理学院数理讲坛（2016年第三十一讲）
报告题目：Refining Lagrange's Four-Square Theorem
报告专家：孙智伟教授（南京大学）
报告时间：6月12日上午10：00—11：00
报告地点：阳明学院 303
报告摘要：Lagrange's four-square theorem asserts that any natural number can be written as x^2+y^2+z^2+w^2 with x,y,z,w integers. This classical result is very famous and it has several different proofs. The speaker recently found that Lagrange's theorem can be further refined by requiring additionally that P(x,y,z,w) is a square (or a cube), where P(x,y,z,w) is a suitable polynomial with integer coefficients. We have proved several results along this line, for example, the polynomials x+y+z,x+y+2z and x^2y^2+y^2z^2+z^2x^2 are okay for our purpose. Moreover, we have formulated lots of surprising conjectures on this topic; for example, we conjecture that one may take P(x,y,z,w)=x+3y+5z with x,y,z nonnegative. A quite mysterious conjecture of the speaker asserts that any natural number can be written as x^2+y^2+z^2+w^2 with x,y,z,w nonnegative integers such that (10w+5x)^2+(12y+36z)^2 is a square. This reveals a surprising connection between Lagrange's theorem and Pythagorean triples. In this talk we will tell the story of such discoveries as well as related new results on partitions of integers motivated by our refinements of Lagrange's theorem.
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, 其研究方向为数论与组合。获得多项荣誉与奖励，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。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。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）在专著或论文中所引用。
欢迎感兴趣的老师和学生参加！