Mathematics

I recently completed my PhD in mathematics at the University of Oklahoma, working under the direction of Greg Muller. My research is focused on cluster algebras, a type of commutative ring with connections to many different areas of mathematics and physics. In my research, I often consider connections to skein algebras, knot theory, decorated Teichmüller spaces, and higher Teichmüller spaces, while using tools from commutative algebra, algebraic geometry, representation theory, and combinatorics.

Papers

Deep Points of Cluster Algebras (arXiv)
Joint with Greg Muller
Int. Math. Res. Not. IMRN, Volume 2025, Issue 4, February 2025, doi:10.1093/imrn/rnaf027
We initiate a systematic study of the deep points of a cluster algebra; that is, the points in the associated variety which are not in any cluster torus. We describe the deep points of cluster algebras of type A, rank 2, Markov, and unpunctured surface type.

In Preparation

Separating Dots with Circles
Joint with Jaewon Min and Greg Muller
We study partitions of dots on the sphere using circles and show that this admits a natural cluster structure. A demo application can be viewed here.
Separating Curves in the Genus 2 Surface and the \( X_7 \) Cluster Algebra
Joint with Greg Muller
We establish a correspondence between separating curves in the genus 2 surface and a subset of cluster variables in the \( X_7 \) cluster algebra. We use this correspondence to show that the separating curve complex of the genus 2 surface is a six-dimensional pseudomanifold.

Dissertation

Deep Points of Cluster Varieties
(Includes portions of the above paper Deep Points of Cluster Algebras.)
We describe the deep points of cluster algebras of unpunctured polygons, unpunctured marked surfaces, punctured polygons, and punctured surfaces with at least two boundary marked points. As a consequence, we classify the deep points of cluster algebras of types \(A_n\) and \(D_n\). We also classify the deep points of the Markov cluster algebra and its upper cluster algebra.
We then study the deep points of cluster algebras of types \(B_n\), \(C_n\), and \(F_4\) via foldings of quivers of types \(A_{2n-1}\), \(D_{n+1}\) and \(E_6\), respectively.

Recreation

OEIS sequence A308821
Semiprimes where the sum of the digits equals the difference between the prime factors.

Old Stuff

For my MS, I wrote a survey paper on the curve complex. My advisor was Leah Childers.