Texas Tech University

Abstracts

**
Baris Coskunuzer, Koc University, Turkey
:
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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.