國立清華大學 數學系

Title: Subelliptic PDE and sub-Riemannian Geometry Time: 10:00am - 12:00pm Date: 5/28(四)、6/4 (四) 、6/11 (四) 、6/25 (四) 、 7/2 (四)、7/9 (四) Place: Lecture Room B, 綜合三館4F (* venue of 6/25 class is T.B.A.) The main purpose of this series lectures is to discuss applications of Hamiltonian mechanics to study sub-Riemannian structures. As we know, Riemannian manifolds and Reimann surfaces are the geometric medium for the problems in classical mechanics. The corresponding medium for quantum mechanics is sub-Riemannian manifolds. The Hamiltonian formalism came into play in the study of sub-Reimannian manifolds and, in particular, has been successfully applied for Heisenberg groups and unit spheres in high dimensional complex Euclidean spaces in the past two decades. The geometry of sub-Riemannian manifolds is defined by the properties of the underlying subspace of the tangent space. We shall use the Hamilton – Jacobi theory of bicharacteristics to study the geometry induced by the sub-Laplacian, which is the sum of squares of underlying basis vector fields of sub-Riemannian manifold. We aim to find the heat kernel for the heat equation with potentials. We shall write the Euler – Lagrange system of equations for the Lagrangian and try to characterize the system qualitatively from the conservation laws point of view. In general, these systems cannot be solved explicitly. However, for simple equations, one may characterize the solutions by finding the first integral of motions. We shall also discuss small time asymptotic of the heat kernels.