Geometry, Topology, and Physics Seminar, Spring 2008

Organizers: Andreas Malmendier and Dave Morrison.
Meets 4:00 - 5:30 p.m. Fridays in South Hall 6635.

Various topics relating geometry, topology, and physics.

Apr. 3/4

### Quantum Curves and Random Matrices

These talks will be a part of the UCSB Distinguished Lectures on "Quantum Curves and Random Matrices" given by the speaker.

Abstract: I will review recent work that relates topological string theory, random matrices and integrable hierarchies, and that leads to a natural quantum' deformation of conformal field theories on Riemann surfaces where algebraic curves are replaced by a non-commutative D-modules.

Thursday: Audio [ mp3, wma ]; Lecture Notes.
Friday: Audio [ mp3, wma ]; Lecture Notes [version 1, version 2].

Apr. 11, 18

Apr. 25

### Calabi-Yau Singularities, III

Abstract: Calabi-Yau threefolds are compact Riemannian manifolds with holonomy SU(3), and as such, are always projective algebraic varieties. As algebraic varieties, there is a natural generalization to singular Calabi-Yau threefolds (although the existence of generalized metrics on these singular spaces is not known). Moreover, in the application of Calabi-Yau threefolds to the study of string compactification, singular Calabi-Yau threefolds play an important role. We will review what is known about the Calabi-Yau singularities' on such spaces, with an eye on recent developments and applications.

Audio [ mp3,wma ]; Lecture Notes.

May 2

### A hyperbolic state sum model for SL(2,C) Chern-Simons theory

Abstract: I will talk about recent work with S. Gukov and J. Lenells. Based on a hyperbolic knot invariant of K. Hikami's, we propose a geometric state-sum-model construction for the partition function of SL(2,C) Chern-Simons theory on hyperbolic three-manifolds. Perturbative coefficients of the partition function can be computed exactly in this construction, and we (successfully) compare the results to the Chern-Simons partition function obtained via geometric quantization.

Lecture Notes [version 1, version 2, version 3].

May 12/13

### The Geometric Langlands Conjecture

These talks will be a part of the UCSB Distinguished Lectures on "The Geometric Langlands Conjecture" given by the speaker.

Monday: Arithmetic and Geometry, Abelian and Non Abelian
Abstract: We will describe the Langlands program in general as a conjectural non abelian analogue of well known "abelian" results: class field theory on the arithmetic side, and the Abel-Jacobi theory of Riemann surfaces and their Jacobians on the geometric side. One powerful approach to the geometric Langlands conjectures involves abelianization via Hitchin's integrable system and its spectral curves.

Tuesday:Algebra and Analysis, Classical and Quantum
Abstract: Recent input from physics suggests that the geometric Langlands conjecture, as formulated by Deligne, Laumon, Beilinson and Drinfeld, can be viewed as a statement in quantum field theory. This has a classical limit which has now been proved, at least generically. The relationship between the classical and quantum versions is deep and mysterious. The quantum version can of course be studied as a deformation of the classical one. But there is tantalizing evidence - from quantum field theory as well as non abelian Hodge theory - that the full quantum version can also be understood as a twistor rotation of the classical version.

Audio [ mp3, wma ]; Transparencies.

May 22

### On Four-Dimensional Einstein Manifolds

Abstract: An Einstein metric is by definition a Riemannian metric of constant Ricci curvature. One would like to completely determine which smooth compact n-manifolds admit such metrics. In this talk, I will describe recent progress regarding a the 4-dimensional case. These results specifically concern 4-manifolds which also happen to carry either a complex structure or a symplectic structure.

Jun 5/6

### Geometric Flows

This talk will be a part of the UCSB Distinguished Lectures on "Geometric Flows and Special Holonomy" given by the speaker.

Thursday: Geometric Flows and Special Holonomy
Abstract: Geometric flows have been considered by several researchers as an approach to constructing solutions of the special holonomy equations. In this talk, I will examine the nature of these `flows', in particular, their well-posedness and stability. The first nontrivial case, namely SU(2) holonomy in dimension 4, already displays many of the interesting features that are encountered in the exceptional cases (G2 holonomy in dimension 7 and Spin(7) holonomy in dimension 8). The study of these features raises some interesting problems in differential geometry even in dimension 3, as will be explained in the lecture.

Audio [ mp3, wma ]; Lecture Notes.

Friday: The Geometry of Riemannian Submersions
Abstract: A considerable amount of work has been done to classify Riemannian submersions and foliations from space forms, and most of the recent work has concentrated on using non-negativity of the curvature to prove global rigidity results. Meanwhile, the local nature of the PDE that define Riemannian submersions has received considerably less attention and, in fact, little appears to be known about the local generality of solutions of the underlying overdetermined PDE. In this talk, I will discuss the nature of this overdetermined PDE system, the cases in which the full local solution space is understood and the prospects for understanding the general case. If there is time, I will also discuss what this information could tell us about the global problem of classifying the Riemannian submersions from space forms.

Audio [ mp3, wma ]; Lecture notes.

Jul 18

### Mathematical Foundations of Orientifolds

Abstract: We give an overview of a project with Dan Freed and Jacques Distler whose goal is a precise and general formulation of orientifold backgrounds of type II string theory. A central theme is the use of twisted equivariant differential generalized cohomology theories. The B-field twists a version of equivariant KR theory, while the RR fields and currents are formulated in terms of a twisted equivariant differential KR theory. One important new result is a formula for the RR charge of an orientifold plane at the K-theory level, which lifts the standard result of Morales-Scrucca-Serone in integral cohomology.

Audio [ mp3, wma ]; Lecture notes.