Problems in Extremal Combinatorics and Finite Geometry

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The PI will develop algebraic tools for solving problems in extremal combinatorics and finite geometry, which are two important subareas of combinatorics. Combinatorics is an area of mathematics primarily concerned with properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc.This project will focus on three types of problems in finite geometry and extremal combinatorics. The PI intends to give improved upper bounds on the sizes of partial spreads, partial ovoids, and 1-systems in some classical polar spaces by using matrix rank argument. These bounds in turn will lead to non-existence proofs of ovoids in certain classical polar spaces. The Erdos-Ko-Rado theorem is a cornerstone in extremal combinatorics. Generalizations of the Erdos-Ko-Rado theorem hold for many different objects that have a notion of intersection. The PI will consider Erdos-Ko-Rado type problems for permutation groups. In the third part of the project, the PI will investigate constructions of difference sets, strongly regular Cayley graphs and related geometric substructures in classical polar spaces.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.