XXth Annual Geometry Festival
SUNY Stony Brook, Stony Brook, NY
April 8-10, 2005
ABSTRACTS OF TALKS
Periodic Solutions of Hamilton's Equations on Tori
Let the torus T^2n carry the standard symplectic structure, and a
Hamiltonian function H of period 1 in the time variable. By the Arnold
Conjecture, proved for the torus by Conley and Zehnder, the Hamiltonian
flow has at least 2n+1 orbits of period 1. Conley and Zehnder also
proved, under the additional assumption that all period 1 orbits are
nondegenerate: If there are only finitely many orbits of period 1, then
there are orbits of arbitrarily large minimal (integer) period. We prove
this statement also holds in the degenerate case. Thus there are always
infinitely many orbits of integer period. This settles a conjecture of
Conley for the torus; this conjecture is still open for other compact
Null Hypersurfaces and
Curvature Estimates in General Relativity
I will talk about recent results, obtained in joint
work with Rodnianski,
concerning conditional pointwise estimates for
the curvature tensor of vacuum solutions of
the Einstein equations. The assumptions we make allow
us to derive, or we simply require, bounds for
the energy and flux of curvature across space-like
and null hypersurfaces. This allows us to control the geometry
of null hypersurfaces and thus to define a `` good''
parametrix for the covariant wave equation.
Using this parametrix we can derive pointwise estimates
for the curvature tensor. One can regard our results
as a General Relativity version of the well known
Beale-Kato-Majda removal of singularities theorem
Singular Structure of Mean Curvature Flow
The lecture will discuss singularities of mean curvature
flow. After a review of background material, the focus
will be on the case of surfaces of positive mean curvature
evolving in 3-space. This is joint work with Toby Colding.
Blowing Up Kahler Manifolds with Constant Scalar Curvature
I will present some recent work with Claudio Arezzo concerning the construction of Kahler metrics with constant scalar curvature on the blow up, at isolated points,
of a manifold M already endowed with a constant-scalar-curvature Kahler metric.
The hypotheses require the vanishing of all
holomorphic vector fields on M.
A Topological View of Gromov-Witten Theory
I will discuss a view of Gromov-Witten theory based on
analogies to classical topology. Theorems can be
proven by localization and degeneration techniques.
One outcome is an approach to the Gromov-Witten
theory of the Calabi-Yau quintic threefold.
The talk represents joint work with D. Maulik.
Non-Linear Waves and Einstein Geometry
The talk will describe interaction between nonlinear waves and geometry in General Relativity.
It will focus on two examples: the new proof of stability of Minkowski space-time for the Einstein vacuum
(and scalar field) equations in harmonic gauge and the Price law for the collapse of a spherically
symmetric self-gravitating scalar field
Methods of Singular Metrics in Algebraic Geometry
Will discuss the application of methods of
singular metrics to problems in algebraic geometry such
as Fujita's conjecture of effective freeness and effective
very ampleness, effective Matsusaka big theorem, effective
Nullstellensatz, deformational invariance of plurigenera,
and finite generation of canonical ring.
Floer Theories in Symplectic Topology and Gauge Theory
I will sketch the basic idea of Floer theory, in particular its
symplectic version for pairs of Lagrangians and its instanton version
for homology 3-spheres.
In a joint project with Dietmar Salamon we define a new Floer theory
for 3-manifolds with boundary, using the instanton equation with
Lagrangian boundary conditions (containing nonlocal conditions on the
The underlying PDE exhibits some unexpected semi-global behaviour.
This can be understood as evidence that the new Floer homology is an
intermediate object between the instanton Floer homology and the
symplectic Floer homology and can thus be used to prove a conjecture
of Atiyah and Floer relating these.