Stanford University Geometry Seminar 2001-02

Unless otherwise noted, all seminars are on Wednesdays 4:00 - 5:00 pm in Room 381-T
(basement of Math Building, Bldg 380).
There is tea at 3:30 on Wednesdays in the third floor lounge of the Math Building.

Winter 2002 Schedule

January 16

No Seminar due to Distingished Lecture

January 23

Steve Bradlow, University of Illinois, Urbana-Champaign
Title: Gauge theory moduli spaces and fundamental group representations.
Abstract: Holomorphic bundles over Riemann surfaces give rise to many interesting moduli spaces. In addition to the moduli space of stable bundles, there are other moduli spaces which parameterize bundles with various types of extra structure. These all have gauge theoretic descriptions - in some cases this reveals them to be moduli spaces of flat connections and thus representation spaces for Riemann surface fundamental groups.

We will give a brief survey of some of these moduli spaces and discuss their relation to representations of the fundamental group of the Riemann surface. In particular, we will describe how the gauge theoretic information can be used to count the number of components in the space of representations into the groups U(p,q).

January 30

Valerio Toledano, MSRI (Special time: 12:00 noon)
 Title: Flat Connections and Quantum Groups
Abstract: In this talk, I will describe how quantum groups serve as a useful means of expressing the monodromy of certain integrable, first order PDE's. A fundamental, and paradigmatic result in this context is the Kohno--Drinfeld theorem which may be stated as follows. Let $\g$ be a complex, simple Lie algebra and $U_{\hbar}\g$ the corresponding Drinfeld--Jimbo quantum group. Its universal $R$--matrix gives, for each $n\in\IN$, a representation of Artin's braid group on $n$ strands $B_{n}$ on the $n$--fold tensor product $V^{\otimes n}$ of any finite--dimensional $U_{\hbar}\g$--module $V$. The Kohno--Drinfeld theorem asserts that this representation is equivalent to the monodromy of the Knizhnik--Zamolodchikov equations, a flat connection on $\{(z_{1},\ldots,z_{n})\in\IC^{n}|z_{i} \neq z_{j}\}$ with values in $V^{\otimes n}$. As Lusztig, Kirillov--Reshetikhin and Soibelman observed, the quantum group $U_{\hbar}\g$ also yields representations of another braid group, namely the generalised braid group $B_{\g}$ of Lie type $\g$. Whereas the $R$--matrix representation is a deformation of the natural representation of the symmetric group ${\mathfrak S}_{n}$ on $n$--fold tensor products, the quantum Weyl group representation of Lusztig {\it et al.} deforms the action of (a finite extension of) the Weyl group $W$ of $\g$ on any finite--dimensional $\g$--module $U$. I will describe a new flat connection on the set of regular elements of a Cartan subalgebra $\h$ of $\g$ with simple poles on the root hyperplanes and values in $U$ \cite{MTL} and prove that, for $\g=\mathfrak{sl}_{n}$, its monodromy is equivalent to the quantum Weyl group representation on $U$ \cite{TL1,TL2}.

February 6

Neshan Wickramasek, Stanford University
Title: Rigidity of tangent cones and singular set of immersed stable minimal hypersurfaces
Abstract:

February 13

Tanya Christiansen, University of Missouri
Title: Resonance counting and manifolds with infinite cylindrical ends
Abstract: We give an introduction to resonances and the problem of bounding their numbers. For some operators with continuous spectrum, resonances are appropriate analogs for eigenvalues. They are also related to long-time behaviour of solutions of the wave equation. We include some results of recent work bounding the number of resonances of the Laplacian on manifold with infinite cylindrical ends.

February 20

Robin Graham, University of Washington, Seattle
Title: Scattering Theory for Poincare Metrics and Q-Curvature
Abstract: This talk will describe recently discovered connections between scattering theory on conformally compact asymptotically Einstein manifolds and conformally invariant differential operators and Branson's Q-curvature in conformal geometry. These connections lead to new definitions of Q-curvature and to new relations between the geometry of the Poincare manifold and the conformal geometry at infinity. The results consist of separate joint work with Maciej Zworski and Charles Fefferman.

February 27

Reiner Schaetzle, Univ. Bonn
Title: Quadratic tilt-excess decay and a strong maximum principle for varifold
Abstract: We prove that integral $\ \n-$varifolds $\ \mu\ $ in codimension 1 with $\ \meansc_\mu \in L^\p_{loc}(\mu), \p > \n, \p \geq 2\ $ have quadratic tilt-excess decay \begin{displaymath} tiltex_\mu(x,\varrho,T_x \mu) = O_x(\varrho^2). \end{displaymath} for $\ \mu-$almost all $\ x\ $ and a strong maximum principle which states that these varifolds cannot be touched by smooth manifolds whose mean curvature is given by the weak mean curvature $\ \meansc_\mu\ $, unless the smooth manifold is locally contained in the support of $\ \mu\ $

March 06

Alex Gamburd, Stanford University
Title: Spectral gap for arithmetic manifolds of infinite volume.
Abstract: A celebrated theorem of Selberg states that for congruence subgroups of the modular group there are no exceptional eigenvalues below 3/16. We will present a new, geometric proof of Selberg's theorem and prove its generalization for infinite index congruence subgroups. For such subgroups with a high enough Hausdorff dimension of the limit set we will establish a spectral gap property and consequently solve a problem of Lubotzky pertaining to expander graphs.

March 13

Fran Presas, Stanford University
Title: Semipositive bundles and Brill-Noether theory
Abstract: We prove a Lefschetz hyperplane theorem for the determinantal loci of a morphism between two holomorphic vector bundles over a complex manifold under a positivity condition. We apply the result to give some homotopy groups of the Brill-Noether loci of a generic algebraic curve.

March 20

Herbert Koch, Visiting UC Berkeley
Title: Unique continuation for elliptic operators: Positive results, counter examples and open questions
Abstract: The talk will be about $L^p$-Carleman inequalities and their relation to strong unique continuation for elliptic operators of second order. Among other things I will discuss sharp results for $L^{n}$ potentials for the gradient and explain strong counterexamples for larger potentials.

Spring 2002 Schedule

Autumn 2001 Schedule

Questions: lni@math.stanford.edu