Research
“In mathematics, the art of asking questions is more valuable than solving problems.”
My research interests are in algebraic topology, in particular computations in equivariant stable homotopy theory.
The study of homotopy theory originates in algebraic topology, with the investigation of algebraic invariants such as (co)homology groups or homotopy groups. By passing from topological spaces to spectra, one can further study stable invariants, and in the presence of a group action, one can study equivariant invariants as well.
While homotopy theory has its origins in algebraic topology, its core tools and ideas have spread and found use in other areas of mathematics. In my research, I apply the computational methods of homotopy theory to answer questions about the modular representation theory of finite groups.
My Research Statement can be found here (last updated October 2023).
What is algebraic topology?
Broadly speaking, topology is a generalization of the study of shapes. These shapes can be abstract and hard to visualize, so we study them though algebra. In particular, in homotopy theory, we use algebraic tools called invariants to describe whether two shapes are the same or different.
Homotopy theory has its origins in algebraic topology, but its core tools and ideas have spread and found use in other areas of mathematics. In my research, I apply the computational methods of homotopy theory to answer questions about the modular representation theory of finite groups.
Click here for a undergraduate level introduction to my research area, written for students that have taken multivariable calculus.
Papers/Preprints
- Endotrivial modules for cyclic $p$-groups and generalized quaternion groups via Galois descent, joint with Jeroen van de Meer.
I gave a talk on this paper at the Workshop on Homotopy Theory and Group Theory at CRM Barcelona. The video recording is available.
Future Directions
In my research, I apply the computational methods of homotopy theory to answer questions about the modular representation theory of finite groups G over a field of characteristic p, where p divides the order of the group. One particvular problem of interest is in computing the group of endo-trivial modules.
Modular representation theory was revolutionized by a focus on an invariant denoted StMod(kG), the stable module category of G. Notably, StMod(kG) has a homotopy-theoretic interpretation as a stable symmetric-monoidal ∞-category. Correspondingly, the group of endo-trivial modules is intepreted as the Picard group of the stable module category, which consists of objects in StMod(kG) that are invertible with respect to the tensor product.
For p-groups, this group of endo-trivial modules was originally computed by work of Carlson-Thévenaz using group cohomology and the theory of support varieties. However, in joint work with van de Meer, I provide new, homotopical proofs for their results for cyclic p-groups and (generalized) quaternion groups using the descent ideas of Mathew-Stojanoska. This provides new insights into the classical representation-theoretic proofs of Carlson-Thévenaz.
In future work, I propose to construct and compute homotopy invariants that have representation-theoretic significance. In particular, I am interested in (1) using homotopy theory to compute endo-trivial modules for other classes of groups, (2) studying other invariants of StMod(kG), and (3) computing endo-trivial modules for variants of StMod(kG) coming from tensor-triangular geometry.