SCIENTIFIC PROGRAMS AND ACTIVTIES
|May 23, 2013|
Geometry and Model Theory Seminar 2010-11
Organizers: Ed Bierstone, Patrick Speissegger
|June 30, 2011||Chris Miller
Oscillatory trajectories, Minkowski dimension and definability.
In recent joint work with A. Fornasiero and P. Hieronymi,
we showed that an expansion of the real field (in the sense
|May 19, 2011||
Patrick Speissegger, McMaster
Non-oscillatory vs. interlaced trajectories
Gal Binyamini, Weizmann Institute
Multiplicity and order of contact for regular and singular
Vincent Grandjean, Fields Institute
In this joint work with Daniel Grieser (Oldenburg, Germany), we want to address a naive and elementary question: Given an isolated surface singularity X in (M,g), can a neighbourhood of the singular point be foliated by geodesics reaching the singular point ? Then could we define an exponential-like mapping at a such point, or if you like get some polar-like coordinates where the half-lines would be replaced by a geodesic ray? And what would be the regularity of such an exponential mapping? Conical singularities of any dimension (Melrose & Wunsch) satisify this property and it can even be done analytically if the setting is real analytic (Lebeau). With D. Grieser, we have exhibited simple classes of examples of non-conical real analytic surfaces with an isolated singularity, and cuspidal-like, in the Euclidean 3-space, such that the geodesics reaching the singular point behave differently according to the considered class.
We obsviously have in mind singular real algebraic sets, or germs or subanalytic sets, as model of the singularities we are interested in dealing with. If times allow I will make some connections with Hardt conjecture about the subanaliticity (or definability in a some larger o-minimal structure) of the inner distance on semi-algebraic sets, and try to explain how the current work could allow to provide counter-examples.
|March 24, 2011||
Dmitry Novikov (Weizmann Institute of Science)
The limit cycles appearing in perturbations of Darboux integrable planar polynomial vector fields are closely connected to the zeros of so-called pseudo-abelian integrals, a generalization of abelian integrals. Therefore, one is interested in upper bounds for the number of zeros of such integrals. Even for an individual system this question is far less trivial than its analogue for abelian integrals. The natural conjecture, generalizing the Varchenko-Khovanskii theorem, claims existence of an upper bound depending on the degrees only. I'll describe the progress toward solving this conjecture.
|Nov. 18, 2010||
Gareth O. Jones (University of Manchester)
Janusz Adamus (University of Western Ontario)
|Oct. 21, 2010||
Jana Mariková (McMaster University)
We let R be an o-minimal field, V a convex subring, and k_ind
the corresponding residue field with structure induced from
I present an approach to Zilber's conjecture (that the complex
exponential field is quasi-minimal) based on an analytic continuation
conjecture for complex functions definable in o-minimal structures.
As an application of the ideas involved here, I prove that
the smallest ring R of analytic germs (at 0, say, in the complex
plane) containing the polynomials and closed under taking
logarithms and raising to real powers has the following property:
every f in R has an analytic continuation along all but finitely
many rays emanating from 0. (If we replace "finitely
many" by "countably many" the result is obvious.)
This is, of course, a purely mathematical result, but I do
not know how to prove it without going via definability in
certain 0-minimal structures.
|Sept. 23, 2010||
Vincent Grandjean (Fields Institute)
Tobias Kaiser (Universität Regensburg)