SCIENTIFIC PROGRAMS AND ACTIVTIES

Wed., Sept. 25, 2013
2:00 p.m.
Room 230

Thurs., May 16, 2013
11:00 a.m.
Room 210

Vincent Grandjean, Universidade Federal do Ceara (Fortaleza, Brazil)
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 the germ.

This is joint work with L. Birbrair (UFC), A. Fernandes (UFC) and A. Gabrielov (Purdue)

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.

Omar Leon Sanchez, University of Waterloo
The model-companion of partial differential fields with an automorphism

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.

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.

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
3:30 p.m.
Room 230

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.

Oct. 11
4:30 p.m.
Room 210

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.