## Research

My research is on algebraic structures in representation theory, and the associated combinatorial constructions. I am particularly interested in applications to Schubert calculus, reductive p-adic groups, (non)symmetric function theory, and representations of quantum groups. Below, I mention some of my favorite areas in more detail..

Click for: Papers Interests Talk Notes Conferences Attended

### Papers

- "Lattice Models, Hamiltonian Operators, and Symmetric Functions", submitted, preprint available here.
- "Frozen Pipes: Lattice Models for Grothendieck Polynomials", joint with Ben Brubaker, Claire Frechette, Emily Tibor, and Katherine Weber, submitted, preprint available here.
- "Arborescences of Covering Graphs", joint with Sunita Chepuri, CJ Dowd, Gregory Michel, Sylvester Zhang, and Valerie Zhang, submitted, preprint available here.
- "Characters of Renner Monoids and Their Hecke Algebras", joint with Jared Marx-Kuo, Vaughan McDonald, John O'Brien, and Alex Vetter,
*International Journal of Algebra and Computation,*, available here.**30**, no. 7, 1505--1535 (2020) - Oral Exam Paper: "Finite Hecke Algebras and Their Characters" (2019), available here.
- "Restricted Symmetric Signed Permutations" (2012), joint with Justin Troyka,
*Journal of Pure Math and Applications,*, available here, slides here.**23**, 179--217 (2012) - Undergraduate Senior Thesis: "Combinatorial Species and Graph Enumeration (2013), joint with Pete McNeely, Tung Phan. and Justin Troyka, available here.

### Interests

#### Lattice Models

Solvable lattice models originated in statistical mechanics. They are 2D rectangular lattices, with edge labels either + or -, and certain allowable configurations around each vertex. By summing over all allowable ``states'', one can use local information to compute the total energy, denoted the ``partition function'' of the model. By modifying the weights and boundary conditions, one may represent special functions such as Schur functions, Whittaker functions, and Grothendieck polynomials. One can then (sometimes) prove identities such as branching rules, (dual) Cauchy formulas, and Littlewood-Richardson rules. Lattice models for Schur functions have a close connection to representation theory through Gelfand-Tsetlin patterns.

The main part of my thesis research (preprint upcoming) explores the connections between solvable lattice models and Hamiltonian operators. I gave a talk on this at the Solvable Lattice Seminar hosted virtually by Stanford University. I am always interested in expressing more (always more!) special functions as lattice models, particularly functions with geometric or representation theoretic importance. I am also exploring the connections between lattice models and Hamiltonian operators.

#### Reductive Groups and Hecke Algebras

Representations of reductive p-adic groups are closely tied to automorphic forms. These representations are difficult to study, and many tools have been developed. Hecke algebras corresponding to various compact open subgroups (e.g. Iwahori, maximal compact) are one of these tools: they are simpler objects whose representations correspond to representations of the p-adic group with a vector fixed by the subgroup.

Finite reductive groups have many uses, a major one being as a prototype for the more complicated p-adic case. More directly, Lusztig's depth zero construction obtains an important class of p-adic representations by inducing (in some sense) from the analogous finite group. The Hecke algebra most often used here is defined relative to the Borel subgroup; the resulting algebra is a trivial deformation of the Weyl group algebra.

#### Reductive Monoide

Reductive monoids are the neglected cousin of reductive groups. Originally studied by Putcha and Renner, they have seen new interest stemming from the work of Braverman, Kazdhan, and Ngo in the Langlands Program. They are Zariski-closed monoids whose unit group is a reductive group. Remarkably, these are exactly the monoids which are regular as semigroups. The representation theory of reductive monoids is closely related to that of reductive groups, and the two studies inform each other. Again, Hecke algebras exist in this setting, and in the finite case the analogous correspondence between Hecke algebra representations and certain representations of the reductive monoid still holds.

#### Alcove Walks

I have recently become interested in alcove walks after running an REU project based on them. One can view a reduced word of an element in an affine Weyl group as a sequence of alcoves, starting at the fundamental alcove and ennding at the element. If we label each step by an element of a field, and allow ourselves to ``fold'' walks, this gives us the flexibility and power to express objects as diverse as Macdonald polynomials, affine Hecke algebras, and intersections of important number-theoretic double cosets in reductive groups.

### Notes From Selected Talks (see CV for a longer list of my talks)

- Hamiltonian Operators and Free Fermionic Lattice Models -- Solvable Lattice Models Seminar, June 2021
- Solvable Lattice Models, Statistical Mechanics, and Symmetric Functions -- Carleton College Colloquium, June 2021
- Crystal Bases -- UMN Student Representation Theory Seminar, August 2020
- Alcove Walks -- UMN Algebra and Combinatorics REU (project introduction), June 2020
- Iwahori Hecke Algebras in Multiple Contexts UMN Student Number Theory Seminar, March 2020; UGA Graduate Student Seminar, July 2020
- Classification of Reductive Algebraic Monoids -- UMN Student Combinatorics Seminar, October 2019
- What is the Biggest (Finite) Number? -- UMN Math Club, October 2019
- What is Number Theory? -- UMN Student Number Theory Seminar, September 2019
- Finite Hecke Algebras and Their Characters -- Oral Exam Talk, March 2019

### Selected Conferences Attended

- New Connections in Integrable Systems (Remote) -- September-October 2020
- Soergel Bimodules and Categorification of the Braid Group Workshop (ICERM, Brown University) -- February 2020
- Abel Conference Honoring Robert Langlands (University of Minnesota) -- November 2018
- Graduate Student Combinatorics Conference (University of Kansas) -- April 2017
- Commutative Algebra Plus (University of Wisconsin-Madison) -- October 2016
- Speaking Sciences (University of Minnesota) -- January 2019

### Organization

- University of Minnesota Student Number Theory Seminar 2019-20. Seminar webpage.