Mathematical Sciences: Cohomology of Arithmetic Groups

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This award supports the research in number theory, arithmetic groups, and automorphic forms of Professor Avner Ash of Ohio State University. Dr. Ash's project centers on the cohomology of the group of n-by-n integral matrices with determinant 1, and will follow two main paths. These are the study of the structure of the cohomology as a module for the Hecke algebra, and computer-aided calculation of cohomology groups in the case where n is 3. 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.