This talk is concerned with locally flat surfaces in 4-space, and instances where the topology of these surfaces is determined by algebraic data. We describe an unknotting theorem for surfaces in the 4-sphere, and how it follows from a more general classification of closed surfaces in 4-manifolds whose complement has fundamental group Z. We then discuss surfaces with boundary and classifications for other fundamental groups. Based on joint work with Lisa Piccirillo and Mark Powell.

For a hyperbolic n-manifold, bending along a totally geodesic hypersurface is a well-studied method of producing deformations of the original hyperbolic structure. However, there are instances where certain computations show that deformations of a hyperbolic structure exist, even in the absence of totally geodesic hypersurfaces to bend along. In a paper of Cooper, Long, and Thistlethwaite, a manifold is called "flexible" if it has this property. But what explains when a manifold is flexible? In this talk, we will explore one potential answer: a generalization of bending, where instead we bend along totally geodesic branched complexes. This is a candidacy talk.

In the recent framework proposed by Ben-Zvi--Sakellaridis--Venkatesh, automorphic periods ought to, very roughly speaking, come in Langlands dual pairs. I will give a short introduction of this prediction and motivate the need to consider certain singular automorphic periods. In particular, I will present an example in joint work with Akshay Venkatesh, where we establish duality using a generalization of L-functions.

I will discuss recent work calculating the top weight cohomology of the moduli space A_g of principally polarized abelian varieties of dimension g for small values of g. The key idea is that this piece of cohomology is encoded combinatorially via the relationship between the boundary complex of a compactification of A_g and the moduli space of tropical abelian varieties. This is joint work with Madeline Brandt, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.

Suppose we have a (known) group acting on the vertices of a (known) graph in an (unknown) way. If we are given "local" information about the group action by the edges of the graph, how can we use such information to recover the "global" group action...? In this talk I will introduce the learning group action problem described above and discuss the conditions under which we can solve for and/or approximate the group action. I will also discuss an application in computer vision that motivates the learning group action problem and, in particular, how learning group actions was used to develop the COVID-19 vaccine.