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

报告简介：Cameron-Liebler line classes are sets of lines in 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 PG(3, 𝑞) having the same number of orbits on points and lines of 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 [1] by Cameron and Liebler, the authors gave several equivalent conditions for a set of lines of 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 ℒ be a set of lines of PG(3, 𝑞) with |𝐿| = 𝑥(𝑞2 + 𝑞 + 1), 𝑥 a positive integer. We say that ℒ is a Cameron-Liebler line class with parameter 𝑥 if every spread of PG(3, 𝑞) contains 𝑥 lines of 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 [2] of a new infinite family of Cameron-Liebler line classes with parameter 𝑥 = (𝑞+1)2/3 for 𝑞 ≡ 2 (mod 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 PG(3, 𝑞), 𝑞 even. This talk is based on joint work with Tao Feng, Koji Momihara, Morgan Rodgers and Hanlin Zou.

References

[1] P. J. Cameron, R. A. Liebler, Tactical decompositions and orbits of projective groups, Linear Algebra Appl. 46 (1982), 91–102.

[2] T. Feng, K. Momihara, M. Rodgers, Q. Xiang, H. Zou, Cameron-Liebler line classes with parameter 𝑥 = (𝑞+1)2/3 , Advances in Math. 385 (2021), 107780.

报告人：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。

报告时间：2022年10月5日（星期三）上午 8:30-10:00.

报告地点：腾讯会议，ID：447 281 800

主办单位：扬州大学数学科学学院

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