GEOMETRY SEMINAR SCHEDULE,
FALL 2004
Tuesday, August 24
organizational meeting
Tuesday, August 31
Dorin Dumitrascu, Department of
Mathematics, University of Arizona, will speak on “C*-algebras and
K-theory for Fredholm manifolds” at 4:00 PM in MATH 402.
Abstract: The purpose of the talk is to indicate the construction of a
C*-algebra A(M)associated in a canonical way to an infinite-dimensional
Fredholm manifold M. Its usefulness is given by the following facts:
(1) under an appropriate spin condition, the operator algebras K-theory
of this C*-algebra is the same as the topological K-theory, as defined
by Mukherjea, and we have a Poincare' duality isomorphism with the
compactly supported K-homology of M; (2) if M is a countable real
Hilbert space, then the construction recovers the C*-algebra used by
Higson, Kasparov and Trout in their proof of the infinite dimensional
Bott periodicity theorem. The notions of C*-algebra and Fredholm
manifold will be introduced at the beginning of the presentation. The
talk reports on some joint work with Jody Trout from Dartmouth College,
and it should be accessible to graduate students.
Tuesday, September 7
Michael Otto, Department of Mathematics, The University of
Arizona, will speak on “Kostant's Convexity theorem and symplectic
geometry” at 4:00 PM in Math 402.
A classical result due to Schur and Horn relates the diagonal
vector of a Hermitian matrix A to the eigenvalues of A. Considering all
Hermitian matrices with the same fixed set of eigenvalues it turns out
that the corresponding set of diagonal vectors describes a convex
polytope. Kostant's convexity theorem generalizes this result to the
framework of semisimple Lie groups. It came as a surprise when Atiyah
found an interpretation of (parts of) Kostant's theorem in terms of
symplectic geometry. We want to discuss various efforts of using
techniques from symplectic geometry to prove Kostant's convexity
theorem in full generality. This talk will be accessible to a wide
audience.
Tuesday, September 14
Michael Otto, Department of Mathematics, The University of
Arizona, will speak on “More on Kostant's convexity theorem and
symplectic geometry” at 4:00 PM in Math 402.
Symplectic convexity theorems like the ones by
Atiyah-Guillemin-Sternberg and Duistermaat can be used to study certain
aspects of the structure of semisimple Lie groups. We will discuss in
some detail the symplectic setup for Kostant's linear and nonlinear
theorems. The ideas involved lead to several other results on
semisimple Lie groups related to Kostant's theorem.
Tuesday, September 21
Paul Bressler, Department of Mathematics, The University of
Arizona, will speak on
“Chiral differential operators” at 4:00 PM in Math 402.
Chiral differential operators (CDO) arise in the
study of spaces of maps of one-dimensional manifolds. Their relationsip
to one-dimensional field
theories is analogous to that of usual differential operators to
zero-dimensional theories (study of spaces of maps of zero dimensional
manifolds).
Tuesday, September
28
Yi Hu, Department of Mathematics, The University of Arizona,
will speak on
“Weight Varieties and their Toric Degeneration” at 4:00 PM in Math 402.
We will explain a recent joint work with Philip Foth on toric
degeneration of quotients
of flag varieties.
Tuesday, October 5
Doug Pickrell, Department of Mathematics, The University of
Arizona, will speak on “The diagonal distribution of the invariant
measure on a compact symmetric space” at 4:00 PM in Math 402.
Given a compact symmetric space U/K (for example a sphere or a complex
projective space), we consider the Cartan embedding of U/K into U. In
this talk we will present a conjecture for the diagonal distribution of
the invariant measure on U/K, relative to a compatible
lower-diagonal-upper decomposition for the complexification of U. This
conjecture is known to be true for many examples. For reasons having to
do with the “large N limit”, it is of interest to understand this
calculation in terms of coordinates arising from the Cayley transform,
x → g=(1-x)/(1+x). This talk will emphasize these concrete examples and
calculations, and hence hopefully will serve as a useful introduction
to symmetric spaces.
Tuesday, October 19
Frederick Leitner, Department of Mathematics, The University of
Arizona, will speak on “The Filtered Tangent Space and Algebra of a
Group in Positive Characteristic” at 4:00 PM in Math 402.
Tuesday, November 9
Xiaofeng Sun, Department of Mathematics, Harvard University,
will speak on “Canonical metrics on the moduli space of Riemann
surfaces” at 4:00 PM in Math 402.
We study two new complete Kahler metrics, the Ricci metric and
perturbed Ricci metric on the moduli space of closed Riemann surfaces
and show that these metrics have nice curvature properties. Both of
these metrics have bounded geometry and have Poincare growth near the
boundary of the moduli space. Furthermore, the holomorphic sectional
curvature and the Ricci curvature of the perturbed Ricci metric are
pinched between negative constants. By using these new metrics we
showed that all known complete metrics on the Teichmuller space and
moduli space are equivalent in the sense that they are quasi-isometric.
This proved the conjecture of Yau about the equivalence between the
Teichmuller metric and the Cheng–Yau–Mok Kahler–Einstein metric and the
conjecture of Bers about the equivalence between the Kobayashi metric
and the Bergman metric. We then investigate the behavior of the Kahler
class of the Kahler–Einstein metric and derive algebro–geometric
properties of the moduli space. As a direct consequence, we showed that
the moduli space is of log general type. More importantly, the
logarithm contangent bundle of the moduli space is Mumford stable.
Also, following Yau's estimates, we show that the Kahler–Einstein
metric has bounded geometry.