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—to be precise, the moduli stack of these isogenies and the closely related moduli stacks of elliptic curves with level structure. I intend to use these stacks to develop an arithmetically global theory of modular-equivariant elliptic cohomology, and with it a theory of topological modular Galois representations.
What I'm currently thinking about
- Obstruction computations in higher algebra
- Spectral Hilbert stacks
- Deformations of almost finitely-presented algebras along square-zero extensions
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
When writing notes for a talk, I sometimes end up with a text that can stand on its own as an exposition of the subject. In such cases, I upload those notes here in the hopes that someone will find them useful.
- 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.
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Department of Mathematics
273 Altgeld Hall, MC-382
1409 W. Green Street, Urbana, IL 61801 USA
Telephone: (217) 333-3350 Fax: (217) 333-9576 |
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