Research

Interests

My research is fundamentally interdisciplinary, and sits in the intersection of number theory, representation theory, combinatorics, and statistical mechanics. Specifically, I study special functions like Whittaker functions and automorphic forms that come from number theory and representation theory. However, I primarily study these functions using a unconventional tool: lattice models, which were originally developed by physicists to study how local particle interactions affect global phenomena such as temperature and phase transfer. These models give me access to a wide range of techniques from combinatorics and statistical mechanics, and illuminate surprising connections with seemingly-unrelated objects like quantum groups. Inspired by the fact that Whittaker functions appear as coefficients of automorphic forms, I also work in analytic number theory more broadly, studying the asymptotic phenomena of coefficients of modular forms and similar objects using more traditional analytic techniques. See my Research Statement for more details.

An evolution of ice-type lattice models: from left to right, the underlying water molecules, the

basic directed graph construction, and the chromatic model from my paper Frozen Pipes below.

One of my favorite parts of this area is that it generates lots of problems for undergraduate research! These are often more underlying combinatorial questions that have deep connections to many different areas of mathematics, so undergrads can engage with the project on any level and we can tailor it together to suit their interests. I've mentored 12 undergrad projects so far, including a solo project with a TRIO McNair scholar, a large group project at the Polymath REU, and several small group projects at the University of Minnesota Algebra and Combinatorics REU in Summers 2019, 2020, 2021, 2022. I believe that undergrad research is a great way to help help students decide if they are interested in continuing on to math grad school. It is also an important focus of my work towards diversity, equity, and inclusion in math, as it helps build the confidence of promising students from underrepresented groups by giving them a piece of math that is uniquely theirs and showing them that their contributions in math are valuable and respected.

Papers

In preparation:

with Sol Friedberg, paper on quasisplit unitary groups and lattice models.

Preprints:

with T. Browning, P. Čoupek, E. Eischen, S. Hong, S. Lee, D. Marcil, Constructing vector-valued automorphic forms on unitary groups, submitted for publication. (arXiv:2408.05198 [math.NT].)
with I. Axelrod-Freed and V. Lang, Measuring the Space of Metaplectic Whittaker functions, submitted for publication. (arXiv:2301.02223 [math.NT].)
Yang-Baxter Equations for General Metaplectic Ice, submitted for publication (arxiv:2009.13669 [math.RT].)

Thesis:

The Flexibility of the Six-Vertex Model in the Study of Special Functions, Thesis (Ph.D.)--University of Minnesota ProQuest LLC, Ann Arbor, MI, 2022. 154 pp. (available here.)

Published:

with M. Gerbelli-Gauthier, A. Hamieh, and N. Tanabe, A Note on Large Sums of Divisor-Bounded Multiplicative Functions, to appear in Proceedings of Women in Numbers. (arXiv:2405.00658 [math.NT].)
with M. Gerbelli-Gauthier, A. Hamieh, and N. Tanabe, Large Sums of Fourier Coefficients of Cusp Forms, to appear in Journal de Théorie des Nombres de Bordeaux. (arXiv:2308.06311 [math.NT].)
with M. Curran, C. Yost-Wolff, S. Zhang, and V. Zhang, A Lattice Model for superLLT Polynomials, Comb. Theory, 3 (2023), pp. Paper No. 3, 52. (arXiv:2110.07597 [math.CO].)
with B. Brubaker, A. Hardt, E. Tibor, and K. Weber, Frozen Pipes: Lattice Models for Grothendieck Polynomials, Algebr. Comb., 6 (2023), pp. 789–833. (arXiv:2007.04310 [math.CO].)
with L. Beneish, p-adic Properties of Certain Half-Integral Weight Modular Forms, J. Number Theory 168 (2016), 413–432. (arXiv:1509.00383 [math.NT].)
with M. Locus, Combinatorial Properties of Rogers-Ramanujan-Type Identities Arising from Hall- Littlewood Polynomials, Ann. Comb. 20 (2016), no. 2, 345–360. (arXiv:1407.2880 [math.CO].)
with J. Awan, E. McMahon, and Y. Li, Demicaps in AG(4,3) and their Relation to Maximal Cap Partitions, Graphs Combin. 38 (2022), no.6, Paper No. 193, 20 pp. (arXiv:2106.14141 [math.CO].)

