题  目：有限集合上的拓扑数

报告人：张影

时  间：2018年4月26日13: 30-14: 30

地  点：3-309

主讲人简介: 张影，苏州大学数学科学学院教授、博士生导师，从事低维流形几何拓扑学的研究；在Acta Arith.、Adv. in Math.、Amer. J. Math.、J. Diff. Geom.、Math. Res. Letters等国际数学杂志发表学术论文，先后主持国家自然科学基金面上项目3项。

Abstract: The numbers T(n) and T0(n) of distinct topologies and T0 topologies, respectively, which can be defined on a finite set of n elements exhibit interesting properties. They are actually the numbers of pre-orders and partial orders, respectively, which can be defined on the same set, and T(n) can be easily calculated by a combinatorial formula involving T0(k), k=0, 1, …, n. So far the effort of computer enumeration can evaluate these numbers only for n not exceeding 18. In around 1980 Z. I. Borevich observed periodicity for T0(n) modulo any positive number m. He proved the result for the case where m is a prime, and obtained periodicity for T(n), while the general case where m is a prime power remains unsolved. In joint work with Xiangfei Li, we have successfully settled the periodicity for T0(n) modulo a prime power. Furthermore, we obtain a powerful formula which gives as a consequence the periodicity of T(n) modulo any prime power.