I will report on recent work in the theory of unbounded functional calculi.

I will give an overview of how time-frequency analysis is used in proving these $L^p$-bounds, with focus on the recently understood setting of functions valued in UMD Banach spaces.

The fundamental gap conjecture for Dirichlet eigenvalue problem on Euclidean domains is a well-known conjecture. In this talk we shall present our progress on Escobar's conjecture, namely, we confirm it for a large class of Riemannian manifolds, including Euclidean domains.

We discuss the uniqueness of solutions of the parabolic evolution equation $\partial\varphi/\partial t+H\varphi=0$ where $H=-\mathrm{div}(C\nabla)$ is a second-order degenerate elliptic operator acting on a domain $\Omega$ in $\mathbf{R}^d$.

In this seminar, Professor Yu Yuan and Professor Young-Heon Kim will discuss special lagrangian equations and optimal transport for dendritic structures.

In this seminar, Professor Huai-Dong Cao will discuss some recent progress on Ricci solitons, especially in dimension four.