Undergraduate Research Students

Solo Projects

Alessandro (Sandro) Orjuela, July 2024 - present: research project on Type D lattice models and their connections to automorphic forms.
- - Sandro is a recent graduate of Bowdoin College (Class of 2024) working with me on his gap year while applying to graduate school for the 2025-26 academic year.
Yifan (Ricky) Li, Boston College, Fall 2023, reading project on lattice models for Hecke-Grothendieck polynomials, examining connections between Ricky's undergrad research at the Polymath Jr REU Summer 2023 and quantum groups. Note: this Polymath project sparked from the Frozen Pipes paper above.
Yaren Euceda Mejia, TRIO McNair Scholar at U Minnesota, Summer 2019 - Spring 2020: “Counting Metaplectic Ice and Modified Alternating Sign Matrices.”
- - Research Presentations Yaren has done:
    - - UMN McNair Symposium, August 2019 (poster)
      - UMN Undergraduate Research Symposium, August 2019 (poster)
      - University of New Mexico McNair Conference, October 2019 (talk)
      - Gulf Coast Undergraduate Research Symposium, November 2019 (talk)
Sara Moore, U Minnesota, reading project on lattice models and special functions, Fall 2019 - Spring 2020.
- - Sara is now a grad student in physics at UC Boulder.

REU Projects

As Lead Mentor:

Ilani Axelrod-Freed and Veronica Lang: "Measuring the Space of Whittaker Functions," UMN Algebra/Combo REU 2022.
- - - Their final report, and their final presentation.
    - Their poster from the JMM 2023 Undergraduate Poster Session.
    - Both Ilani and Veronica won Honorable Mentions for the AWM 2023 Schafer Prize for research including this project! They are both now in grad school for math, Ilani at MIT and Veronica at UMichigan.

Elisabeth Bullock, Noah Caplinger, Ariana Chin, Nyah Davis and Gahl Shemy: "Lattice models and puzzles for dual weak symmetric Grothendieck polynomials," UMN Algebra/Combo REU 2021.
- - - Their final report, and their final presentation.
    - They intended to present at the 2022 Joint Mathematics Meetings, but sadly, the meetings were postponed due to COVID and then intersected with their spring semesters.
    - All five of them are now in grad school for math! Elisabeth is at MIT, Noah at UChicago, Ariana at UCLA, Nyah at Rice, and Gahl at UMichigan. In addition, Elisabeth and Nyah won NSF graduate student fellowships!

As Co-Mentor:

Polymath Jr. REU: "Generalized Alternating Sign Matrices," Summer 2020. (see https://geometrynyc.wixsite.com/polymathreu for more information about the Polymath REU).
Lingxin Cheng, Eli Fonseca, and Erin Herman-Kerwin: "Schur Function identities and The Yang-Baxter Equation," UMN Algebra/Combo REU 2020.
Kedar Karhadkar: "Lattice Models, Differential Forms, and the Yang-Baxter Equation," UMN Algebra/Combo REU 2020.
Neelima Borade, Lingxin Cheng, and Eli Fonseca: "Bent Lattice Models." UMN Algebra/Combo REU 2020.
Michael Curran, Calvin Yost-Wolff, Sylvester Zhang, and Valerie Zhang: "Ribbon Lattices and Ribbon Function Identities," UMN Algebra/Combo REU 2019.
Quang Dao, Nathan Kenshur, Feiyang Lin, Christina Meng, Zack Stier, and Calvin Yost-Wolff: "Metaplectic Analogues of Gelfand-Graev Models," UMN Algebra/Combo REU 2019.

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