Mathematical Sciences: Integral Quadratic and Hermitian Forms

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This award supports the research in quadratic forms of Professor Donald James of the Pennsylvania State University. Dr. James's project is to study quadratic and Hermitian forms over integer rings of number fields, with special emphasis on the theory of classification, diagonalization, and representation. Except for counting, number theory, which is the study of the properties of the whole numbers, is the oldest branch of mathematics. In modern days, problems in number theory have furnished the driving force to creation of new mathematics in the fields of pure algebra, analysis, and geometry; some of the most recent and most astonishing applications of number theory have appeared in theoretical computer science and coding theory. The theory of quadratic forms is an aspect of number theory that is almost as ancient as the whole field itself, and was advanced especially far by the great nineteenth-century mathematician C.F. Gauss. This subfield of number theory retains its vitality even today, and has proved to have applications both in physics and in many branches of mathematics.