# Differential Geometry Seminar Schedule for Spring 2000

Note that the geometry group is running a K-theory seminar this qaurter at Friday 3:30 pm, small seminar room at ITP.

# Fridays 3:15 - 4:15pm, SH 6617

### 3/10 3-4pm Christine Escher, Oregon State Univ., Corvallis Classifying families of manifolds"

Abstract: About thirty years ago differential geometers were asking questions about the topological nature of a fundamental class of manifolds, namely homogeneous spaces. Of particular interest was the question whether any two homeomorphic homogeneous spaces necessarily had to be diffeomorphic. In 1988 examples of seven dimensional homogeneous spaces which are homeomorphic but not diffeomorphic were found. The interest in this class of homogeneous spaces containing examples which are homeomorphic but not diffeomorphic arises from problems in differential geometry and theoretical physics; in particular these families include Einstein manifolds. In this talk I will describe a specific family of seven dimensional manifolds which occur naturally in differential geometry and were considered independently by theoretical physicists in connection with Kaluza-Klein theories. I will give a complete classification up to homeomorphism and diffeomorphism of these generalized Einstein-Witten manifolds.

### 3/10 4:10-5:10pm Weiping Li, Oklahoma State Univ., Stillwater Knot and Link Invariant"

Abstract: We show that the representation varieties of $\pi_1(S^2 \setminus (S^2 \cap L))$ (a link $L$ in $S^3$) with different conjugacy classes in $SU(2)$ along meridians are symplectic stratified varieties from the group cohomology point of view. We obtain an invariant of links (knots) from intersection theory on such a variety. We also study a $SL_2({\C})$-character variety of a knot $K$ in $S^3$ with fixed holonomy $\mu + \mu^{-1}$ along the meridian of $\pi_1(S^3 \setminus K)$ ($\mu \in {\C}^*$), and discuss its relation with $A$-polynomial.

# Fridays 4:00 - 5:00pm, SH 6623

### 10/15 Siye Wu, UCSB Supersymmetric Gauge Theories"

Abstract: We give an overview of classical and quantum aspects of supersymmetric gauge theories. This is the first of three talks on non-perturbative quantum field theory and geometry.

The subsequent talks will focus on monopoles and duality, and applications to four manifold theory.

All the talks shall be accessible to graduate students.

### 11/5 Kyungho Oh, University of Missouri, visiting UCSB Mirror Symmetry II"

Abstract: This is a continuation of Kyungho Oh's talk given in Arithmetic and Geometry Seminar on Oct. 22nd. He will sketch the topological mirror construction due to Mark Gross and discuss other related issues if time permits.

### 12/3 Frank Packard, Paris 12, visiting Stanford University Linear and nonlinear aspects of Ginzburg-Landau vortices"

Abstract: We present a joint work with Tristan Riviere concerning existence and uniqueness questions for Ginzburg-Landau vortices.
More precisely, we describe precisely some branches of critical points of the Ginzburg-Landau functional $E(u) = \int |\nabla u|^2 + \frac{1}{2 \e^2} \int (1 - |u|^2)^2,$ as the parameter $\e$ tends to $0$, here $u$ is a complex valued function defined in some bounded domain of ${\mathbb R}^2$.
In particular we prove that, provided $\e$ is small enough, all solutions of $\Delta u + \frac{u}{\e^2} (1- |u|^2) =0,$ which are defined in the unit ball and have boundary data given by $u = e^{i \theta}$ are " radialy symmetric", which means that they are of the form $u = S (r) \, e^{i \theta}$.
Applications to the gauge invariant Ginzburg-Landau functional are also given.