# Research

# Algebraic Topology Research

I work in the field of algebraic topology, where we study topological spaces by tagging these spaces with some kind of algebraic object that encodes important information. Algebraic topologists have also used tools from topology to study objects in algebra; this is the story of algebraic K-theory for rings.

Algebraic K-theory lies at the intersection of algebraic topology, algebraic geometry, and number theory, making it an invariant of great interest to many mathematicians. However, algebraic K-groups are notoriously difficult to compute. In the trace methods approach to K-theory, we study more computationally approachable invariants as approximation.

As an equivariant homotopy theorist, my work is grounded in a generalization of this story where we concern ourselves with a version of K-theory for objects that have a group action on them, specifically the C2-action of involution. I study an approximation of this equivariant K-theory called Real topological Hochschild homology (THR). My Ph.D. thesis develops some of the first computational tools for Real topological Hochschild homology (THR) and identifies the algebraic structures structures present in THR.

As part of a Women in Topology IV team with Agnès Beaudry, Clover May, Sabrina Pauli, and Liz Tatum, I'm also working on a project about the equivariant Thom isomorphism and Steenrod operations for equivariant cohomology.

In collaboration with Sarah Klanderman, Emily Rudman, and Danika Van Niel, I'm working on a project to develop new Hochschild invariant for polynomial rings.

Below is a video version of my research statement (aimed at an audience of algebraic topologists) kindly hosted by the Junior Mathematician Research Archive. This video was created for an employment networking event sponsored by the electronic Computational Homotopy Theory (eCHT) online research community.

## Talks and Slides

Building a bridge from algebra to topology and back - various locations (Spring 2023)*

K-theory of CWG Catgeories - MIT Talbot Workshop (June 2022)

Computational tools for Real topological Hochschild homology - Graduate Student Geometry and Topology conference, Georgia Tech University (April 2022)

An introduction to Hochschild (co)homology - graduate course in group cohomology (December 2021)

Taking topology to the Supreme Court: an introduction to the mathematics of gerrymandering - Calvin University colloquium (November 2021)*

Computational tools for Real topological Hochschild homology - Binghamton University Graduate Combinatorics, Algebra, and Topology Conference (March 2021)

Quillen's higher algebraic K-theory - graduate course on communicating classical results in topology (March 2021)

* denotes talk for an undergraduate audience