Research
My main area of research is Computational and Applied Algebraic Geometry. I use and develop methods from algebraic geometry, commutative algebra,
algebraic combinatorics, and symbolic/numeric computation to understand and solve problems in
industrial and applied mathematics. My research has been focused on addressing problems arising
in biology, geometric modeling, physics, and statistics.
I am a member of the SIAM Activity Group in Algebraic Geometry.
Below I describe some of my research projects. However, for a more up to
date list of my constributions please visit my list of publications.
Algebraic Statistics
Developer of the algstat package for the free software environment for statistical computing and graphics R.
The algstat package provides functionality for
algebraic statistics in R. Current applications include exact inference in log-linear models
for contingency table data, analysis of ranked and partially ranked data, and general
purpose tools for multivariate polynomials.
Developer of the
GraphicalModels package for Macaulay2.
The Macaulay2 package GraphicalModels contains algorithms for the algebraic study of graphical models associated to undirected, directed and mixed graphs, and associated collections of conditional independence statements.
Identifying causal effects in graphical models website.
The long-standing identification problem for causal effects in graphical models has many partial results. This website records one of the first efforts to perform a systematic study of this problem. This study was based on computer algebra methods to either prove that a causal effect can be identified, generically identified, or show that the effect is not generically identifiable.
Small Phylogenetic Trees website. Mentioned in the SIAM News, Volume 40, Number 6, July-August 2007.
Biochemical Reaction Networks
2015-2017 AIM SQuaRE on "Ideals in Algebraic Systems
Biology" and 2020 and AIM SQuaRE on "Algebraic Geometry of Chemical
Reaction Networks".
Since 2015, Elizabeth Gross,
Heather Harrington, Nicolette Meshkat,
Anne Shiu and I have studied the algebro-geometric properties of biological
chemical reaction networks. Specifically focusing on certain
polynomial ideals arising from systems biology. Our joint work is
supported through the American Institute of Mathematics SQuaRE
program.
Geometric Modeling
Algebraic Geometry in Algebraic Statistics and Geometric Modeling
project. In this collaborative between faculty and students at SHSU and TAMU focused on a study of some common algebraic geometric structures in algebraic statistics, geometric modeling, and algebraic geometry.
Sandpile Groups
Research Advisor of the MSRI Undergraduate Program (MSRI-UP) 2016 program on sandpile groups.
Minicourse for Undergraduates: An Introduction to the Theory of Sandpiles at the Modern Math Workshop 2015. October 28 - 29, 2015.
Research Advisor for three undergraduate research programs on the topic of "Sandpile groups" as part of the NSF/MCTP 2011, 2013, and 2014
PURE Math program at the
University of Hawai'i at Hilo.
Schubert Calculus
Frontiers of reality in Schubert calculus project.
Systems Biology
Developer of the REACT: Evolutionary Algorithm for Discrete Dynamical System Optimization algorithm. This algorithm is now part of the TURING platform for algorithms and analysis pipelines focused on time- and state-discrete dynamical systems.
Theoretical Neuroscience
Developer of the NeuralIdeals package for SageMath. The package NeuralIdeals
provides methods to study algebraic and combinatorial properties of neural codes.
