
XVIIIth Annual Geometry Festival
Duke University, Durham, NC
Thursday March 13 – Sunday March 16, 2003
ABSTRACTS OF TALKS

Bennett Chow
,
Harnack estimates of LiYauHamilton type for the Ricci flow

We will survey some works on differential Harnack type estimates
for the Ricci flow. Starting from the work of LiYau on the heat equation,
Hamilton's matrix estimate for the Ricci flow, the geometric spacetime
approach (joint with SunChin Chu), a generalization (joint with Dan Knopf),
and its recent extension due to Bing Cheng.


Anda Degeratu
,
Geometrical McKay Correspondance

A CalabiYau orbifold is locally modeled on C^{n}/G
where G is a finite group of SL(n,C). One way to handle this type of
orbifolds is to resolve them using a crepant resolution of singularities.
We are interested in studying the topology of the crepant resolution.
This can be expressed in terms of the finite group G via the McKay
Correspondance.
Here are some
open problems discussed in the talk.


Ron Donagi
,
Griffiths' intermediate Jacobians,
integrable systems, and string theory

Griffiths showed how to associate to a smooth projective variety X its
intermediate Jacobian J(X), a complex torus which depends holomorphically
on X and which encodes a great deal of information about X and especially
about the algebraic cycles in it. When X is a CalabiYau threefold, the
Griffiths intermediate Jacobian occurs as the fiber of a natural
integrable system associated to the moduli of X. This leads to numerous
connections with string theory, which is typically compactified on such
CalabiYaus. In this talk we hope to show that this family of intermediate
Jacobians (together with various related integrable systems) is a valuable
tool for understanding mathematically a wide range of string theory
contexts. These include SeibergWitten integrable systems, superpotentials
and their connection to Hilbert schemes and other moduli problems, and
certain aspects of mirror symmetry, large N duality, heterotic/Ftheory
duality, FourierMukai transforms in the presence of Bfields, and more.


John Etnyre
,
Legendrian knots in high dimensions

I will discuss what is known about Legendrian submanifolds
in R^{2n+1} for n > 1.
Specifically I will describe Legendrian Contact Homology
for such Legendrian submanifolds and produce many examples.
If time permits I will also discuss various
intriguing constructions of Legendrian submanifolds.


Joeseph Harris
,
Are Cubics Rational?

The geometry of cubic polynomials, and in particular the rationality
of cubic hypersurfaces, has been a catalyst for new developments in
algebraic geometry for two centuries. The discovery of the
irrationality of cubic curves, for example, led to the study of
abelian integrals, which was central to much of 19th century
mathematics; while the rationality of cubic surfaces in many ways
gave birth to the subject of birational geometry. In the 20th
century, Clemens' and Griffiths' proof of the irrationality of cubic
threefold provided not just the first example of a counterexample to
Luroth's theorem in higher dimensions, but a magnificent example of
how Hodge theory could be used to settle algebraogeometric questions.
In this talk, I'd like to review these developments and then to focus
on the next, and currently unsettled, case: the rationality of cubic
fourfolds. Here evidence obtained by Brendan Hassett and others
suggests that cubic fourfolds may also play a pivotal role, providing
among other things an answer to the questions of whether rationality
is an open or a closed condition in smooth families. I'll discuss the
current state of our knowledge, and what is conjectured.


Claude LeBrun
,
Zoll Manifolds, Complex Surfaces, and Holomorphic Disks

I will describe some recent joint work
with Lionel Mason concerning compact surfaces whose geodesics
are all simple closed curves. Our approach not only yields
completely new proofs of all the main classical results concerning
the Riemannian case, but also gives equally strong results
concerning general affine connections.
In contrast to previous work on this subject, our approach is twistortheoretic, and depends fundamentally on the
fact that, up to biholomorphism, there is only one complex structure
on CP^{2}.


John Morgan
,
Variations of Hodge structure for 1parameter families
of CalabiYau threefolds

Abstract forthcoming


Madhav Nori
,
A modified Hodge conjecture

A conjecture of Hodge type will be stated. From
this conjecture it would follow that the kernel of the
AbelJacobi invariant defined by Griffiths is independent of
the embedding of the base field into the complex numbers.


Justin Sawon
,
Twisted FourierMukai transforms for holomorphic symplectic
manifolds

In his thesis, Caldararu described twisted FourierMukai
transforms for elliptic fibrations. In this talk I will describe how
certain holomorphic symplectic manifolds can be deformed to integrable
systems, i.e. fibrations by abelian varieties. These are higher
dimensional analogues of elliptic K3 surfaces, and twisted FourierMukai
transforms once again arise.
Here are some
open problems discussed in the talk.


Wilfried Schmid
,
Automorphic distributions, Lfunctions,
and functional equations

Traditionally the analytic continuation and functional
equation for Lfunctions of automorphic representations are
derived from the socalled Whittacker expansion of automorphic
forms. After a brief discussion of this approach and the technical
problems associated with it, I shall introduce the notion of an
automorphic distribution. I shall then describe how automorphic
distributions provide an alternative approach to the study of
Lfunctions. This is joint work with Steve Miller.


Jeff Viaclovsky
,
Fully nonlinear equations and conformal geometry

I will discuss some fully nonlinear
PDEs which first arose in the
conformal brach of the variational
equivalence problem studied by Bryant
and Griffiths. They are conformally
invariant, and have turned out to
be very useful in conformal geometry.
I will discuss some recent joint
work with Matt Gursky on fourmanifolds
with positive scalar curvature, and on the
existence of metrics with constant Qcurvature.


Claire Voisin
,
Kcorrespondences and intrinsic pseudovolume forms

We introduce the notion of Kcorrespondence, and show that many
CalabiYau varieties carry a lot of selfKisocorrespondences,
which furthermore satisfy the property of multiplying the canonical
volume form by a constant of modulus different from 1. This leads
to the introduction of a modified KobayashiEisenman pseudovolume
form, for which we are able to prove many instances of the
Kobayashi conjecture.
Here are some
open problems discussed in the talk.


