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. I am currently working on describing ramified level structures on oriented elliptic curves, with the long-term goal of constructing an arithmetically global version of modular-equivariant topological modular forms and understanding the appearance of Langlands-type phenomena in chromatic homotopy theory.
What I'm currently thinking about
- Describing ramification in terms of dualizing lines and linearizations
- Stratifying moduli stacks of isogenies by ramification
- Spectral Lie theory of oriented abelian varieties
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.
- 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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