps. (Joint with Philipp Hieronymi.)
I will give some applications of this result to the study of expansions of the real field by trajectories of vector fields.

 

 

Room 230, on a Thursday announced below, for one talk 2-3pm and a second talk 3:30-4:30pm. Please subscribe to the Fields mail list to be informed of upcoming seminars.

 

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