I am a fourth-year graduate student in Mathematics at Auburn University studying under Hal Schenck. I earned a B.S. in Mathematics with minors in History and Classics at the University of Kentucky. While at the University of Kentucky, I conducted research with the University of Kentucky Math Lab and also as part of the Multimodal Vision Research Laboratory.
My primary research interests lie in commutative algebra and topological data analysis. However, I've been finding that a lot of side projects that interest me come from combinatorial optimization. A copy of my CV can be found here along with my contact information.
Publications
We study the Artinian reduction \( A \) of a configuration of points \( X \subset \mathbb{P}^n \), and the relation of the geometry of \( X \) to Lefschetz properties of \( A \). Migliore initiated the study of this connection, with a particular focus on the Hilbert function of \( A \), and further results appear in work of Migliore–Miró-Roig–Nagel. Our specific focus is on Betti tables rather than Hilbert functions, and we prove that a certain type of Betti table forces the failure of the Weak Lefschetz Property (WLP). The corresponding Artinian algebras are typically not level, and the failure of WLP in these cases is not detected in terms of the Hilbert function.PDF
Repeat-visit airborne lidar is a powerful tool for change detection in urban and rural environments. In this work, we present a learning-based approach that addresses one of the key challenges in comparing point cloud scans of the same region: handling geometric differences caused by varying sensor position. Our approach is to perform shape modeling through ray casting with a point cloud neural network. Recent work on learning-based shape modeling has been based on the assumption that an explicit surface representation is available, which is not the case for airborne lidar datasets. Our key insight is that by using a ray casting approach we can perform shape modeling directly with lidar measurements. We evaluate our method both quantitatively and qualitatively on learned surface accuracy and show that our method correctly predicts surface intersection even in sparse regions of the input cloud.PDF
Preprints
The connected sum construction, which takes as input Gorenstein rings and produces new Gorenstein rings, can be considered as an algebraic analogue for the topological construction having the same name. We determine the graded Betti numbers for connected sums of graded Artinian Gorenstein algebras. Along the way, we find the graded Betti numbers for fiber products of graded rings; an analogous result was obtained in the local case by Geller. We relate the connected sum construction to the doubling construction, which also produces Gorenstein rings. Specifically, we show that a connected sum of doublings is the doubling of a fiber product ring.PDF
We introduce the MatrixSchubert package for the computer algebra system Macaulay2. This package has tools to construct and study matrix Schubert varieties and alternating sign matrix (ASM) varieties. The package also introduces tools for quickly computing homological invariants of such varieties, finding the components of an ASM variety, and checking if a union of matrix Schubert varieties is an ASM variety.PDF | Code
Flight Route Estimation
How well can you estimate the path of a LiDAR sensor given a scan of the terrain?
MatrixSchubert
A Macaulay2 package with functions for investigating ASM and matrix Schubert varieties.