Research

Preprints

1. Computing Jet Differentials and the Green-Griffiths-Lang Conjecture with Ryan Contreras, Eric Reidl and Jaziel Torres. arxiv version

2. Identifiability of phylogenetic networks and quintet concordance factors joint with Maize Curiel, Bryan Currie, Bryson Kagy, Udani Ranasinghe, and John A. Rhodes. arxiv version

Published/Accepted Articles

1. Mixed volume and ramification points of sparse homotopies with Jonathan D. Hauenstein. To appear in LNCS 16677. A version of this article is available here.

2. The well-poised property and torus quotients with Christopher Manon. Appeared in Communications in Algebra 2026. arxiv version

3. The Pfaffian Structure of CFN Phylogenetic Networks with Elizabeth Gross, Benjamin Hollering, Samuel Martin, and Ikenna Nometa. Appeared in Journal of Mathematical Biology 2026. arxiv version

4. Routing functions for parameter space decomposition to describe stability landscapes of ecological models with Kyle J.-M. Dahlin, Elizabeth Gross, and Jonathan D. Hauenstein. Bulletin of Mathematical Biology, Volume 87, article number 177, 2025. arxiv version

5. Smooth connectivity in real algebraic varieties with Jonathan D. Hauenstein, Hoon Hong, and Clifford Smyth. Numerical Algorithms, 100(1), 63-84, 2025. arxiv version

6.  Computing Implicitizations of Multi-Graded Polynomial Maps with Benjamin Hollering. Journal of Symbolic Computation, 132, 2026. arxiv version. This is accompanied by our Macaulay2 package MultigradedImplicitization.m2 available in the latest M2 distribution.

7.  Multi-graded Macaulay Dual Spaces with Jonathan D. Hauenstein.  Journal of Algebra and Its Applications, 24(10), 2550246, 2025. arxiv version

8.  A Fano compactification of the $Sl_2(C)$ free group character variety with Christopher Manon. Published in Geom Dedicata 218, 17 (2024). arxiv version

9.  Invariants for level-1 phylogenetic networks under the Cavendar-Farris-Neyman Model with Benjamin Hollering and Christopher Manon. Advances in Applied Mathematics Volume 153, February 2024, 102633. arxiv version.

10. Generalized Cut Polytopes for Binary Hierarchical Models with Jane Ivy Coons, Benjamin Hollering, and Aida Maraj. Published in Algebraic Statistics Vol. 14 (2023), No. 1, 17–36. arxiv version

11.  Reliability of markerless motion capture systems for assessing movement screenings with Jonathan D. Hauenstein, Alan Huebner, John P. Wagle, Emma R. Cobian, Caroline Hills, Megan McGinty, Mandy Merritt, Sam Rosengarten, Kyle Skinner, Michael Szemborski, and Leigh Wojtkiewicz. Appeared in Orthopaedic Journal of Sports Medicine 2024.

Software

1. MultigradedImplicitization.m2 -- This is a M2 package designed to solve the implicitization problem for multigraded ring homomorphisms. It is available with the latest M2 distribution and on github.

2. ConnectedComponents.jl -- This is a Julia package currently under development with David K. Johnson. It computes the smoothly connected components in a real algebraic variety by implementing the algorithms developed in Smooth connectivity in real algebraic varieties. The code is available here.

3. SymbolicCoalescentModel.m2 -- This code accompanies the preprint, Identifiability of phylogenetic networks and quintet concordance factors. The main thrust of this package is to compute gene tree probabilities on a specified species network. As of now, this is the only software capable of doing this for arbitrary networks symbolically. The code is available here.

4. Mixed Volume and Ramification Points -- This project accompanies the paper, Mixed Volume and ramification points of sparse homotopies. It provides functionality for computing certain Chern classes in toric varieties using mixed volumes. It is implemented in Macaulay2 and Python. It is available here.

5. Multi-graded Macaulay Dual Spaces -- This project accompanies the paper, Multi-Graded Macaulay Dual Spaces. It is implemented in Macaulay2 and computes multi-graded components of Macaulay dual spaces of an ideal recursively. It is available here.

6. Routing functions for parameter space decomposition – This project accompanies the paper, Routing Functions for Parameter Space Decomposition to Describe Stability Landscapes of Ecological Models. The symbolic parts of the computation were done in Macaulay2 though the numerical path tracking and system solving were implemented in Julia. The code is available at here.

Here are two examples that were computed using ConnectedComponents.jl. 

First, here is a 3RPR robotic mechanism where one leg length is fixed. The configuration space for this mechanism is a real algebraic variety with 2 smoothly connected components. The mechanisms below interpolates between configurations in the same smoothly connected component. There is no smooth path between any configuration of the mechanism on the left to any configuration of the mechanism on the right.

Second, we have the Clebsch cubic with it's 27 lines plotted. The 27 lines (plus the line at infinity) divide the Clebsch cubic into 141 regions. There is a point in each region in the picture below.