




Nonlinear and Geometric Analysis / Analyse non linéaire et
géométrique (Robert McCann, Organizer)
 ANTHONY BLOCH, University of Michigan, Department of Mathematics, Ann Arbor,
Michigan 480191109, USA
The geometry and analysis of symmetric and discrete rigid body
flows

In this talk I will discuss the geometry, symplectic structure, and
dynamics of certain mechanical flows on cross products of Lie groups.
These flows are in coupled Lax pair and in particular coupled double
bracket form. I will describe how such systems give rise to a
symmetric formulation of the generalized rigid body equations. I will
also describe a discrete version of the symmetric rigid body equations
and indicate the link of this work to the discrete generalized rigid
body equations of Moser and Veselov. Finally I will mention some
infinitedimensional generalizations. This is work with P. Crouch,
J. Marsden and T. Ratiu.
 ADRIAN BUTSCHER, Max Planck Institute for Gravitational Physics, Golm, Germany
The conformal constraint equations

One method of studying the asymptotic structure of spacetime is to
apply Penrose's conformal rescaling technique. The Einstein equations
then yield an underdetermined nonlinear system of second order PDEs for
an unphysical metric and a conformal factor (the product of
which gives the metric on the physical spacetime). This method is
problematic for the study of the geometry of the spacetime at null
infinity because the principal parts of the field equations degenerate
where the conformal factor vanishes. However, these difficulties can
be avoided by means of a technique of Friedrich, which replaces the
Einstein equations in the unphysical spacetime by an equivalent, yet
formally regular, system of equations called the conformal
Einstein equations. This system itself implies a set of constraint
equations for initial data hypersurfaces, called the conformal
constraint equations, in the unphysical spacetime for which no direct
method of finding solutions exists at present. This talk outlines a
strategy for finding perturbative solutions of the equations and
presents positive results under certain simplifying assumptions.
 RUSTUM CHOKSI, Simon Fraser University, British Columbia
On some recent methods for capturing multiple scales in
material microstructures

In the vast materials literature, scale analysis of microstructures is
usually undertaken by first assuming a rigid ansatz for the geometry
of possible structures. In this talk, I will discuss two analytical
methods for capturing certain features of minimizing structures which
are not geometrybased (ansatz dependent). I will illustrate the
application of these methods to microphases in diblock copolymer
melts.
 JIM COLLIANDER, University of Toronto, Toronto, Ontario
Illposedness for defocussing NLS

Standard tricks for showing illposedness for nonlinear dispersive wave
equations exploit special travelling wave or blowup solutions. Such
special solutions are unavailable in the defocussing setting, leaving
open the possibility that wellposedness extends to rougher functions
than in the focussing case. We construct a family of solutions of
defocussing NLS using ideas from scattering theory and show
illposedness in Sobolev spaces H^{s}(R) for s < 0. This
talk will report on work in progress with M. Christ and T. Tao.
 JUSTIN CORVINO, Department of Mathematics, Brown University, Providence, Rhode
Island, USA
Scalar curvature deformation and applications

The Einstein Constraint Equations form an underdetermined elliptic
system for a metric and a symmetric (0,2)tensor on a
threemanifold. For the vacuum timesymmetric case, the equations
reduce to a single condition, that the scalar curvature of the metric
vanish. In this talk we outline a procedure which takes advantage of
the underdetermined nature of the equations to localize deformation of
the scalar curvature: generically, small compactly supported
deformations of the scalar curvature can be realized as the scalar
curvature of small compactly supported deformations of the metric. As
an application we discuss a gluing construction of solutions of the
constraints which are identically Schwarzschild outside a compact
region (``near infinity''). These methods have been extended for the
full constraints (joint with Rick Schoen), where the Kerr solution is
the model at infinity.
 PANAGIOTA DASKALOPOULOS, CaliforniaIrvine, California, USA
Gauss curvature flow with flat sides: all time regularity
of the interface

We study the freeboundary problem associated to the degenerate Gauss
Curvature Flow: we show that if the initial surface is weakly convex
with a flat side, then the surface becomes smooth, up to the interface,
at time t > 0, and remains so up to the vanishing time of the flat
side.
 AILANA FRASER, Brown University, Providence, Rhode Island, USA
The fundamental group of manifolds with positive isotropic
curvature

