Integral Quadratic Forms and Related Topics

[摘要]陈伟桥这笔拨款支持首席研究员和他的同事们对积分二次型的算术理论及相关课题的研究。他们将致力于正定二次型的表示理论，紧型单连通代数群的几乎强逼近性质，以及具有不适当自同构群的四元二次型的二次分类。这项研究涉及当变量被整数替换时，决定哪些整数可以是二次型的值所产生的数论问题。它属于数论领域，数论是数学中最古老的分支之一。如今，二次型等数论已成为通信科学、数据传输与处理、纠错码构造、信息技术密码学中不可缺少的工具。

AbstractWai Kiu ChanThis grant supports the research of the principal investigator andhis colleagues on the arithmetic theory of integral quadraticforms and related topics. They will work on the representationtheory of positive definite quadratic forms, the almost strongapproximation properties of simply connected algebraic groups ofcompact types, and the classification of quaternary quadraticforms with improper automorphism groups by their degree two thetaseries.Quadratic forms are homogenous polynomials of degree two in manyvariables. This research concerns the number theoretic problemswhich arises from deciding which integers can be values of thequadratic forms when the variables are replaced by integers. Itfalls into the area of number theory which is among the oldestbranches of mathematics. Nowadays, quadratic forms and otherkinds of number theory become indispensable tools in communicationsciences, data transmission and processing, construction of errorcorrecting codes, and cryptography in information technology.