Analytic and Algebraic Methods in Discrete Geometry

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This award supports the PI's research in building mathematical tools to solve combinatorial problems in Discrete Geometry. Discrete geometry studies fundamental geometric objects, such as points, lines and circles, and their combinatorial properties. Thus most combinatorial problems in Discrete Geometry are quite visual and can be presented in a simple manner: What is the minimum total width of strips that cover a unit disk in the plane? Or, what is the maximum number of lines through the origin pairwise separated by the same angle? In contrast to the simple appearances of these problems, many of these problems, including the ones considered in this project, remain unsolved for an extended period or have been partly solved only recently following great efforts. This project aims to further develop the toolbox of available approaches, most of which are analytic or algebraic in nature. Simultaneously, the project integrates these research problems and themes into educational and outreach activities that extend from the high school level to the graduate.The first topic of this project concerns extensions and variations of Tarski's plank problem: How to efficiently cover a convex body in the plane using planks? Generalizations of Tarski’s plank problem are central to geometric analysis and convex geometry, and continue to generate interest in the geometric and analytic aspects of coverings of a convex body. The second topic concerns equiangular lines —- a collection of lines through the origin pairwise separated by the same angle. Investigation of the equiangular lines problem turns out to be fruitful and unearthed many challenging problems in algebraic graph theory. Solutions to some problems have practical consequences in Operational Research, Quantum Computation and Communication Theory. Previous foundational work, including some done by the PI, has shown that tools and insights from Ramsey Theory, Extremal Combinatorics, Probabilistic Methods, Spectral Graph Theory, Algebraic Combinatorics, Topological Combinatorics, and Algebraic Geometry can be helpful in solving the problems in this project.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.