|
Le vendredi 28 avril 2006 / Friday, April 28, 2006 The talk focuses on two central problems in spectral geometry. The first one is to study the asymptotic distribution of Laplace eigenvalues on a Riemannian manifold. The second one is to understand geometric invariants of a manifold that are determined by the spectrum. I will discuss some recent advances on these problems, highlighting links between spectral geometry and other areas of mathematics, such as number theory, dynamical systems and combinatorics. 15:30 - 16:00 Le vendredi 28 avril 2006 / Friday, April 28, 2006 Schrödinger flow is a Hamiltonian flow for mappings from a Riemannian manifold into a Kahler manifold with the energy as the Hamiltonian. It is also known as «Schrödinger map». One does not know if finite energy solutions of the Schrödinger flow can develop a singularity in finite time. I will talk about a joint project with Stephen Gustafson and Kyungkeun Kang in which we search for blow-up solutions in the class of equivariant Schrödinger flow from $R^2$ to $S^2$ with degree $m \ge 1$ and energy close to harmonic map energy. We relate the (hypothetical) blow-up to the vanishing of the length scale of the nearest harmonic map. We also show that, when $m \ge 3$, the solution converges locally to a harmonic map at time infinity and does not blow up. The cases $m=1,2$ are open. Renseignements / Information : Un cocktail suivra les conferences au Salon Maurice l'Abbe (6245). |