报告题目：Numbers of finite topologies and p-group actions
报 告 人：张影 教授
所在单位：苏州大学
报告时间：2024年11月18日 10:30-11:30
报告地点：吉林大学正新楼209
报告摘要：Let T(n) (respectively, T_0(n)) denote the number of distinct topologies (respectively, T_0-topologies) which can be defined on a finite set of n points. We obtain arithmetic properties of T(n) and T_0(n) by constructing certain p-group actions on suitable subsets of the topologies. First, we obtain a formula which expresses T(n+p^k) modulo p^m in terms of the numbers of certain finite topologies, and thus establish the periodicity of T(n) modulo p^m; precisely, we have, for p ≥ 3, T(n+p^{p+m-1}) ≡ T(n+p^{m-1}) modulo p^m, and for p=2 similar but slightly sharper formulas. By the same methods we prove that, for any prime power p^k, T_0(n+p^k) ≡ T_0(n+p^{k-1}) modulo p^k, answering an unsolved problem in the work of Z. I. Borevich in around 1980. In another aspect, we obtain recursive relations for T_0(n) and T(n) modulo p^m. This is joint work with Xiangfei Li.
报告人简介：张影，1985.9-1999.5 吉林大学数学系本研学习、任教，1999.5-2004.7 新加坡国立大学读研，2006.3-2007.2 巴西国家数学所(IMPA)博士后，2009.6至今，苏州大学数学科学学院教授。从事几何拓扑学研究。
