Alex Dunbar

Research Interests

My research interests are in applied algebraic geometry, real algebraic geometry, optimization, and machine learning. I am particularly interested in interactions between real algebraic geometry and semidefinite programming as well as applications of tropical geometry in machine learning.

Publications and Preprints

1. "Pythagoras Numbers for Ternary Forms" (with Greg Blekherman and Rainer Sinn ). ArXiv Preprint
2. "A Topological Approach to Simple Descriptions of Convex Hulls of Sets Defined by Three Quadrics" (with Greg Blekherman). ArXiv Preprint
3. "Alternating Minimization for Regression with Tropical Rational Functions" (with Lars Ruthotto ). Algebraic Statistics . Published Version Code ArXiv Preprint
4. "Relaxations and Duality for Multiobjective Integer Programming" (with Saumya Sinha and Andrew J. Schaefer). Mathematical Programming. Published Verison Optimization Online Preprint ArXiv Preprint

In Preparation

1. "Tensor-Tensor Products, Group Representations, and Semidefinite Programming", with (Elizabeth Newman)

Conference Presentations

1. "Alternating Minimization for Regression with Tropical Rational Functions" SIAM Conference on Mathematics of Data Science 2024 ( Poster )
2. "A Topological Approach to Simple Descriptions of Convex Hulls of Sets Defined by Three Quadrics". MEGA (Effective Methods in Algebraic Geometry) 2024 (slides)
3. "Tensor-Tensor Products, Group Representations, and Semidefinite Programming". JM Invariant @ 60 Conference on Tensor Invariants in Geometry and Complexity Theory 2024, Poster Session
4. "Alternating Minimization for Regression with Tropical Rational Functions". SIAM Conference on Applied Algebraic Geometry 2023, Minisymposium on Real Algebraic Geometry and Optimization.
5. "Alternating Minimization for Regression with Tropical Rational Functions". AMS Spring Southeastern Sectional Meeting 2023, Special Session on Algebraic Methods in Algorithms
6. "An Alternating Minimization Algorithm for Regression with Tropical Rational Functions". Combinatorial, Computational, and Applied Algebraic Geometry 2022 (CCAAGS-22), Poster Session
7. "Relaxation and Duality for Multiobjective Integer Programming". INFORMS Annual Meeting 2020, Undergraduate Operations Research Prize Session A

Upcoming Conferene Travel

1. International Conference on Continuous Optimization (ICCOPT) 2025

I am teaching Advanced Calculus (MATH 211) at Emory in Fall 2024.

Some interactive graphs in Desmos which I used to supplement my Calculus I lecture notes can be found here.

Previous Teaching:

• Instructor, Advanced Calculus (MATH 211), Fall 2023
• Instructor, Calculus I (MATH 111), Spring 2023
• Instructor, Calculus I (MATH 111), Fall 2022
• TA, Linear Algebra (MATH 221), Spring 2022
• 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. In the 2023-24 academic year, I co-directed the program with Shilpi Mandal. For more information about the program, see the DRP website.

Past reading topics include:

• Nonlinear Algebra, Spring 2024, based on Invitation to Nonlinear Algebra by Michałek and Sturmfels
• Convex Geometry, Spring 2023, based on Semidefinite Optimization and Convex Algebraic Geometry, edited by Blekherman, Parrilo, and Thomas and Convex Optimization by Boyd and Vandenberghe
• Concrete Algebra, Fall 2022, based on Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea
• 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