Integral Quadratic Forms and Related Topics

[摘要]陈维桥（音译）：本基金资助项目负责人及其同事对积分二次型的算术理论及相关课题进行研究。他们将研究正定二次型的表示理论，紧型的单连通代数群的几乎强逼近性质，以及根据二阶级数对具有不适当自同构群的四元二次型的分类。二次型是多变量的二阶齐次多项式。本研究涉及数论问题，当变量被整数代替时，如何决定哪些整数可以是二次型的值。它属于数论领域，数论是数学最古老的分支之一。如今，二次型和其他类型的数论已成为通信科学、数据传输和处理、纠错码的构建和信息技术中的密码学中不可或缺的工具。

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.