Algebraic and Geometric Structures in Extremal Combinatorics

The aim of this project is to better understand mathematical structures that are 1) discrete, and 2) are of geometric and algebraic nature. The examples of discrete structures include networks, matrices and arrangements of convex sets. There are a number of instances where the optimal discrete objects necessarily possess non-trivial algebro-geometric properties. This project is devoted to understanding this phenomenon. Students will be mentored as part of the project. The specific problems include algebraic and geometric questions related to the Turan problems and combinatorial questions in finite geometry. Particular attention will be devoted to algebraic constructions, especially in mixed characteristic. The potential impacts include a new method for testing conjectures in discrete geometry and better locally decodable codes. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.