This award supports the research in automorphic forms of Professor Don Blasius of the University of California at Los Angeles. Dr. Blasius's project is to complete the construction of Galois representations for Maass forms of Galois type by extending to coherent cohomology certain methods of interpre- tation that have been established for holomorphic forms; to stabilize the Trace Formula for GSp(4), beginning with the Fundamental Lemma; and to study certain lifting problems involving Abelian varieties, in the setting of Shimura varieties, applying his work on p-adic properties of Hodge classes. 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.
