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Graduate Colloquium Spring 2008

Thursday - 4:00 p.m.
Jack Baskin Engineering Room 301A
Refreshments served at 3:40
For further information please contact Alex Castro at acastro@math.ucsc.edu

May 7, 2008

Moduli spaces of bundles and generalized theta functions

Dr. Dragos Oprea, Stanford University

The Jacobian of any compact Riemann surface carries a natural theta divisor, which can be defined as the zero locus of an explicit function, the Riemann theta function. I will describe a generalization of this idea, which starts by replacing the Jacobian with the moduli space of bundles over a Riemann surface (or a higher dimensional base). These moduli spaces also carry theta divisors, described via "generalized" theta functions. In this talk, I will describe recent progress in the study of generalized theta functions.

May 14, 2008

The Laplacian Method in Combinatorics

Dominic Dotterrer, Stanford University

Any poset has an associated chain complex of modules whose homology represents the combinatorial geometry of the poset. In this setting, the Laplace-Beltrami operator carries more information than just cohomology groups; the first positive eigenvalue of the Laplace-Betrami is a measure of "phase-changes" in cohomology. This idea comes directly from the Cheeger constant or isoperimetric inequalities in Riemannian geometry and has been studied extensively in network design, measures of graph connectivity, and mixing of random walks. I would like to convince you that the natural picture of the Laplacian is as a random walk operator on cycles of the homology of truncated chain complexes and show you how to use it for your favorite (co)chain complexes.

May 21, 2008

Ricci flow, Kaehler-Einstein manifolds, and numerical geometry

Prof. Matthew Headrick, Stanford University

Ricci flow is a flow on the space of metrics on a real manifold, or on the space of Kaehler metrics on a complex manifold. It arises in physics in the context of certain two-dimensional quantum field theories, and is used in mathematics to prove theorems about topology. After reviewing the basic properties of Ricci flow and explaining its connection to physics, I will explain how it can be used as a numerical tool for computing Kahler-Einstein metrics on Calabi-Yau and del Pezzo manifolds.