Early Career Talks

Carolyn R. Abbott (Brandeis University)

Title: Boundaries, Boundaries, and more Boundaries

Abstract: It is possible to learn a lot about a group by studying how it acts on various metric spaces. One particularly interesting (and ubiquitous) class of groups are those that act nicely on negatively curved spaces, called hyperbolic groups. Since their introduction by Gromov in the 1980s, hyperbolic groups and their generalizations have played a central role in geometric group theory. One fruitful tool for studying such groups is their boundary at infinity. In this talk, I will discuss two generalizations of hyperbolic groups, relatively hyperbolic groups and hierarchically hyperbolic groups, and describe boundaries of each. I will describe various relationships between these boundaries and explain how the hierarchically hyperbolic boundary characterizes relative hyperbolicity among hierarchically hyperbolic groups.  This is joint work with Jason Behrstock and Jacob Russell.

Anthony Conway (The University of Texas at Austin)

Title: Locally Flat Surfaces in 4-manifolds

Abstract: The talk will survey classifications of locally flat surfaces in 4-manifolds and, time permitting, briefly describe the main tools that enter the proofs.

Colleen Delaney (University of California, Berkeley)

Title: An Efficient* Classical Algorithm for some Quantum Invariants of 3-manifolds

Abstract: I will explain how the Turaev-Viro-Barrett-Westbury state sum TQFT invariants of 3-manifolds that arise from Tambara-Yamagami fusion categories can actually be computed in polynomial time on a classical computer, provided that there is a bound on the first Betti number. On the other hand, if we don’t insist on a bound on the first Betti number, then the invariants are NP-hard to compute. Time permitting I will interpret these results in a physical context. This talk is based on joint work with Clément Maria and Eric Samperton.

Antoine Song (California Institute of Technology)

Title: Minimal Surfaces, Hyperbolic Surfaces and Randomness

Abstract: Minimal surfaces and hyperbolic surfaces are both "optimal 2d geometries" which are ubiquitous in differential geometry. The first kind is defined by an extrinsic condition (the mean curvature vanishes), while the second kind is defined by an intrinsic condition (the Gaussian curvature is equal to -1). I will discuss a surprising connection between the two geometries coming from randomness. The main statement is that there exists a sequence of closed minimal surfaces in Euclidean spheres, constructed from random permutations, which converges to the hyperbolic plane. This result came from my attempt to bridge minimal surfaces and unitary representations. I will introduce this circle of ideas and mention some general questions.

Iris Yoon (Wesleyan University)

Title: How Topology Reveals Structure in Neuroscience Data

Abstract: We live in an exciting time where new data is generated at an exponential rate. Such data explosion necessitates the development of novel methods for studying large, noisy, and complex data. One interesting aspect of data is its shape and structure. In this talk, I will discuss recent developments in applied topology that studies the structure of data. In particular, I will show how constructions in topology reveal interesting structures and insights in neuroscience.

Allen Yuan (Institute for Advanced Study/Northwestern University)

Title: Higher Algebra and p-adic Cohomology Theories

Abstract: Higher algebra is a topological enlargement of ordinary algebra where generalized cohomology theories play the role of abelian groups.  I will begin by explaining the origins of higher algebra in classical algebraic topology.  Then, I’ll discuss some recent applications of this circle of ideas to the study of p-adic cohomology theories in arithmetic geometry such as the prismatic cohomology of Bhatt—Scholze.  In the end, I will touch on joint work with Devalapurkar, Hahn, and Raksit about the prismatization of commutative ring spectra.