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Nonlinear and Geometric Analysis / Analyse non linéaire et géométrique
(Robert McCann, Organizer)

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ANTHONY BLOCH, University of Michigan, Department of Mathematics, Ann Arbor, Michigan  48019-1109, 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 infinite-dimensional 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 geometry-based (ansatz dependent). I will illustrate the application of these methods to microphases in diblock copolymer melts.

JIM COLLIANDER, University of Toronto, Toronto, Ontario
Ill-posedness for defocussing NLS

Standard tricks for showing ill-posedness for nonlinear dispersive wave equations exploit special travelling wave or blow-up solutions. Such special solutions are unavailable in the defocussing setting, leaving open the possibility that well-posedness 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 ill-posedness 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 three-manifold. For the vacuum time-symmetric 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, California-Irvine, California, USA
Gauss curvature flow with flat sides: all time regularity of the interface

We study the free-boundary 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 quarter-pinched 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 non-simply 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 Cⁿ. 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 Monge-Ampère equation.

STEVE GUSTAFSON, ETH Zuerich (Theoretische Physik ETH Hoenggerberg CH-8093 Zuerich) and University of British Columbia
Effective dynamics of magnetic vortices

Various time-dependent Ginzburg-Landau 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 widely-spaced 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 3-folds

We will discuss some of the recent work done in trying to understand singularities of special Lagrangian 3-folds. 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 Calabi-Yau 3-folds. This talk will concentrate on describing what is known about special Lagrangian cones in complex 3-space. 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 system-the Neumann equation which describes motion in a quadratic potential constrained to a sphere.

ROBERT JERRARD, University of Illinois at Urbana-Champaign, Illinois, USA
Vortex density models

I discuss some recent results concerning limiting behavior sequences of minimizers of the Ginzburg-Landau functional as the Ginzburg-Landau constant and the applied magnetic field tend to infinity in an appropriate fashion. (joint work with H. M. Soner)

DAN KNOPF, University of Wisconsin-Madison, Madison, Wisconsin, USA
Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons

(joint work with Mikhail Feldman (UW-Madison) and Tom Ilmanen (ETH-Zürich)).

We establish the existence of two new families of Kähler-Ricci solitons on complex line bundles over CP^n-1. Our results complement examples found by Cao. Moreover, by combining one of our examples with one of his, we construct a smooth self-shrinking solution of the Ricci flow for t < 0 that has a unique isolated singularity at t=0 and that becomes a smooth self-expanding solution for t > 0. This solution has the effect of blowing down a CP^n-1 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 well-posedness of Schroedinger flows on Kaehler manifolds with 1 and higher dimensional, compact and non-compact, spatial domains. In particular, we will prove that the 1-dimensional periodic Schroedinger flow is globally well-posed via a geometric-analytic approach which makes use of an approximation by an equation of Landau-Lifshitz 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
Long-time results for thin film equations

I will discuss long-time results for the thin film equation

ht = - (hn hxxx)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 self-similar solution is believed to be the relevant exact solution. Carrillo and Toscani have recently proven that for the n=1 case

ht = - (h hxxx)x

strong solutions of the initial value problem converge to a self-similar 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 two-dimensional 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 Cⁿ 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 well-posed 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 so-called Weil-Petersson 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 pseudo-holomorphic curves in symplectic 4-manifolds. 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 non-compact symplectic manifolds, such as symplectic cobordisms, where finite ``energy'' pseudo-holomorphic curves can have infinite length and can continue indefinitely along the ends of the ambient manifold.