Abstract: The study of embedded surfaces in hyperbolic 3-manifolds has led to several major advances in the fields of geometry, topology, and geometric group theory. In this talk we address the higher dimensional analogue of codimension-1 embedded submanifolds with a focus on arithmetic manifolds of small volume.

Here is the link to the talk recording. 

Abstract: One of the biggest questions in low-dimensional topology is to understand when homeomorphic smooth four-manifolds are diffeomorphic.  Surprisingly, these are not the same!  For example, Taubes constructed uncountably many four-manifolds homeomorphic to $\mathbb{R}^4$, none of which are diffeomorphic.  On the other hand, the smooth 4D Poincare conjecture predicts that a four-manifold homeomorphic to the four-sphere is diffeomorphic to the four-sphere.  This talk will discuss how to construct new invariants of four-manifolds coming from Floer homology which can tell apart homeomorphic four-manifolds in a number of cases. This is joint work with Adam Levine and Lisa Piccirillo.

Here is the link to the talk recording.

Abstract: Imagine a surface $X$ obtained from taking the plane $\mathbb{R}^2$ and deleting every point with integer coordinates. Which surfaces arise as finite-sheeted covers of $X$? Which surfaces can $X$ cover by finitely-many sheets?

This talk will be a tour through the intriguing landscape of infinite-type surfaces, with a particular focus on finite-sheeted covers. We will answer the particular questions above, as well as many others like them, and see that they are not as innocent as they may first appear! Parts of the talk involve joint work with Alan McLeay.

Here is the link to the talk recording.