FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
and Model Theory Seminar 2012-13
at 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, 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.
Wed., Sept. 25, 2013
|Jacob Tsimerman, Harvard University
Thurs., May 16, 2013
Vincent Grandjean, Universidade Federal do Ceara (Fortaleza,
Lipschitz contact equivalence classes of analytic functions
do not have moduli
To make a long story short, the (smooth) contact equivalence
of two mappings was introduced (1960's) by J. Mather in
his tremendous work about the stability of smooth mappings.
For two germs of analytic functions it just means that the
ideals generated by each of the functions are the same.
Later, some notions appeared of topological (resp. bi-Lipschitz)
contact equivalence refining (resp. coarsening) various
classifications of smooth mappings with singularities. Recently
Birbrair-Costa-Fernandes-Ruas found a very simple equivalent
definition of bi-Lipschitz contact equivalence between germs
of Lipschitz functions. The subsequent work by Ruas-Valette
shows that a similar simple equivalent definition exists
for the bi-Lipschitz contact equivalence for germs of Lipschitz
mappings. They also prove that germs of polynomial mappings
of given bounded degree admit only finitely many bi-Lipschitz
contact equivalence classes.
The result I will present concerns only germs of plane
continuous subanalytic functions. We show that they can
be associated with a finite combinatorial object, called
a Hölder diagram, as a consequence of the Bierstone-Milman-Parusinski
rectilinearization theorem. The main result is that this
object completely classifies germs of plane Lipschitz subanalytic
functions under the subanalytic bi-Lipschitz contact equivalence,
which implies that there are only countably many such classes.
I will try to present, in the slightly simpler case of germs
of real analytic functions, the main ideas to understand
the combinatorial object (Hölder Diagram) encoding
This is joint work with L. Birbrair (UFC), A. Fernandes
(UFC) and A. Gabrielov (Purdue)
|Wed., March 6, 2013
Rahim Moosa, University of Waterloo
Algebraic reductions of hyperkaehler manifolds; model theory
In a 2010 paper, Campana, Oguiso, and Peternell make some
observations about the structure of the algebraic reduction
map on a nonalgebraic hyperkaehler compact complex manifold.
Anand Pillay and I have given a model-theoretic treatment
of some of this material, leading both to an abstract model-theoretic
generalisation as well as a slight improvement of the complex-goemetric
result. I will report on this work.
|Wed., Feb. 13, 2013
Omar Leon Sanchez, University of Waterloo
The model-companion of partial differential fields with
We explain how one can characterize the existentially closed
models in terms of differential-algebraic varieties, and
then show that this class is elementary using characteristic
sets of differential prime ideals.
|Wed., Feb. 13, 2013
Patrick Speissegger, McMaster University
Are all non-oscillatory trajectories of three-dimensional
real analytic vector fields o-minimal?
A few years ago, Rolin, Sanz and Schäfke constructed
a real analytic vector field in real 5-space that has a
non-oscillatory trajectory that is not o-minimal. In real
2-space, no such trajectory exists, but the question remains
open in 3-space and 4-space. The construction used for the
example in 5-space cannot work in 3- or 4-space; together
with Le Gal and Sanz, we are trying to prove that no examples
exist in 3-space. In this talk, I will outline what we have
learned so far.
|Thurs., Nov. 29
|Hadi Seyedinejad (Western
Fibre families of complex analytic mappings
Fibres of a morphism between complex spaces form a family
that encodes much information regarding the behaviour of
the morphism. In fact, the study of fibres leads us to efficient
testing methods for specific properties of the map, like
openness and flatness. I will survey some results from my
PhD project, that are mostly efforts to extend certain previous
criteria to the general setting of maps over the singular
targets. I will also discuss the long term goal of my approach,
which is to classify different
non-regular (e.g., non-open) mappings in terms of singularities
in their family of fibres.
Thurs., Nov. 29
|Gal Binyamini (University of Toronto)
Complexity of Noetherian functions
Noetherian functions are functions which satisfy certain
systems of differential equations. They are defined in a
manner analogous to the Pfaffian functions, but without
imposing a triangularity condition. The global finiteness
properties of the Pfaffian class do not carry over to the
Noetherian class. However, Khovanskii and Gabrielov have
conjectured that local analogs of these finiteness properties
remain. In the first part of the talk I will introduce the
Noetherian functions and some old results on their finiteness
properties. In the second part I will describe some recent
progress (joint with Dmitry Novikov) toward the general
conjecture mentioned above.
David Marker (University of Illinois at Chicago)
Model Theory and Complex Exponentiation?
As the integers are definable in the complex exponential
field it's model theory has long been ignored. Still there
are interesting open questions about definability. Zilber
proposed a novel model theoretic attack. We will survey
Zilber's work and later developments.
| Oct. 11
| Philipp Hieronymi (University
of Illinois at Urbana-Champagin)
Tame Geometry: A tale of two spirals
As part of Back2Fields