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

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.



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. 



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. 



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:

Seminar talks presented

Undergraduate Seminar (Fitchburg State)

Graduate Student Algebra Seminar (University of Missouri-Columbia):


Graduate Student Seminar (University of Missouri-Columbia):