Keynote Speaker:Xiang Qing

Abstract:

The Erd{\H o}s-Ko-Rado (EKR) theorem is a classical result in extremal set theory. It states that when $k<n/2$, 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 {\it 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.

Speaker Introduction:

Xiang Qing is now a Chair Professor of Southern University of Science and Technology, and he is mainly engaged in the research of combinatorial design, finite geometry, coding theory and addition combinatorics. He graduated from The Ohio State University with a doctorate in 1995. He was awarded the Kirkman Medal by the Institute of Combinatorics and its Applications in 1999. He once served as Bateman Instructor of California Institute of Technology, Tenured Professor of University of Delaware, and Chair Professor of Zhejiang University.

Inviter:

Zhao Lilu Professor of School of Mathematics

Time:

16:00-17:00 on November 10 (Tuesday)

Location:

Tencent Meeting, Meeting ID: 460 732 686

Hosted by: School of Mathematics, Shandong University