Tuesday, Jan. 22, 10am, 7218
Gauge theory moduli spaces and fundamental group representations.
Steve Bradlow (University of Illinois, Urbana-Champaign/UCSD)
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).
Tuesday, Jan. 29, 10am, 7218
The Kontsevich integral, the Duflo isomorphism, and Wheeling
Justin Roberts (UCSD)
The Kontsevich integral is an invariant of framed knots in the
three-sphere, which is universal for the class of Vassiliev and
quantum invariants. It takes values in an algebra whose generators and
relations are described pictorially using so-called Jacobi
diagrams.
The Duflo isomorphism is an algebra isomorphism between the invariant
parts of the symmetric algebra and universal enveloping algebra of any
Lie algebra. It turns out to have a purely abstract Jacobi-diagrammatic
formulation, the "wheeling" theorem of Bar-Natan, Le and D. Thurston,
which can be proved using knot theory and the Kontsevich
integral. I'll try to explain something of this rather mysterious
theory.
Tuesday, Feb. 5, 10am, 7218
Patterns of circles: uniformization and rigidity
John He (UCSD)
Methods for studying circle patterns on the
Riemann sphere are discussed.
Tuesday, Feb. 12, 10am, 7218
Tuesday, Feb. 19, 10am, 7218
Tuesday, Feb. 26, 10am, 7218
Suspension Flows are Quasigeodesic
Diane Hoffoss (USD)
A flow on a manifold M is a decomposition of M into a collection
of coherently oriented 1-manifolds; one can think of a flow as
the integral curves of a nowhere-zero vector field on M. A
quasigeodesic flow is one in which each flow line is uniformly
efficient in measuring distances in its relative homotopy class.
Not all flows can be isotoped to be quasigeodesic, nor do all
manifolds admit a quasigeodesic flow.
Every hyperbolic 3-manifold which fibers over the circle admits a
flow called the suspension flow. We show that such a flow can be
isotoped to be quasigeodesic, thus producing a large class of
manifolds admitting a quasigeodesic flow.
Tuesday, March 5, 10am, 7218
Index theorems: even vs. odd
Xianzhe Dai (UCSB)
The celebrated Atiyah-Patodi-Singer index theorem generalizes the
Atiyah-Singer index theorem to manifold with boundary. Here the
topology, geometry and anlysis are all linked in a single formula. But
this formula is mostly restricted to even dimensional manifolds. We
will talk about an odd dimensional analogue of the
Atiyah-Patodi-Singer index formula. This is joint work with Weiping
Zhang.
Tuesday, March 12, 10am, 7218
TBA
Dylan Thurston (Harvard)