Problems in Extremal Combinatorics and Finite Geometry

PI将开发用于解决极值组合学和有限几何问题的代数工具，这是组合学的两个重要子领域。组合数学是一个数学领域，主要关注有限结构的性质。它与数学的许多其他领域密切相关，并有许多应用，从逻辑到统计物理，从进化生物学到计算机科学等。本项目将集中在有限几何和极值组合学中的三种类型的问题。PI利用矩阵秩变元给出了一些经典极空间中部分散布、部分卵形和1-系的大小的改进上界。这些界限反过来又会导致卵形体在某些经典极空间中的不存在性证明。Erdos-Ko-Rado定理是极值组合学的基石。Erdos-Ko-Rado定理的推广适用于许多具有交集概念的不同对象。PI将考虑置换群的Erdos-Ko-Rado型问题。在该项目的第三部分，PI将研究差集，强正则Cayley图和经典极空间中相关几何子结构的构造。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估而被认为值得支持。

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.