April 25, 2014

Geometry and Model Theory Seminar


November 7, 2003
Ilia Zharkov, Harvard University
Integral Kähler affine structures
Integral Kähler affine structures on spheres with singularities in codimension 2 arise in the metric limit of maximally degenerate Calabi-Yau families. I will describe some examples of such structures and discuss an analog of the Calabi conjecture in this affine setting as well as several other open questions.

November 21, 2003
Kiumars Kaveh, University of British Columbia
Vector Fields and Cohomology Ring of Toric Varieties
The general results of Carrell and Lieberman give an interesting relationship between the zero scheme of a vector field and the cohomology ring of the underlying variety. We explain these results in the case of toric varieties. We give an interesting connection with Brion's description of the polytope algebra.

December 12, 2003
Georges Comte, University of Nice
Local invariants and regularity conditions in real geometry
For any germ of subanalytic sets (or definable sets), we define two finite sequences of new numerical invariants. The invariants of the first sequence are obtained by localizing the classical Lipschitz-Killing curvatures (using Thom-Mather lemma). The invariants of the second sequence are of "polar" nature (connected to the geometry of the discriminants of projections of the germ). We prove that each invariant of one sequence is a linear combination, with universal coefficients, of invariants of the second one. Then we prove that our sequences (as mappings depending on the base-point of the germ) are continuous along Verdier strata of a closed set.

January 16, 2003
Dan Miller, University of Toronto
A Preparation Theorem for Weierstrass Systems
The globally subanalytic sets can viewed as the definable sets of the real ordered field with restricted analytic functions. By adding division and all nth roots into the language, Van den Dries, Macintrye and Marker showed a very strong quantifier elimination result: all globally subanalytic functions are piecewise given by terms in this expanded language. Their proof used model theoretic techniques. Lion and Rolin then supplied a purely geometric proof by showing that the globally subanalytic function have a certain preparation theorem. This talk discusses an adaptation of Lion and Rolin's proof which shows that their preparation theorem holds for the collection of definable functions in any expansion of the real orderd field by a system of restricted analytic functions closed under differentiation, composition and Weierstrass preparation.

January 30, 2004 -- 10:30 a.m.
Yoav Yaffe, McMaster University
A version of Hensel's lemma for partial differential fields
The concept of a valued partial differential field (VPDF) arises quite naturally when one studies differential equations on an algebraic variety which has a "very small" dimension. The first step of analyzing VPDFs, from a model theoretic point of view, is attempting to find a model completion, i.e. determine whether the class of existentially closed VPDFs is an elementary class.

February 27, 2004 -- 10:30 a.m.
Chris Miller, The Ohio State University
Expansions of o-minimal structures by logarithmic spirals
Let M be an o-minimal expansion of the field of real numbers and S be a logarithmic spiral. It turns out that the expansion (M,S) has a reasonable model theory relative to that of M provided that M defines restricted exponentiation and restricted sine but defines no irrational power functions (e.g., if M is the expansion of the real field by all globally subanalytic sets). I will outline a model-theoretic proof of this fact (no "standard" proof is known, as far as I know) and discuss some consequences for the definable sets in (M,S).

March 5, 2004
Raf Cluckers, Ecole Normale Superieure
Constructible motivic functions and motivic integration
I will present a totally new framework of motivic integration for a new class of functions, which is joint work with F. Loeser. It generalizes the motivic integrals introduced by Kontsevich, Denef, and Loeser to a relative framework (i.e. depending on parameters). It avoids the use of a completional process and it can be specialised to any of the classical constructions of (non-relative) motivic integrals (both classical and arithmetic integration), as well as to (relative) p-adic integrals for p big enough. In this setting a very general change of variables formula is obtained.

I will explain why this construction works, how completion can be avoided, how a general change of variables is proved, and how a Fubini theorem is obtained. Most of this relies on cell decomposition for valued Henselian fields. I will also explain Serre's construction of an induced measure on an analytic subset of affine space over the p-adic numbers.

This framework opens the way for studying (relative) motivic oscillatory integrals which specialize to (relative) p-adic oscillatory integrals for p big enough.

March 12, 2004 -- 10:30 a.m.
Jean-Philippe Rolin, Université de Bourgogne
New results on oscillating curves
We prove that, under certain conditions, the trajectories of analytic real vector fields belong to an o-minimal structure. This result gives easy examples of pairs of o-minimal structures which do not admit any common o-minimal expansion. Moreover, it can be shown that an analytic vector field in Euclidean 3-space can be desingularized "along" any non-oscillating trajectory.

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