Speaker: Dave Anderson, University of Washington

Title: Positive products, effective classes: how group actions turn geometry into combinatorics

Abstract: Suppose X is a compact complex manifold. Any complex subvariety Y determines a class in the (singular or de Rham) cohomology ring of X -- and classes that arise this way are called "effective". Given a class in cohomology, how can you tell when it is effective? If y and z are two effective classes, is the product y*z effective?

In general, these questions lead to some of the deepest problems in algebraic and complex geometry. However, when a group acts transitively on X, the answers are simple. I'll focus on the (typical) example where X is the manifold of complete flags in an n-dimensional vector space. Here the geometry yields to concrete algebra, in the form of a presentation of the cohomology ring, along with a basis of effective classes. The structure constants for multiplication in this effective basis are of particular interest, and thanks to the Kleiman-Bertini transversality theorem, they are known to be positive. I'll also describe some recent generalizations of this positivity phenomenon, where the structure constants are *polynomials* with positive coefficients. (This is joint work with Stephen Griffeth and Ezra Miller.)

Positivity leads to a fundamental problem in the subject: What is a combinatorial formula for these structure constants? To a surprising degree, this question is unresolved. I'll conclude with a brief survey of some of the cases where the answer is known.

Place: Science and Technology Building I, Room 242

Refreshments will be served before the talk at 3:00 p.m. in Room 222.

Department of Mathematical Sciences
George Mason University
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Fairfax, VA 22030-4444
http://math.gmu.edu/
Tel. 703-993-1460, Fax. 703-993-1491