Covers of the Integers by Residue Classes and their Extensions to Groups
数学公众报告
报告题目(Title)：Covers of the Integers by Residue Classes and their Extensions to Groups
报告人(Speaker)：孙智伟教授 （南京大学）
地点(Place)：后主楼1124
时间(Time)：2024 年 1 月 4 日（周四） 16:00--17:00
邀请人(Inviter)：徐敏
报告摘要
A system $A=\{a_s+n_s\mathbb Z\}_{s=1}^k$ of $k$ residue classes is called a cover of $\mathbb Z$ if any integer belongs to one of the $k$ residue classes. This concept was introduced by P. Erd\H os in the 1950s. Erd\H os ever conjectured that A is a cover of $\mathbb Z$ whenever it covers 1,…,.
In this talk we introduce some basic results on covers of $\Z$ as well as their elegant proofs. We will also talk about covers of groups by finitely many cosets, give a proof of the Neumann-Tomkinson theorem, and introduce progress on the Herzog-Sch\H oheim conjecture and the speaker's disjoint cosets conjecture. The generalized spectrum of a graph G consists of the spectrum of G together with that of its complement, and G is determined by its generalized spectrum if any graph having the same generalized spectrum as G is isomorphic to G. In this talk, I shall report some recent developments on the topic of generalized spectral characterizations of graphs, with an emphasis on a recent work on the generalized spectral characterizations of mixed graphs based on Eisenstein integers.
主讲人简介
孙智伟，现为南京大学数学系教授、博士生导师，数学系数学与应用数学专业主任, 中国数学会组合与图论专业委员会副主任，其研究方向为数论与组合数学。他获过多项荣誉与奖励，例如：教育部首届青年教师奖（2000）、国家杰出青年科学基金（2005-2008）与国务院政府特殊津贴（2010）。他在数论与组合、代数的交叉领域有许多创新成果, 迄今已在国外著名数学期刊《Trans. Amer. Math. Soc.（美国数学会汇刊）》等杂志上发表了两百多篇学术论文， 还著有《数论与组合中的新猜想》、《Fibonacci数与Hilbert第十问题》等书。在限定未知数个数的整数环上Hilbert第十问题方面，他保持着世界最佳记录。他还提出了许多原创性数学猜想（如1-3-5猜想），引起了国际同行的关注与研究。
