Fall 2001 UCSD Topology/Geometry Seminars
Tuesday, Oct. 9, 10am, 6438
Most Knot Energies are Minimized by Circles
Jason Cantarella (University of Georgia, Athens GA)
For the past ten years, there has been a great deal of interest in
studying the behavior of various energy functionals on knots and
links. The motivating example for much of this work has been the
Mobius-invariant energy. For this energy, one can use
Mobius-invariance to see easily that the curve of least energy is the
round circle. This gives rise to a natural question; are other
energies also minimized by circles? Since the other functionals will
not be Mobius-invariant, a proof must come from different ideas. In
this talk, we prove that a broad class of knot energies are uniquely
minimized by round circles. We also find some knot energies that are
not minimized by round circles. In the last part of the talk, we'll
discuss the prospects for extending some other results known for the
Mobius energy to other knot energies.
Tuesday, Oct. 23, 10am, 7218
Moduli spaces in gauge theory and algebraic geometry
Steve Bradlow (University of Illinois, Urbana-Champaign/UCSD)
This will be an introductory survey in which we describe a collection
of moduli spaces associated to holomorphic bundles. We will show how
they can all be described variously as spaces of solutions to gauge
theoretic equations, or as GIT quotients, or as symplectic reduced
spaces. We will describe some of their interesting geometric features
and indicate some applications.
Tuesday, Oct. 30, 10am, 7218
Gerbes and Homotopy Quantum Field Theory
Simon Willerton (Heriot-Watt University, Edinburgh/UCSD)
In this talk I will try to give some feel of how gerbes are
generalizations of line bundles and will explain some joint work with Paul
Turner on the relationship between gerbes and Turaev's notion of homotopy
quantum field theory.
Tuesday, Nov. 6, 10am, 7218
Rozansky-Witten invariants and extended TQFT
Justin Sawon (Oxford University)
The fact that a ribbon category can be used to construct a modular
TQFT in three-dimensions has become part of the folklore of the
subject. A traditional example of such a category is the category of
representations of a quantum group at a root of unity, and the
corresponding 3-manifold invariants are quantum invariants. In our
attempts to understand the Rozansky-Witten invariant of 3-manifolds we
have discovered that the derived category of coherent sheaves on a
holomorphic symplectic manifold also has a ribbon structure. In this
talk I will outline the construction of a TQFT based on this derived
category, following the extended TQFT approach. This leads to a
completely new kind of TQFT, without any of the semisimplicity
inherent in the usual algebraic examples. This is joint work with
Justin Roberts and Simon Willerton.
Tuesday, Nov. 13, 10am, 7218
Skein theory and the Murphy operators
Hugh Morton (Liverpool University)
The Murphy operators (Jucys-Murphy elements) in the Hecke algebra
$H_n$ of type $A$ are explicit commuting elements with the property
that symmetric functions of them belong to the centre of the
algebra. They can be represented by simple tangles in the Homfly skein
theory version of $H_n$. I shall review some examples of Homfly-based
skeins which have nice algebraic interpretations, and define
geometrically a homomorphism from the skein of the annulus to the
centre of each algebra $H_n$. I show how to represent the $m$th power
sum of the Murphy operators in $H_n$ by a tangle in a way which is
essentially independent of $n$, and consider some possible extensions
to other skein-based algebras.
Tuesday, Nov. 20, 10am, 7218
Hyperbolic polyhedra and scissors congruences.
Yana Mohanty (UCSD)
I will describe a new set of formulas for the volume of hyperbolic
tetrahedra that Greg Leibon obtained from a geometrical construction.
I will then show how this formula can be used to answer the hyperbolic version
of a question posed by Justin Roberts of finding an explicit
construction which shows that the Regge symmetry is a scissors congruence.
This is a work in progress.