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Lecture by Dave Anderson

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Dr. Dave Anderson (Paris) will visit the Graduiertenkolleg from October 9th to 12th, 2012. He will give a lecture series in Algebraic Geometry.

What
  • Single Talk
When Oct 09, 2012
from 10:00 AM to 12:00 PM
Where Eckerstraße 1, SR 404
Contact Name Stefan Kebekus
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Dave Anderson will deliver four lectures on "Equivariant Methods in Algebraic Geometry". 

Lecture 1: Introduction to equivariant cohomology

When an algebraic group acts on a variety X, the equivariant cohomology ring provides a useful tool for understanding the topology of X: for instance, many calculations can be reduced to computations at fixed points, even when they are isolated, since the equivariant 
cohomology ring of a point is nontrivial. In this talk, I'll give an example-based introduction to the subject, focusing on torus actions. Time permitting, I'll also describe positivity theorems for the equivariant product of Schubert classes on a flag variety.

Lecture 2: Arc spaces and equivariant cohomology

A group action on a smooth complex variety X induces a natural action on the arc space of X, an infinite-dimensional space parametrizing germs of curves in X. In joint work with Alan Stapledon, we developed a new perspective on the equivariant cohomology of X, by replacing X with its arc space. Under certain hypotheses, these infinite-dimensional varieties allow us to obtain a geometric basis (over the integers!) for equivariant cohomology, as well as geometric representatives for cup products as intersections. I'll explain how this leads to a new invariant of singularities, and illustrate our approach with examples from toric varieties and flag varieties.

Lecture 3: Degeneracy loci and Schubert polynomials

Many interesting varieties arise as degeneracy loci: the set of points where a map of vector bundles drops rank, or equivalently, the set of points where two vector bundles intersect more than necessary. The problem of finding formulas for the cohomology classes of these loci dates to the 19th century, but has experienced a surge of interest in the last few decades. The answer will be a universal polynomial in the Chern classes of the vector bundles involved, and is closely related to the equivariant classes of Schubert varieties in G/B, where G is semisimple algebraic group. I'll survey recent progress in understanding these polynomials in classical types.

Lecture 4 Divisors on Bott-Samelson varieties [subject to change]

Given a sequence of roots, one can construct a corresponding Bott-Samelson variety. These varieties are basic tools in representation theory and geometry of G/P's; for instance, the Bott-Samelson varieties corresponding to reduced sequences resolve singularities of Schubert varieties. Partly because of these connections, a good understanding of line bundles and divisors, including descriptions of the nef and effective cones, is of interest. In this talk, I'll explain what is known about these questions, what I would like to know, and how much I know of what I'd like to know.

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