Research

My research broadly falls within the fields of algebraic topology, algebraic geometry, and homotopy theory. In particular, I am interested in understanding chromatic homotopy theory using tools from spectral arithmetic geometry, such as elliptic cohomology. My current research concerns isogenies of oriented elliptic curves and the construction and properties of Hecke operators on elliptic cohomology. Ultimately, I hope to develop a theory of topological modular Galois representations, which will play a role in the Chromatic Langlands Program.

What I'm currently thinking about

• Dualizing sheaves for the moduli of orientation data of a formal group
• The underlying classical stack of the moduli stack of isogenies between oriented elliptic curves

Papers

• Complex Orientations are Partial Strictifications of the Unit (arXiv).
• Finite Manifolds and Minimal Finite Models of Closed Surfaces (pdf). Written for the University of Chicago Math REU, Fall 2018.

Selected Talk Notes

• Higher Local Class Field Theory (notes). Given as part of an advanced topics class on local class field theory taught by Yuan Liu in Fall 2025.
• Isogenies of Oriented Elliptic Curves (slideshow). Given at the AMS Sectional Meeting at the University of Denver on August 23rd, 2025.
• Computing Pullbacks of ∞-Topoi; Or, How I Learned to Stop Worrying and Love Gabriel-Ulmer-Duality (handwritten notes). Given at the Graduate Student Homotopy Theory Seminar on March 3rd, 2025.
• A Deligne-Mumford Stratification of Elliptic Cohomology with Level Structure (notes). Given at the Graduate Student Homotopy Theory Seminar on September 30th, 2024.
• Multiplicative Structures in Equivariant Homotopy Theory (notes). Given at the Equivariant Homotopy Theory Learning Seminar on March 26th, 2024.
• Higher Galois Descent for Chromatic Localizations of K-theory (notes). Given at Jeremiah Heller's Telescope Conjecture Learning Seminar on February 21st, 2024.
• Orientations of Formal Groups (notes). Given at the Graduate Student Homotopy Theory Seminar on February 16th, 2024.
• (Not An) Introduction to Topological Modular Forms (notes). Given at Dan Berwick-Evans's Twisted Equivariant K-Theory and Topological Modular Forms learning seminar on September 15th, 2023.
• The Chromatic Fracture Square: Disassembly and Reassembly (notes). Given at the Chromatic Homotopy Theory Reading Seminar on April 26th, 2023.
• Towards a Chromatic Langlands Program (notes). Given at the Graduate Student Homotopy Theory Seminar on March 10th, 2023.
• Elliptic Cohomology and Conformal Field Theories (notes). Originally given on May 19th, 2022 as part of an advanced topics class on TQFT taught by Gabriele La Nave. Given in a revised form as a two-part talk at the Graduate Student Geometry and Topology Seminar on September 16th and 23rd, 2022.
• An Informal Introduction to Formal Groups (notes). Given at the Graduate Student Commutative Algebra and Algebraic Geometry Seminar on April 12th, 2022.
• Simplicial Localizations and How to Find Them (notes). Given at the Graduate Student Homotopy Theory Seminar on October 8th, 2021.
• Finite Spaces and Finite Models (notes). Given at the Graduate Student Homotopy Theory Seminar on September 14th, 2020.

Other Mathematical Writings

• A Spectral Nakayama's Lemma (pdf). A lemma I proved for my thesis, but didn't end up needing. I'm planning to write up a proper paper on its applications when I find the time.