In this talk, I will explore the relationship between a domain's shape and its first Laplace eigenvalue, with emphasis on the domains which minimize, or are more generally critical points, for this eigenvalue, for their given volume.

In recent years much attention has turned to rigidity, and almost-rigidity, problems involving lower scalar curvature bounds. In this talk, I'll discuss some contributions to this area, including some new stability theorems for spheres. Some of this is joint work with Davi Maximo, and some is joint with Paul Sweeney Jr.

Symplectic manifolds show up in many areas of mathematics, including topology, geometry, representation theory, and dynamics. In the talk I will give an introduction to symplectic manifolds and then discuss embeddings of 3-manifolds into symplectic 4-manifolds. This is joint work with Ali Daemi and Mike Miller Eismeier.

One of the main approaches to topological data analysis is to study discrete invariants of (multiparameter) persistence modules. In this talk, we discuss recent research which utilizes techniques from (relative) homological algebra to interpret classical examples of invariants and define new invariants. The Hilbert function and (generalizations of) the rank invariant serve as our main…

On a Riemannian manifold with boundary, the X-ray transform integrates a function or a tensor field along all geodesics through the manifold. The reconstruction of the integrand of interest from its X-ray transform is the basis of important inverse problems in seismology and medical imaging. For this transform, statements of injectivity and microlocal stability in the interior of the domain were established some…