Abstracts for Rivière-Fabes Symposium on Analysis and PDE
April 23-25, 2010
All talks in Vincent Hall 16
Lecture 1 - Conservation laws for conformally invariant problems and the Noether theorem in the absence of symmetry
Abstract 1: We will review applications of Noether's Theorem for conformally invariant Lagrangians of maps from a Riemann surface into a symmetric manifold. We will explain how the existence of conserved quantities, issued from the symmetry of the target, - the so called Noether Currents - play a decisive role in the analysis of critical points to 2-dimensional conformally invariant Lagrangians. Once the symmetry assumption is dropped, Noether Theorems a-priori does not apply anymore. However we will exhibit the survival of Noether Currents for general target, beyond the symmetry assumption. These generalized Noether Currents will play again a central role in the analysis of 2-dimensional conformally invariant Lagrangians of maps into general manifold and will permit us in particular to prove the Heinz-Hildebrandt regularity conjecture.
Lecture 2 - A PDE version of the constant variation method and the sub-criticality of Schroedinger Systems with antisymmetric potentials
Abstract 2: We will explain how the approach we developed in the first talk in order to find conservation laws for critical points to conformally invariant problems can be systematized and applied to a large family of equations : Linear Schroedinger Systems with antisymmetric potentials of various orders. This leads to a series of new compactness and regularity results for PDEs which are a-priori critical but happen in fact to have a subcritical behaviors. We will present some applications of these results to problems from geometric non-linear analysis.
2 Lectures - Large data wave maps
Abstract 1: I will describe recent work, joint with Jacob Sterbenz, on the wave map equation in 2+1 dimensions. This is an energy critical problem, i.e. the energy of the wave map is invariant with respect to the natural scaling of the problem. For initial data with large energy we establish a dichotomy between global existence and scattering, on one hand, and soliton-like concentration on the other hand.
Lecture 1 - Homogenization of interface velocities in periodic and random media
Abstract 1: We will discuss several free boundary problems where the free boundary moves in heterogeneous environment with oscillatory boundary velocities. The stability of the interface in the homogeniation limit, as well as the properties of the limiting interface will be discussed. The key tool is maximum principle-type arguments as well as strong averaging properties of the media. We will also discuss similarities and differences with viscosity solutions method used to homogenization of nonilnear PDES.
Lecture 1 - Wiener's "closer of translates" problem and Piatetskii-Shapiro uniqueness phenomenon.
Abstract 1: Wiener characterized cyclic vectors (with respect to translations) in lp(Z)and Lp(R) (p=1,2) in terms of zero sets of Fourier transform. He conjectured that a similar characterization should be true for 1 < p < 2. I will discuss this conjecture.
Joint work with Nir Lev.
Lecture 1 - Ancient solutions to the Ricci flow and Ricci solitons
Abstract 1: We will give a classification of ancient solutions to the Ricci flow on compact surfaces. We show the contracting spheres and Angenant ovals are the only possibilities. We will also discuss some classsification results for Ricci solitons and some geometric properties of those.
Lecture 1 - Traveling Fronts in Combustible Media
Abstract 1: Traveling fronts are special solutions of reaction-diffusion equations which model phenomena such as propagation of species in an environment or spreading of flames in combustible media. In this talk we will address questions of existence, uniqueness, and stability of traveling fronts in general inhomogeneous media. We will show that in certain circumstances they are global attractors of the corresponding parabolic evolution, thus describing long time dynamics for very general solutions of the PDE. We will also present examples of media in which no traveling fronts exist.