Discrete Geometry and Convexity

This project focuses on developing new mathematical tools to address open problems in discrete geometry. Discrete geometry, as a branch of mathematics, involves analyzing structures within sets of geometric objects, including points, lines, and circles. The central direction of this project is the study of covering and intersection properties of convex domains such as balls. The PI aims to find new connections between discrete geometry and other mathematical fields. Undergraduate students will be mentored as part of this project. In the long term, this project aims to explore combinatorial properties of coverings and intersection patterns of convex bodies in higher-dimensional spaces. In the short term, the focus lies on addressing fundamental problems at the interface of discrete geometry and combinatorial convexity, including plank covering problems and Tverberg-type problems. The former direction addresses various variations of Tarski's plank covering conjecture such as affine, polynomial, spherical, and hyperbolic. The second direction is devoted to Tverberg graphs, as well as the colorful and dual Tverberg conjectures. To settle these problems, the PI plans to apply and further develop methods from discrete and convex geometry, combinatorics, linear algebra, functional analysis, and algebraic topology, with a particular focus on optimization techniques. 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.