WAGS / Spring 2006

Western
Algebraic
Geometry
Seminar

Abstracts

Jean-Louis Colliot-Thelene, Universite Paris-Sud

The study of rational points on (connected) linear algebraic groups and their principal homogeneous spaces often breaks up into two parts : the study of semisimple, simply connected groups, then reductions to the latter case by means of Galois cohomology.

In this context, two classical presentations of arbitrary linear algebraic groups have been used. We propose a third one ("flasque resolutions"), which is simultaneously more flexible and more natural than these earlier two. This is used to:

• define and study the algebraic fondamental group of a connected linear group,
• compute the Brauer group of smooth compactifications of such a group,
• compute R-equivalence on the set of rational points
• measure the lack of weak approximation
• study the Hasse principle for homogeneous spaces

If time permits, we shall also say something about not necessarily principal homogeneous spaces.

Jordan Ellenberg, University of Wisconsin

We will discuss a lot of things that people believe, and a few things that people know, about the distribution of K-points of bounded height on varieties over global fields K; we will try to emphasize the connection between arithmetic questions about Batyrev-Manin conjectures and geometric (or even topological!) ones about moduli spaces.

Kevin Purbhoo, University of British Columbia

In 1962, A. Horn conjectured a recursive solution to the Hermitian sum problem, which asks: if we know the eigenvalues of two n x n Hermitian matrices, what can we say about the eigenvalues of their sum? The conjecture was initially quite mysterious, but over the last decade, it has been shown not only that Horn's conjecture is true, but that it has connections and implications in several different areas of mathematics. Among these implications is a strange recursive aspect to vanishing problems in the cohomology ring of Grassmannians. I will talk about some of the geometry behind this fact, and some of its generalisations.

Yongbin Ruan, University of Wisconsin

Many aspects of birational geometry have a strong symplectic flavor. For example, uniruledness is a symplectic property. More than ten years ago, we already knew that extremal rays play an important role in understanding the symplectomorphism group. With the build-up of machinery in Gromov-Witten theory, we want to revisit some old questions and propose some new directions. This is joint work with Tian-Jun Li.

Constantin Teleman, Cambridge University

This talk describes joint work with Chris Woodward on the "abelian reduction" formula for index or integration moduli of holomorphic G-bundles over a Riemann surface. An application is the proof of Newstead's conjecture on the vanishing of high Chern classes of these moduli spaces. The methods suggest a "splitting principle" for gauged Gromov-Witten theory which seems compatible with standing conjectures in the subject.

Michael Thaddeus, Columbia University

The Strominger-Yau-Zaslow version of mirror symmetry calls for a Calabi-Yau and its mirror to be fibered by dual families of special Lagrangian tori. We will exhibit a large class of examples where such dual families exist for straightforward reasons. They are quotients of an abelian variety by a finite group acting by automorphisms and translations. The orbifold Hodge numbers can be calculated and seen to satisfy the mirror relationship. However, a complicating feature is the "turning on of the B-field": we must work with a flat U(1)-gerbe and take cohomology with the corresponding local coefficients.