Mathematical Sciences: Arithmetic of L-Values

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This award supports the mathematical research of Professor Glenn Stevens of Boston University. Dr. Stevens's project is concerned with special values of L-functions. These functions are number-theoretic constructs that are studied by methods from complex analysis. Dr. Stevens plans to embed the theory of Shintani decompositions for totally real number fields of degree n into the geometry of the Borel-Serre completion of the symmetric space of GL(n), and to use the extra structure afforded by this geometry and the action of GL(n,Q) on it, to study the arithmetic properties of special values of partial zeta functions. Non-Euclidean plane geometry began in the early nineteenth century as a mathematical curiosity, but by the end of that century, mathematicians had realized that many objects of fundamental importance are non-Euclidean in their basic nature. The detailed study of non-Euclidean plane geometries has given rise to several branches of modern mathematics, of which the study of Modular and Automorphic Forms is one of the most active. This field is principally concerned with questions about the whole numbers, but in its use of Geometry and Analysis, it retains connection to its historical roots.