Organizational meeting, 4:00 PM in MATH 402.

This is the first in a series of introductory talks on basic topics, which will hopefully constitute a gentle and coherent introduction to Calabi-Yau manifolds, and possibly mirror symmetry. In this first talk I will approach the subject from the point of view of Riemannian geometry. Later Philip Foth will approach the subject from the point of view of algebraic geometry. The basic prerequisites for this first talk are knowledge of what a Riemannian metric and a Lie group are, and an intuitive idea of what curvature means. I tentatively plan to explain 1) the notion of holonomy, 2) Berger's theorem on the classification of holonomy groups, and 3) how Calabi-Yau manifolds fit into this picture. Along the way I will mutter some words about how this fits into the string theory view of the universe.

This will a continuation of last week's lecture. I will spend about 10 minutes reviewing some of the basic ideas which we discussed last week, in terms of concrete calculations (covariant derivative, parallel translation, holonomy). I will then review the statement of Berger's theorem, and try to more carefully explain the connection between special holonomy and geometric structure. In the second half of the talk, we will discuss the connection between curvature and holonomy (the Ambrose-Singer theorem), the symmetries of the curvature tensor, and how this is used in Simon's proof of Berger's theorem.

In this talk we will start with a short introduction to some aspects of noncommutative geometry as introduced by A. Connes and then describe some applications. One will be to show how duality in topology leads to a version in noncommutative geometry related to some ideas in physics. A second application, if there is time, will cover recent related work, joint with Xiang Tang, on invariants for operators on manifolds with a Riemannian foliation.

Liouville theorem says that there are few conformal mappings in three and more dimensions. The same can be said about vector fields of conformal deformations (Ahlfors, 1974). In this case conformality appears in the form of a system of linear PDEs. When classifying Weyl connections on tori with nonpositive sectional curvatures we arrive at somewhat similar system of nonlinear PDEs. We will discuss a general (and well known to experts) approach to systems of PDEs that works in both cases and (hopefully) solve the nonlinear problem.

A Kaehler manifold is at the same time a complex, Riemannian and symplectic manifold. It therefore has a rich geometrical structure. We discuss basic properties of such manifolds. The talk will be elementary. Students are especially encouraged to attend.

This is an introductory talk with basic definitions and examples, from the point of view of algebraic geometry. If time allows, we will also talk about the Kaehler cone and deformations of complex structures.

This talk will give an overview of the definition and some basic structure results of Deligne-Mumford stacks. Applications of stacks to the study of moduli spaces and to Brauer groups will be described.

We will investigate the use of weighted triangulations as a discrete analogue of Riemannian geometry. We will then introduce discrete Laplacian operators, which are particularly weighted Laplacians on the 1-skeleton of a metric (Euclidean) triangulation in the sense of Laplacians on graphs. We will investigate some of the properties of these Laplacians, including an interesting optimality result for weighted Delaunay triangulations originally proven by Rippa for only Delaunay triangulations.

In this continuation talk we will explore more deeply discrete Riemannian structures and especially discrete Laplacians using some tools of Euclidean geometry and techniques for working with weighted Delaunay triangulations. It should be accessible to those who were unable to attend the previous talk.

Consider an algebraic action of a connected complex reductive algebraic group on a complex polarized projective variety. In this talk, we first introduce the nilpotent quotient, the quotient of the polarized projective variety by a maximal unipotent subgroup. Then, we introduce and investigate three induced actions: one by the reductive group, one by a Borel subgroup, and one by a maximal torus, respectively. Our main result is that there are natural correspondences among quotients of these three actions. In the end, we mention a possible application to the moduli spaces of parabolic bundles over algebraic curves for further research.

Tuesday, 25 April 2006

Semail Ülgen, Department of Mathematics, The University of Mississippi, will speak on “K-Exactness of Group C*-Algebras” at 4:00 PM in Math 402.