As it is well known, the heat kernel is a robust tool in the study of Atiyah-Singer index theorem and harmonic analysis. A natural question arising in this sort of application is how the heat kernel behaves on various geometric objects. This is generally well understood through Riemannian geometry on complete non-compact manifolds in a series of pioneering work due to Cheng-Li-Yau, Cheeger-Yau and Li-Yau. Following this manner, the speaker and his collaborator manage to understand the exact heat kernel bound on Cartan-Hadamard asymptotically hyperbolic manifolds. Such manifolds are of great interest and importance in a variety of mathematical subjects (conformal geometry, scattering theory, and spectral theory) as well as theoretical physics (AdS-CFT correspondence and general relativity). The speaker and his collaborator’s approach is microlocal and based on the resolvent on AH manifolds constructed in the celebrated work of Mazzeo-Melrose as well as its high energy asymptotic due to Melrose-Sa Barreto-Vasy. This is a joint work with A. Hassell.
About the speaker
Dr Xi Chen obtained his PhD at the Australian National University in 2015. He is currently a Postdoctoral Fellow of Shanghai Centre for Mathematical Sciences at Fudan University. He works on geometric microlocal analysis and its interactions with linear dispersive and wave equations, geometric analysis, and harmonic analysis.
