The partial differential equations and analysis seminar is the research seminar associated with the applied and nonlinear analysis, and the analysis and geometry programs.

The partial differential equations and analysis seminar is the research seminar associated with the applied and nonlinear analysis, and the analysis and geometry programs.

Consider families of circular means where the radii are restricted to a given subset of a compact interval. One is interested in the $L^p$ improving estimates for the associated maximal operators and related objects.

This seminar will present an algorithm for decomposing immersions satisfying a weaker inequality into a finite number of handles using a surgically modified (high codimension) mean curvature flow.

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.