A central theme in Riemannian geometry is understanding the
relationships between the curvature and the topology of a Riemannian
manifold. Positive isotropic curvature (PIC) is a natural and much
studied curvature condition which includes manifolds with pointwise
quarterpinched sectional curvatures and manifolds with positive
curvature operator. By results of Micallef and Moore there is only one
topological class of compact simply connected manifolds with PIC;
namely any such manifold must be homeomorphic to the sphere. On the
other hand, there is a large class of nonsimply connected manifolds
with PIC. An important open problem has been to understand the
fundamental group of manifolds with PIC. In this talk we describe a new
result in this direction. The techniques used involve minimal
surfaces.
 BO GUAN, Department of Mathematics, University of Tennessee, Knoxville,
Tennessee 37996, USA
Regularity of pluricomplex Green functions

Let W be a bounded smooth strongly pseudoconvex domain in
C^{n}. In this talk we will prove the C^{1,1} regualrity of
the pluricomplex Green function of W with pole at infinity.
This is done by solving an extrior Dirichlet problem for the
homegeneous (degenerate) complex MongeAmpère equation.
 STEVE GUSTAFSON, ETH Zuerich (Theoretische Physik ETH Hoenggerberg CH8093 Zuerich)
and University of British Columbia
Effective dynamics of magnetic vortices

Various timedependent GinzburgLandau equations model dynamics in
superconductors (and arise as a model in particle physics). The
dominant feature of typical solutions is the presence of localized,
topological structures called vortices. We present results for several
equations, showing that for long times (and appropriate initial data),
solutions remain close to a configuration representing widelyspaced
vortices. We identify the approximate dynamic law satisfied by the
vortex centres. This is joint work with I. M. Sigal.
 FENGBO HANG, Princeton University and Institute of Advanced Study
Density problem for Sobolev mappings

We shall discuss the important role of obstruction theory for
various questions on Sobolev mappings, with special emphasis
on density problems.
 MARK HASKINS, Johns Hopkins University, Baltimore Maryland, USA
Singularities of special Lagrangian 3folds

We will discuss some of the recent work done in trying to understand
singularities of special Lagrangian 3folds. Understanding these
singularities is a key ingredient in the ongoing effort to determine
the precise relationship between special Lagrangian torus fibrations
and mirror symmetry for CalabiYau 3folds. This talk will
concentrate on describing what is known about special Lagrangian cones
in complex 3space. In the case of cones on tori there is a close
connection with integrable systems. We discuss the geometry of the
simplest cones which arise in this manner. In this case the cones
arise from some special solutions of a classical finite dimensional
completely integrable Hamiltonian systemthe Neumann equation which
describes motion in a quadratic potential constrained to a sphere.
 ROBERT JERRARD, University of Illinois at UrbanaChampaign, Illinois, USA
Vortex density models

I discuss some recent results concerning limiting behavior sequences of
minimizers of the GinzburgLandau functional as the GinzburgLandau
constant and the applied magnetic field tend to infinity in an
appropriate fashion. (joint work with H. M. Soner)
 DAN KNOPF, University of WisconsinMadison, Madison, Wisconsin, USA
Rotationally symmetric shrinking and expanding gradient
KählerRicci solitons

(joint work with Mikhail Feldman (UWMadison) and Tom Ilmanen
(ETHZürich)).
We establish the existence of two new families of KählerRicci
solitons on complex line bundles over CP^{n1}. Our results
complement examples found by Cao. Moreover, by combining one of our
examples with one of his, we construct a smooth selfshrinking solution
of the Ricci flow for t < 0 that has a unique isolated singularity at
t=0 and that becomes a smooth selfexpanding solution for t > 0. This
solution has the effect of blowing down a CP^{n1} as
t\nearrow 0 and replacing it by a point for t > 0.
 PETER PANG, National University of Singapore, Singapore
Nonlinear Schroedinger flows and equations

In this talk, we will discuss global and local wellposedness of
Schroedinger flows on Kaehler manifolds with 1 and higher
dimensional, compact and noncompact, spatial domains. In particular,
we will prove that the 1dimensional periodic Schroedinger flow is
globally wellposed via a geometricanalytic approach which makes use
of an approximation by an equation of LandauLifshitz type and a
conservation law.
We will also mention the relationship between the Schroedinger flow on
hermitian symmetric spaces and generalized nonlinear Schroedinger
equations with values in Lie algebras.
 MARY PUGH, Department of Mathematics, University of Toronto, Toronto,
Ontario M5S 3G3
Longtime results for thin film equations

I will discuss longtime results for the thin film equation
h_{t} =  (h^{n} h_{xxx})_{x}. 

