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

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.

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.

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.

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.

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.