Erdős-Ko-Rado Type Theorems for Permutation Groups
数学专题报告
报告题目(Title)：Erdős-Ko-Rado Type Theorems for Permutation Groups
报告人(Speaker)：向青 教授 （南方科技大学）
地点(Place)：腾讯会议 ID：176361275
时间(Time)：2022 年 11 月 30 日（周三） 15:30--16:30
邀请人(Inviter)：吕本建、王恺顺
报告摘要
The Erdős-Ko-Rado (EKR) theorem is a classical result in extremal set theory. It states that when $n/2>k$ any family of $k$-subsets of $\{1,2,\ldots,n\}$, with the property that any two subsets in the family have nonempty intersection, has size at most ${n-1\choose k-1}$; equality holds if and only if the family consists of all $k$-subsets of $ {1,2,\ldots,n\}$ containing a fixed element.
Here we consider EKR type problems for permutation groups. In particular, we focus on the action of the $2$-dimensional projective special linear group $PSL(2,q)$ on the projective line $PG(1,q)$ over the finite field $\mathbb{F}_q$ where $q$ is an odd prime power. A subset $S$ of $PSL(2,q)$ is said to be an intersecting family if for any $g_1,g_2\in S$, there exists an element $x\in PG(1,q)$ such that $x^{g_1}=x^{g_2}$. It is known that the maximum size of an intersecting family in $PSL(2,q)$ is $q(q-1)/2$. We prove that all intersecting families of maximum size must be cosets of point stabilizers for all odd prime powers $q>3$. This talk is based on joint work with Ling Long, Rafael Plaza, and Peter Sin.
主讲人简介
向青，现为南方科技大学数学系讲席教授。向青教授于1995获美国 Ohio State University博士学位。他的主要研究方向为组合设计、有限几何、编码理论和加法组合。在国际组合数学界最高级别杂志《J. Combin. Theory Ser. A》，《J. Combin. Theory Ser. B》，《Combinatorica》, 以及顶尖的数学综合期刊《Advances in Math.》，《Trans. Amer. Math. Soc.》等重要国际期刊上发表学术论文99篇。主持完成美国国家自然科学基金、中国国家自然科学基金海外及港澳学者合作研究基金等科研项目10余项。正在主持中国国家自然科学基金重点项目一项，以及海外资深研究学者基金一项。曾在国际学术会议上作大会报告或特邀报告60余次。