For the problem on the interval [0,L] with periodic or Neumann
boundary conditions, an initial profile with mean value [`(h)]
relaxes to the constant solution [`(h)] as t ® ¥
Baretta, Bertsch, and dal Passo 1994, Bertozzi and Pugh 1996. For the
problem on the line (¥,¥) the selfsimilar solution is
believed to be the relevant exact solution. Carrillo and Toscani have
recently proven that for the n=1 case
h_{t} =  (h h_{xxx})_{x} 

strong solutions of the initial value problem converge to a
selfsimilar solution. Here, I will present work on the n ¹ 1
case.
 JESSE RATZKIN, University of Utah, Salt Lake City, Utah 84112, USA
An end to end gluing contstruction for constant mean curvature
surfaces

Embedded noncompact constant mean curvature surfaces with finite
topology have a definite asymptotic structure. This allows one to
truncate two such surfaces by cutting off a choosen end of each and
patching them together along these ends, provided the asymptotics
match. In doing so one obtains a surface whose mean curvature is
globally close to one, which is called an approximate solution
(to the equation H=1). I will discuss the problem of deforming this
approximate solution to a constant mean curvature surface, which
involves solving a quasilinear elliptic PDE.
This is part of a large joint project with Rafe Mazzeo, Frank Pacard
and Daniel Pollack.
 RINA ROTMAN, Courant Institute of Mathematical Sciences, New York
University, New York, New York, USA
The length of a shortest closed geodesic

The subject of talk will be explicit upper bounds for the length of a
shortest closed geodesic l(M) on a compact Riemannian manifold M.
Also, in the special case when M is diffeomorphic to the
twodimensional sphere we prove much stronger upper bounds l(M) £ 8Ö{Area(M)} and l(M) £ 4diam(M) that improve previous
results by C. B. Croke. This proof involves some geometric measure
theory.
 VLADEN TIMORIN, Department of Mathematics, University of Toronto, Toronto,
Ontario M5S 3G3
Four dimensional geometry of circles

We classify all Kahler metrics in an open subset of C^{n}
whose geodesics are circles.
 AGNES TOURIN, Toronto
Optimal glider flying

A joint work with Robert Almgren will be presented. We propose a
stochastic optimal control model applying to glider flying. The goal
is to find the optimal strategies that permit to reach a target in
minimum time under uncertain lift. We will derive the nonlinear
Partial Differential Equation associated with this problem, show that
it is wellposed and how it can be solved numerically.
 SUMIO YAMADA, University of Alabama at Birmingham
Harmonic mappings into Teichmueller spaces

Having been developed by complex analysts, Teichmüller spaces have
not been extensively used in geometric analysis. In this talk I would
like to show that a Teichmüller space equipped with the socalled
WeilPetersson distance function has various features which can be used
to study behaviors of energy minimizing maps into the space. Mappings
into a Teichmüller space appears naturally whenever one looks at
manifolds which fibers over some base with its fibres being Riemann
surfaces of a fixed topological type.
 CARMEN YOUNG, The Fields Institute, Toronto, Ontario
Another look at Gromov compactness in dimension 4

We will discuss a new proof of Gromov compactness for
pseudoholomorphic curves in symplectic 4manifolds. The proof is
based upon a regularity theorem of Taubes for positive cohomology
assignments, the latter being a kind of generalized intersection number
for closed sets. One advantage of this new proof is that it adapts
well to proving Gromov compactness in noncompact symplectic
manifolds, such as symplectic cobordisms, where finite ``energy''
pseudoholomorphic curves can have infinite length and can continue
indefinitely along the ends of the ambient manifold.

