Research
My research lies at the intersection of representation theory, algebraic combinatorics, and invariant theory, where I use quivers to better understand certain classes of finite-dimensional algebras.
Recently I have also become interested in using quiver invariant theory to investigate problems in persistent homology and neural networks.
Truthfully, I haven't encountered an area of math that I've disliked. I'm always amazed at the connections between different areas, and so I make it a point to acquaint myself with as much math as possible. Making connections between seemingly disparate branches or topics in math has provided me a richer view of the tapestry of the subject and I believe is essential in making deeper insights.
For instance, quivers are essential in studying finite-dimensional algebras, yet also play a key role in understanding maximal Cohen-Macaulay modules in commutative algebra. There is also a deep connection between the bounded derived categories of quiver representations and those of coherent sheaves on an algebraic variety. Understanding the (representation theoretic, commutative, homological) algebra then provides a better understanding of the geometry and combinatorics, and conversely, which is why I'm also very interested in homological commutative algebra.
For more, you can read my research statement here.
Papers, preprints, and descriptions
Generalized Littlewood-Richardson coefficients for branching rules of GL(n) and extremal weight crystals. Algebraic Combinatorics, Vol. 3, 2020.
My PhD thesis and some of my current work uses quiver representation theory to understand multiplicities arising from certain branching rules of GL(n) and extremal weight crystals. Littlewood-Richardson coefficients are ubiquitous in mathematics, yet notoriously difficult to calculate explicitly or work with, in general. Quiver theory allows a novel approach to classical combinatorial questions as well as related questions, like the multiplicities' complexity, which is vital in Geometric Complexity Theory.
Membership in moment cones, quiver semi-invariants, and generic semi-stability for bipartite quivers (w. Calin Chindris and Daniel Kline) (submitted to computational complexity)
We construct a polytopal description for weight spaces of semi-invariants for complete bipartite quivers and their flag extensions. By doing so, we prove that there is a strongly polynomial time algorithm for solving the generic semi-stability problem for representations of these quivers.
Rank characters for generalized persistence modules. (w. David Meyer) Preprint expected by February 2023.
Persistent homology, which is an area of Topological Data Analysis, has recently seen a surge of influence by representation theory. With David Meyer, we provided a way of determining if a generalized persistence module defined over any poset cannot be decomposed into convex submodules, and if it can, then what the decomposition would be.
Generalized Littlewood-Richardson coefficients for branching rules of classical groups. (w. Terry Yang) (In-progress)
In this project we describe certain branching rules of orthogonal and symplectic groups in terms of symmetric quivers and study their combinatorial properties using both quiver invariant theory and LR hives. By doing so, we can find the Horn inequalities which describe when these numbers are nonzero, produce a polytopal description of these numbers and show that determining that they're nonzero can be done in polynomial time, and describe various combinatorial behavior of these multiplicities (for instance, certain cases when they're equal to one and conjectural evidence for the behavior of their stretched polynomials).
Terry Yang participated in this research as an undergraduate at Bucknell University.
Course Notes
Here are some course notes I've typed during previous semesters.
Quiver Invariant Theory (Spring 18)
Local cohomology (Fall 17)
Quiver invariant theory (Spring 16)
Presented talks:
Decomposing convex persistence modules with rank characters, University of Iowa Algebra Seminar, October 2021.
Generalized Littlewood-Richardson coefficients for branching rules of GL(n), AMS Sectional Meeting, May 2021.
Generalized Littlewood-Richardson coefficients for branching rules of GL(n), Sixth Conference on Geometric Methods in Representation Theory, University of Iowa, November 2018.
Generalized Littlewood-Richardson coefficients for branching rules of GL(n) and extremal weight crystals. Joint Mathematics Meetings. San Diego, CA, January 2018.
Seminar talks presented
Undergraduate Seminar (Fitchburg State)
Modern applications of the Perron-Frobenius Theorem, Undergraduate Seminar, Fitchburg State, October 25, 2018.
Graduate Student Algebra Seminar (University of Missouri-Columbia):
Auslander-Buchweitz approximations for MCM modules, November 27, 2017
Plethysm for sl_2(C) II, October 2, 2017.
Plethysm for sl_2(C) II, September 25, 2017.
Character theory of finite groups, April 18, 2017.
Calculating polynomial invariants of finite groups, February 21, 2017.
Tilted algebras and generalization of Gabriel's Theorem, November 17, 2016.
Derived Categories II, September 22, 2016.
Derived Categories I, September 15, 2016.
Fundamental theorems of invariant theory, March 24, 2016.
Computing Littlewood-Richardson coefficients via quivers, November 15, 2015.
Littlewood-Richardson coefficients, September 23, 2015.
Relative K-theory and excision, February 25, 2015.
K_0 of a ring, February 18, 2015.
The Stone Representation Theorem, October 16, 2014.
Tilting theory and torsion pairs, October 2, 2014.
Hilbert's 14th problem: Old and new results, May 8, 2014.
Spectral sequences II, September 17, 2013.
Spectral sequences I, September 10, 2013.
Goldman domains, February 19, 2013.
Hilbert-Jacobson rings, February 12, 2013.
Graduate Student Seminar (University of Missouri-Columbia):
The Perron-Frobenius Theorem, February 9, 2018
Rogers-Ramanujan and other partition identities, October 20, 2017.
Counting with Burnside's Lemma, February 17, 2017.
The Littlewood-Richardson Rule, November 4, 2016.
Towards a classification of complex semisimple Lie algebras, April 22, 2016.
Irreducible representations of the symmetric group, April 10, 2015.
Categorical Banach Space Theory, November 8, 2013