Alex Dunbar

I'm a second-year PhD student in Mathematics at Emory University. I graduated from Rice University in 2020 with a BS in Mathematics and a BA in Computational and Applied Math. I am coadvised by Victoria Powers and Greg Blekherman (Georgia Tech).

My research interests are broadly in algebraic methods for optimization and machine learning. Most recently, this has taken the form of sums of squares methods for optimization.

Current Projects

My work explores intersections between algebraic geometry and optimization. Right now, I am working on applying tropical methods for data fitting problems and understanding sums of squares certificates on varieties.

Sums of Squares on Varieties

Sums of squares (SOS) certificates for nonnegative polynomials has been a longstanding area of research in real algebraic geometry. One recent motivation for this is that the search for an SOS certificate can be phrased as a semidefinite feasibility problem. We are interested in better understanding the relatoinship between SOS polynomials and nonnegative polynomials in the coordinate ring of a variety.

Tropical Methods for Data Fitting

Recent work has shown that rational functions over the tropical semiring have a broad representation power. We are interested in exploring algorithms for fitting tropical rational functions to data and comparing these approaches to traditional approaches in data science.

Publications and Preprints

"Relaxations and Duality for Multiobjective Integer Programming" (with Saumya Sinha and A.J. Schaefer). Submitted.

Invited Talks

"Relaxation and Duality for Multiobjective Integer Programming". INFORMS Annual Meeting 2020, Undergraduate Operations Research Prize Session A

I am currently a TA for Linear Algebra (MATH 221) at Emory.

Previous Teaching:

TA, Numerical Analysis (MATH 315), Fall 2021
Grader, Numerical Analysis (MATH 315), Spring 2021
Grader, Mathematical Statistics I (MATH 361), Fall 2020
Grader, R for Data Science (STAT 405, Rice University), Fall 2018

Emory Mathematics Directed Reading Program

I have been a graduate student mentor for the Emory Mathematics DRP since Fall 2021. In Spring 2022, I joined the steering comittee to help organize the program. For more information about the program, see Chris Keyes's website.

Past reading topics include:

Computational Algebra Part II, Spring 2022, based on Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea
Computational Algebra, Fall 2021, based on Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea

Below are some informal expository notes I prepared for talks at the Emory Graduate Student Algebra and Number Theory Seminar (RANT)

Positivity in Real Algebraic Geometry and Moment Problems (notes)
Gröbner Bases and Integer Programming (slides)

When I am not doing math, I enjoy distance running. In college, I competed on the Rice Cross Country and Track & Field (5000m, 10000m) teams. Since graduating, I have participated in a variety of races on roads and trails. (Picture is from 2022 Lousiana Half Marathon)