报告题目：Cameron-Liebler Line Classes, Tight Sets and Strongly Regular Cayley Graphs

报告人：向青教授，南方科学技术大学

报告时间：2022年6月10日下午16：00-17：00

报告地点：龙洞校区行政楼610

主持人：何伟骅

报告摘要：Cameron-Liebler line classes are sets of lines in ${\rm PG}(3,q)$ having many interesting combinatorial properties. These line classes were first introduced by Cameron and Liebler in their study of collineation groups of ${\rm PG}(3,q)$ having the same number of orbits on points and lines of ${\rm PG}(3,q)$. During the past decade, Cameron-Liebler line classes have received considerable attention from researchers in both finite geometry and algebraic combinatorics. In the original paper \cite{camlie} by Cameron and Liebler, the authors gave several equivalent conditions for a set of lines of ${\rm PG}(3,q)$ to be a Cameron-Liebler line class; later Penttila gave a few more of such characterizations. We will use one of these characterizations as the definition of Cameron-Liebler line class. Let ${\mathcal L}$ be a set of lines of ${\rm PG}(3,q)$ with $|L|=x(q^2+ q+1)$, $x$ a positive integer. We say that ${\mathcal L}$ is a Cameron-Liebler line class with parameter $x$ if every spread of ${\rm PG}(3,q)$ contains $x$ lines of ${\mathcal L}$. It turned out that Cameron-Liebler line classes are closely related to certain subsets of points (tight sets) of the Klein quadric.We will talk about a recent construction in \cite{efhhh} of a new infinite family of Cameron-Liebler line classes with parameter $x=(q+1)^2/3$ for $q\equiv 2 \pmod 3$. When $q$ is an odd power of 2, this family of Cameron-Liebler line classes represents the first infinite family of Cameron-Liebler line classes ever constructed in ${\rm PG}(3,q)$, $q$ even. This talk is based on joint work with Tao Feng, Koji Momihara, Morgan Rodgers and Hanlin Zou.

专家简介：向青，现为南方科技大学数学系讲席教授，数学系副主任。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98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。