报告人信息：Emmanuel Trélat is full professor at Sorbonne Université and is the director of Laboratoire Jacques-Louis Lions. He is the Editor in Chief of the journal ESAIM: Control Calc. Var. Optim. and is Associate Editor of several other journals, including SIAM Review. He has been awarded a number of prizes, among which the Felix Klein Prize in 2012, and an invited speaker invitation to ICM in 2018. He is also a member of Academia Europaea. His research interests range over control theory in finite and infinite dimension, optimal control, stabilization, geometry, numerical analysis. He also collaborates with industrials, for example he is the inventor of the software that is used for the control guidance of the Ariane launchers.
Abstract: In a series of works on sub-Riemannian geometry with Yves Colin de Verdière and Luc Hillairet, we study spectral properties of sub-Riemannian Laplacians, which are hypoelliptic operators. The main objective is to obtain quantum ergodicity results, what we have achieved in the 3D contact case. In the general case we study the small-time asymptotics of sub-Riemannian heat kernels. We prove that they are given by the nilpotentized heat kernel. In the equiregular case, we infer the local and microlocal Weyl law, putting in light the Weyl measure in sR geometry. This measure coincides with the Popp measure in low dimension but differs from it in general. We prove that spectral concentration occurs on the sheaf generated by Lie brackets of length r-1, where r is the degree of nonholonomy. In the singular case, like Martinet or Grushin, the situation is more involved but we obtain small-time asymptotic expansions of the heat kernel and the Weyl law in some cases. Finally, we give the Weyl law in the general singular case, under the assumption that the singular set is stratifiable.
