Title: Recent Progress on Fractional GJMS OperatorsAbstract: The Fractional GJMS operators are one-parameter family of conformally invariant operators, defined by the renormalized scattering operators on the conformal infinity of Poincare-Einstein Manifolds. These operators provide a bridge to transfer the information of interior Einstein geometry to the boundary conformal geometry. In this talk, I will first […]

Title: L^p bounds for spectral projectors on hyperbolic surfacesAbstract: In this talk I will present L^p boundedness results for spectral projectors on hyperbolic surfaces, focusing on the case where the spectral window has small width. I will show that the negative curvature assumption leads to improvements over the universal bounds of C. Sogge, thus illustrating […]

Title: Oscillatory integrals on manifolds and related Kakeya and Nikodym problems. Abstract: This talk is about oscillatory integrals on manifolds and their connections to  Kakeya and Nikodym problems on manifolds. There are two types of manifolds that are particularly interesting: manifolds of constant sectional curvature and manifolds satisfying Sogge's chaotic curvature conditions. I will discuss these two […]

Title: Maximal Subellipticity Abstract: The theory of elliptic PDE stands apart from many other areas of PDE because sharp results are known for very general linear and fully nonlinear elliptic PDE.  Many of the classical techniques from harmonic analysis were first developed to prove these sharp results; and the study of elliptic PDE leans heavily on […]

Title: On the isoperimetric profile of the hypercube  Abstract: The isoperimetric problem in the hypercube is a strikingly simple topic which is not yet completely understood. We will review the known facts on the problem, focusing on how the gaussian isoperimetric inequality provides a lower bound for the isoperimetric profile of the hypercube.We will exploit this […]