FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
and Model Theory Seminar 2013-14
the Fields Institute
Organizers: Ed Bierstone, Patrick Speissegger
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,
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
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.
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.