Research
I am interested in algebraic topology and algebraic geometry. In particular much of my research has focused on the interplay between enumerative geometry and homotopy theory, particularly motivic and equivariant homotopy theory.
The $S_4$-orbits of the 27 lines on the Clebsch surface.
Papers and Preprints:
- $\mathbb{A}^1$-Brouwer degrees in Macaulay2, with N. Borisov, F. Espino, T. Hagedorn, Z. Han, J. Lopez Garcia, J. Louwsma, G. Ong, and A. Tawfeek.
13 pages, 2023. Submitted. - Equivariant enumerative geometry,
38 pages, 2023. Submitted. - An enriched degree of the Wronski map,
24 pages, 2023. Submitted. - Residue sums of Dickson polynomials over finite fields, with J. Harrington, M. Litman, T.H.W. Wong,
18 pages, 2021. Accepted pending revisions. - Lifts, transfers, and degrees of univariate maps, with S. McKean,
Mathematica Scandinavica 129(1), 2023. - Bézoutians and the $\mathbb{A}^1$-degree, with S. McKean, S. Pauli,
Algebra & Number Theory 17(11), 2023. - Homotopy Mackey functors of equivariant algebraic $K$-theory,
Journal of Pure and Applied Algebra 226(8), 2022. - An introduction to $\mathbb{A}^1$-enumerative geometry,
In Lecture Notes in Mathematics, vol 2292. Springer, 2021. - A note on twisted group rings and semilinearization,
Communications in Algebra, 49:8, 3380-3386, 2021. - The trace of the local $\mathbb{A}^1$-degree, with R. Burklund, S. McKean, M. Montoro, M. Opie,
Homology, Homotopy and Applications 23(1):243-255, 2021. - Zeros of newform Eisenstein series on $\Gamma_0(N)$, with V. Jakicic,
J. Number Theory (190):109-130, 2018. - On consecutive $n$th roots of unity modulo $q$, with J. Harrington, S. Kannan, M. Litman,
J. Number Theory (174):494-504, 2017.
Software:
- A1BrouwerDegrees.m2, a Macaulay2 package for computing local and global $\mathbb{A}^1$-Brouwer degrees, and manipulating the associated symmetric bilinear forms. With N. Borisov, F. Espino, T. Hagedorn, Z. Han, J. Lopez Garcia, J. Louwsma, G. Ong, and A. Tawfeek.
Other writing:
- Constructing the Unstable Motivic Homotopy Category using $(\infty,1)$-Categories, arXiv:1810.00094.
- The Generalized Poincaré Conjecture Using $s$-Cobordism, for the Penn Grad Student Seminar, Fall 2018.
Notes:
Attached are expository sets of notes from various talks I've given. They are likely riddled with errors - please email me if you find any.- Diagonalizing symmetric bilinear forms.
- Algebraic vector bundles, Penn grad geometry/topology seminar, 2022.
- Ambidexterity, Talbot 2021
- Euler characteristics of real algebraic manifolds, Penn grad geometry/topology seminar, 2021.
- Hopf algebroids, Penn chromatic homotopy seminar, 2021.
- Complex orientations, Penn chromatic homotopy seminar, 2021.
- Dwyer-Kan localization, Penn infinity-categories seminar, 2020.
- Colimits in quasicategories, Penn infinity-categories seminar, 2020.
- $K$-theory of infinity categories, Penn algebraic $K$-theory seminar, 2020
- Homotopy coherent nerve, Penn topology proseminar, 2020.
- Bundles, for quals, 2019.
Conference and course notes:
Disclaimer: Any errors found in these notes should be attributed to me, not the original lecturer. If you find any typos or have suggestions, please feel free to contact me.- School on Motives and Stacks, September 2019, minicourses by Hoskins, Østvæer, Rydh. Notes available upon request.
- European Autumn School in Topology, September 2019, minicourses by Barthel, Salvatore. Notes available upon request.
- PIMS Workshop on Arithmetic Topology, June 2019, minicourses by Ellenberg, Farb, Galatius, Ho, Vakil/Landesman, Wickelgren.
- An Introduction to $\textbf{A}^1$-homotopy theory from an infinity-categorical viewpoint, June 2018, (notes from lectures of Strunk and and Tamme, Homotopy Theory Summer Berlin 2018).