Texas Geometry and Topology Conference

 February 18-20, 2011
Texas Tech University





Travel Lodging






Baris Coskunuzer, Koc University, Turkey : Generic uniqueness of area minimizing disks for extreme curves


Abstract : In this talk, we will start with an overview of the Plateau problem in 3-manifolds. Then, we will give a sketch of the proof of the following statement: For a generic nullhomotopic simple closed curve C in the boundary of a compact, orientable, mean convex 3-manifold M with trivial second homology, there is a unique area minimizing disk D embedded in M where the boundary of D is C. The same statement is also true for absolutely area minimizing surfaces, too.

Jeanne Clelland, University of Colorado at Boulder : Backlund transformations and Darboux integrability for nonlinear wave equations


Abstract :   The nonlinear wave equation

z_{xy} = e^z

is called Liouville's equation. It has two important properties:

1) It is integrable by the method of Darboux at second order;
2) It has a Backlund transformation relating its solutions to those of the wave equation z_{xy} = 0.

Each of these properties provides a (different) method for solving Liouville's equation via ODE methods. In this talk we show that the fact that Liouville's equation has both of these properties is no accident: specifically, we prove that a second-order Monge-Ampere equation for one function of two variables is connected to the wave equation by a Backlund transformation (of a specific type) if and only if it is integrable by the method of Darboux at second order. The proof relies on a geometric formulation of a Backlund transformation as a certain type of exterior differential system and a careful study of its associated differential invariants. This is joint work with Thomas Ivey of the College of Charleston.

Annalisa Calini, College of Charleston :  Integrable evolution of closed vortex filaments: finite-gap solutions and their linear stability


Abstract : Finite Gap Solutions

Richard Palais, University of California Irvine :  Geometry and Computer Graphics: Partnership, Interaction, and Challenges


Abstract :   Mathematical Visualization can be seen as a marriage of Geometry and Computer Graphics, with each partner gaining from and contributing to the partnership.
In this talk I will first say a little about "how it works", then explain some of the major goals (e.g., being able to "see" higher dimensional objects and processes), and finally explain how some fairly advanced geometric concepts can lead to improvements in various Computer Graphics algorithms, that then allow us to see geometric objects better or in a new way. I will be using 3D-XplorMath, a Mathematical Visualization program that I helped develop, to illustrate my talk. It is freely available at http://3D-XplorMath.org

Anthony Bloch, University of Michigan :  Geometry of Nonholomic Systems


Abstract :  The geometry that arises in the study of nonholonomic systems

Dragos Oprea, University of California San Diego :  The moduli space of stable quotients


Abstract :   I will introduce the moduli space of stable quotients of the trivial sheaf over stable curves, which compactifies the space of maps from smooth curves to Grassmannians. I will discuss the connection between stable quotient invariants and Gromov-Witten theory. This is based on joint work with Alina Marian and Rahul Pandharipande.

Shantanu Dave  University of Vienna, Austria :  Asymptotic analysis of regularizations that know about singularities


Abstract :   This talk is an introduction to generalized function algebras and their applications in non-smooth global analysis.

We first study some structural properties of usual Schwartz distributions that play an important role in  linear PDEs with smooth coefficients. We then proceed to look for a setup that would work for certain (linear) PDEs with non-smooth coefficients and some nonlinear PDEs. We also briefly outline a few applications.

Two aspects of a distribution determine many of its analytical properties:

1)  its regions of non-smoothness (singular support and Wave-front set), and

2)  its Sobolev regularity.

We shall motivate our discussion by describing a geometrical approximation procedure for distributions via smooth functions where the singularity and regularity information can be encoded in the asymptotics of the approximating net.  Such a construction, whether via spectral theory or otherwise, leads to the notion of algebras of generalized functions, which basically provide a framework (like distributions do) to treat certain kinds of PDEs. One of their main benefits is that they function as a book-keeping device that tracks regularity and singularity structures.

We shall work with a simple spectral theoretic example and also provide some historical background on a note by L. Schwartz that led to the construction of Colombeau algebras, as well as on some older constructions as described by Guillemin-Sternberg.

Alexander A. Voronov (University of Minnesota and IPMU) :  Higher Categories and TQFTs


Abstract :   The goal of the talk is to describe categorical formalism for higher dimensional, a.k.a. extended, Topological Quantum Field Theories (TQFTs) and present them as functors from a suitable category of cobordisms with corners to a linear category, generalizing 2d open-closed TQFTs to higher dimensions. The approach is in the spirit of monoidal categories (associators, interchangers, Mac Lane's pentagons and hexagons), in contrast with the simplicial (weak Kan and complete Segal) approach of Jacob Lurie's. This is a joint work with Mark Feshbach.