

AhlforsBers Colloquium
Stony Brook 1998
FRANCIS BONAHON  University of Southern California
Bending Measured Laminations of Hyperbolic 3Manifolds
Abstract: (joint work with JeanPierre Otal)
Let M be a compact
3manifold with nonempty boundary. If the topological necessary conditions
are satisfied, the hyperbolization theorem of Thurston and the AhlforsBers
double uniformization theorem provide many geometrically finite hyperbolic
structures on the interior of M. An invariant of such a geometrically
finite hyperbolic structure is the bending measured lamination of its
convex core, which is a measured geodesic lamination on the boundary of M.
We will consider the inverse problem of characterizing which measured
geodesic laminations on the boundary of M can be realized as bending
measured laminations of geometrically finite hyperbolic structures on the
interior of M. The problem is completely solved when M has incompressible
boundary. The problem is much more difficult when the boundary of M is
compressible, and we will discuss some additional necessary (conjecturally
sufficient?) conditions in this case.
LENNART CARLESON  MittagLeffler Institute
The Hexagonal Lattice in Complex Analysis
DARYL COOPER  U. C. Santa Barbara
The Orbifold Theorem
Abstract: The proof of Thurston's Orbifold theorem
requires a Margulis type theorem describing the
thin part of a hyperbolic 3manifold with conetype
singularities. This requires the classification of
noncompact Euclidean conemanifolds. We will
describe a strong version of the Soul Theorem for
these objects.
HERSHEL FARKAS  Hebrew University of Jerusalem
Riemann Surfaces and Theta Functions
BRIAN GREENE  Columbia University
Exotic Properties of Spacetime in String Theory
PHILLIP A. GRIFFITHS  Institute for Advanced Study
K_{2}(C) and Complex Function Theory
Abstract:
The study of meromorphic functions on a compact Riemann
surface leads to the Jacobian variety, which is the fundamental
invariant of the Riemann surface. For a triangle viewed as
a degenerate Riemann surface of genus one, the Jacobian variety
becomes C*, which to algebraists is known as
the group K_{1}(C).
For a complex algebraic structure the analogue of the Jacobian
variety is not understood and presents one of the most
fundamental challenges to Hodge theory.
However, for the tetrahedron viewed as a degenerate quartic
surface the analogue for surfaces of genus one curves
the Jacobian can be constructed and it turns out
to be the group K_{2}(C).
EMILY HAMILTON  Emory University
Geometrical Finiteness for Hyperbolic Orbifolds
DENNIS HEJHAL  University of Minnesota and Uppsala
Kernel Functions, Poincaré Series, and LVA
NIKOLA LAKIC  Cornell University
Asymptotic Teichmüller Space
FENG LUO  Rutgers University
SL(2,C) Representations of the Surface Groups
Abstract:
Each hyperbolic metric on a surface can be constructed as follows.
Assign to each subsurface homeomorphic to a 1holed torus or a
4holed sphere a hyperbolic metric
so that when two such subsurfaces overlap in an annulus or a 3holed sphere,
they overlap geometrically. The hyperbolic metric on the surface
is constructed as the result of gluing of all hyperbolic metrics on
the subsurfaces. This principle of reconstruction is extended to
SL(2,C) linear representations (and their characters) of
the fundamental groups of the surfaces.
YAIR MINSKY  S.U.N.Y. at Stony Brook
Kleinian Groups and the Complex of Curves
SCOTT A. WOLPERT  University of Maryland
Ramanujan, Coefficient Sums and Automorphic Quantum Ergodicity
