# Texas Geometry and Topology Conference

## Feb. 22-24, 2008 Texas Tech University

Abstracts

Ian Anderson, Utah State : Differential Geometry with Maple

Abstract : In this talk I will describe a new suite of Maple packages for symbolic computations in Differential Geometry, Lie Groups and Lie Algebras. The basic features of the software will be demonstrated.

Ian Anderson, Utah State : Superposition Formulas for Differential Systems

Abstract : In this talk I will use the  concept  of  a superposition formula for an exterior differential system to obtain a far-reaching generalization of  Darboux's  integration method for partial differential equations. This concept leads to deep insights into this classical method; uncovers the fundamental geometric invariants of Darboux integrable systems;  provides for their algorithmic integration; and has applications well beyond  those currently found in the literature.

Sun-Yung Alice Chang, Princeton : Compactness of Conformally Compact Einstein Manifolds of Dimension 3+1

Abstract : In recent years, there has been great interest to study regularity properties of metrics which are conformal to Einstein metrics. In this talk, I will report some recent joint work with Sophie Chen and Paul Yang to study such metrics in the Ads conformally compact Einstein setting. The main issues are: Bach flat metrics on manifolds with boundary, suitable set up of matching boundary conditions, ε regularity results, and the order of the ALE ends.

Abstract : A spacetime $(X^{m+1}, g)$ is a Lorentz manifold  and two points $x_1, x_2\in X$ are causally related if there is a nonspacelike curve between them. To $x\in X$ one associates the sphere of all null-geodesics through $x$ called the sky $S_x$ of $x.$ Low observed that if the link $(S_x, S_y)$ in the space of all null geodesics of a globally hyperbolic spacetime is nontrivial, then $x,y$ are causally related. We generalize the linking number to such links whose components are nonzero homologous. We show that often two events are causally related if and only if the generalized linking number $alk(S_x,S_y)\neq 0.$ For $y$ in the future of $x$ we interpret $alk(S_x,S_y)$ as the algebraic number of times an observer travelling to $x$ along a timelike curve sees the light  rays from $x.$ We show that $x,y$ in a nonrefocussing globally hyperbolic spacetime are causally unrelated if and only if $(S_x, S_y)$ can be unlinked by an isotopy through skies. Low showed that if a globally hyperbolic $(X^{m+1}, g)$ is nonrefocussing then its Cauchy surface $M$ is compact, we show that the universal cover of $M$ is also compact. The techniques used to construct the invariants give rise to graded a Poisson algebra on bordism group of mappings of garlands that are copies of manifolds glued together at some points.

Richard Melrose, MIT : Adiabatic Limits and Morse Decomposition

Abstract : I will describe the semiclassical limit and the adiabatic degeneration of operators corresponding to a fibration and briefly explain some applications. As time permits I will indicate how to extend the discussion to minimally singular (i.e. Morse) fibrations.

Emma Previato, Boston University : Theta Functions and Isospectral Manifolds

Abstract : Click for PDF

Ken Richardson, TCU : Desingularizing Compact Lie Group Actions

Abstract : Click for PDF

Colleen Robles, Texas A&M : Rigidity of Projective Homogeneous Varieties

Abstract : The problem of identifying homogeneous varieties from their local differential geometry dates back to Monge, and has been studied by Fubini, Griffiths and Harris, Hwang and Yamaguchi, and others.  I will describe recent work with J.M. Landsberg that establishes a general recognition theorem.  The key component is the resolution of exterior differential systems by Lie algebra cohomology.

Adam Sikora, SUNY Buffalo : Quantizations of Character Varieties and Knot Theory

Abstract : We will define three different deformation-quantizations of the $G$-character varieties of the torus, for every reductive group $G$.

We prove that all of them are in the direction of Goldman bracket.Motivated by the work of Frohman and Gelca, we will conjecture that all of them are isomorphic to each other and we will report on our progress towards proving that conjecture.

In the second part of the talk we will explain the significance of our conjecture to knot theory. In particular, we will show that it relates the Witten-Reshetikhin-Turaev $\mathfrak g$-quantum invariants of a knot $K$ to the $A$-polynomial of $K$ and to other invariants of $S^3\setminus K$ (like the orthogonal ideal of $K$).