【摘要】Due to the efforts of Fermat, Euler, Legendre and Gauss, it is known what natural numbers can be written as the sum of two squares or three squares. Lagrange's four-square theorem proved in 1770 states that each natural number can be expressed as the sum of four squares.In the talk we will first review classical results on sums of two or three or four squares. Then we turn to the speaker's recent discoveries which refine Lagrange's four-square theorem or the Gauss-Legendre theorem on sums of three squares. In particular we will introduce our results refining Lagrange's four-square theorem as well as recent progress on the speaker's 1-3-5 conjecture which states that each natural number can be written as x^2+y^2+z^2+w^2 with x,y,z,w nonnegative integers such that x+3y+5z is a square.

【报告人简介】孙智伟，1965年10月生。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, 其研究方向为组合数论。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(2011)。他是《Journal of Combinatorics and Number Theory（组合与数论杂志）》的创刊主编(2009年至今), 《中国数学会通讯》现任编委（2016-）。曾多次应邀去美国、欧洲、香港、台湾等地访问讲学或担任客座教授。他在数论与组合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SCI杂志上发表了一百多篇学术论文。其工作被一些著名数学家（如 Fields奖获得者T. Tao与著名组合学家N. Alon）在专著或论文中引用。他还提出了许多原创性数学猜想，引起一些国际著名数学家（如Fields奖获得者J. Bourgain）的关注与研究。