Mathematical Sciences: Arithmetic Representation by Integral Quadratic Forms

点击翻译按钮获取中文摘要

This award supports mathematical research involving quadratic and hermitian forms defined over the integers of algebraic number fields, their connections in algebraic geometry, and Fuchsian groups and the related hyperbolic geometry. There are two specific goals of this research. The first is to determine necessary and sufficient conditions for the local primitive representation of one form by another integral quadratic form. The second is to study analogues for hermitian forms of Milnor's theorem on the classification and structure of unimodular indefinite quadratic forms over the rational integers. This research falls into the general mathematical area of number theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.