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 10:00 - 11:00am, SH 4607

4/10 10-11am Yongbin Ruan, visiting CalTech, U. Wisconsin at Madison ````New" geometry and topology of orbifolds"

5/5 10-11am Maria Noronha, CSU at Northridge ``Homogeneous submanifols of codimension 2"

5/26 10-11am Daryl Cooper, UCSB ``F-strutures in the Orbifold Theorem"

Differential Geometry Seminar Schedule for Winter 2000

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

1/21 Xianzhe Dai, UCSB ``Family index theorem and adiabatic limit"

1/28 Yuhan Lim, UCSB ``Non-abelian Seiberg-Witten invariants for 3-manifolds"

2/4 Xianzhe Dai, UCSB ``Family index theorem and adiabatic limit, Part II"

2/11 cancelled

2/18 Rugang Ye, UCSB ``Mirror Symmetry and Special Lagrangian Submanifolds"

2/25 Rugang Ye, UCSB ``Mirror Symmetry and Special Lagrangian Submanifolds, Part II"

3/3 Guofang Wei, UCSB ``Ricci Curvature and Hausdorff Convergence"

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.

3/21 2-3pm Mark Haskins, Texas University, Austin ``Special Lagrangian cones in dimension 3"

Differential Geometry Seminar Schedule for Fall 1999

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

10/1 ``Organizational Meeting"

10/8 Rick Ye, UCSB ``An introduction to neural networks"

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.

10/22 Siye Wu, UCSB ``Supersymmetric Gauge Theories II: Monopoles and Duality"

10/29 Siye Wu, UCSB ``Supersymmetric Gauge Theories III: Applications to four manifold theory"

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.

11/12 Siye Wu, UCSB ``Supersymmetric Gauge Theories IV: Seiberg-Witten Invariants"

11/19 cancelled

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.

12/10 Meilin Yau, UCSB ``Tight contact structures on 3-manifolds"

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