## About Me

I am a fifth year PhD student in Mathematics and an NSF Graduate Research Fellow at the University of Pennsylvania. **I am on the job market this fall.** I am co-advised by Mona Merling and Kirsten Wickelgren. I am interested in enumerative geometry and algebraic topology, particularly $\textbf{A}^1$-homotopy theory and equivariant homotopy theory. My CV is here.

tbraz at math.upenn.edu

I co-founded the Penn chapter of the Directed Reading Program with Mona Merling, which I now co-organize with Marielle Ong. We were recently featured in an article in Omnia.

### Papers and Preprints:

**Equivariant enumerative geometry**,

36 pages, 2022. Preprint.**An enriched degree of the Wronski**,

23 pages, 2021. Preprint.**Residue sums of Dickson polynomials over finite fields**, with J. Harrington, M. Litman, T.H.W. Wong,

18 pages, 2021. Submitted.**Lifts, transfers, and degrees of univariate maps**, with S. McKean,

32 pages, 2021. To appear in*Mathematica Scandinavica*.**Bézoutians and the $\mathbb{A}^1$-degree**, with S. McKean, S. Pauli,

30 pages, 2021. To appear in*Algebra & Number Theory*.**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: Neumann F., Pál A. (eds)*Homotopy Theory and Arithmetic Geometry – Motivic and*Lecture Notes in Mathematics, vol 2292. Springer, Cham. 2021.

Diophantine Aspects.**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.

The $S_4$-orbits of the 27 lines on the Clebsch surface.

### 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.- 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).