Research

My research is about studying problems in Applied Algebraic Geometry motivated by Algebraic Statistics. 

Applied Algebraic Geometry is an inclusive field involving industry, scientists, and mathematicians. Our main conference SIAM AG happens every two years and the top journal is SIAM Applied Algebra and Geometry (SIAGA). Students who work with me are encouraged to participate in this community.  One way to do this stems from the global lockdown in 2020: I coorganized our community’s monthly webinar, SIAM SAGA, which has recordings archived. Another way is to attend our Applied Algebra Seminar or to join the SIAM Algebraic Geometry Activity Group (free memberships are often available for students).

Algebraic Statistics is a branch of mathematical statistics that focuses on the use of algebraic, geometric and combinatorial methods in statistics. In Fall 2023, I will coorganize a semester long program at the new NSF institute IMSI (Chicago) titled Algebraic Statistics and Our Changing World: New Methods for New Challenges (Send me an email if interested). Also, I have organized special sessions with Algebraic Statistics Themes at the AMS Sectional meetings, the SIAM Annual Meeting, and Joint Statistics Meeting. Many of my students begin working with me on problems in this area and branch out into others. 

More specifically, my research involves three key aspects that intertwine and motivate one another. The first aspect is to determine algebraic degrees of optimization problems. Many optimization problems can be phrased as solving a polynomial system whose roots correspond to critical points. Several of my results are in the context where the objective function measures Euclidean distance or likelihood. The number of critical points the associated objective on a given model is an invariant called the Euclidean Distance (ED) degree and Maximum Likelihood degree respectively. Using a combination of tools including numerical nonlinear algebra, symbolic computation, and singularity theory, I compute these invariants. Some applications of this work is to view the ED degree as an Euler characteristic and then use this point of view to prove a formula for the number of critical points for the n-view triangulation problem. 

To solve global optimization problems, the second aspect of my research is to develop new techniques using numerical algebraic geometry. For example, with collaborators I developed a fiber product homotopy method to study multiparameter eigenvalue problems (MEP), I introduced new algorithms for solving decomposable sparse polynomial systems, and  created a toolkit for studying multiprojective varieties. Homotopy continuation underlies these methods, and one strength of algebraic geometry comes from the notion of degree. Knowing the degree of an optimization problem allows the practitioner to make provable guarantees about finding a global optimum. 

This leads to the third aspect of my research. I put the theoretical ideas into practice. In a recent collaboration, we do density estimation for n-dimensional Gaussian mixtures using the tools of algebraic geometry. These are large scale problems that have hundreds of thousands of unknown parameters that can be accurately recovered using our method. More broadly, I look to apply my techniques in other applications like kinematics, data science, and computer vision. This is in addition to building community by supplying implementations. 

If you are a student who is interested in any of these aspects, then I am happy to chat with you.

• I am available for directed reading courses on Ideal, Varieties, and Algorithms (introduction to computational algebraic geometry), Algebraic Statistics,  Solving polynomial systems, and related references. 

• Expectations for directed readings: I expect to meet once to twice a week for ~25 minutes. During this time you are encouraged to tell me what you found interesting and to present a worked out example. This is followed up with notes in overleaf. If you like, I am also happy to give feedback on your notes. 

• Undergraduate research: I strongly encourage students to be familiar with  Ideal, Varieties, and Algorithms OR to have worked through this crash course with HomotopyContinuation.jl before committing to a research project. If you need help getting started it might be better to do a reading course that leads into a project.

• In 2020, I ran the Collaborative Undergraduate Research Laboratory (CURL):  Applications and outline are found here. 

• In Spring 2023, with Aviva Englander (Graduate Student) I am mentoring three undergraduates in an MXM project on Algebraic Systems Biology.