Cycles in Locally Symmetric Spaces of Orthogonal and Unitary Type and Modular Forms

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Principal Investigator: Jens FunkeProposal Number: 0305448Institution: New Mexico State UniversityTitle: Cycles in locally symmetric spaces of orthogonal and unitary type and modular formsAbstract:This project is concerned with the utilization of the theory of dual pairs and the theta correspondence to study the connections between geometrically defined cycles in certain locally symmetric spaces and automorphic forms. One major theme of the proposal is to extend the theta lift introduced by Kudla and Millson to the full cohomology of locally symmetric spaces of orthogonal and unitary type and to cohomology with non-trivial coefficients. The other major theme is based on recent work of the PI in collaboration with Bruinier establishing a duality result between the Kudla-Millson lift and the singular theta lift first introduced by Borcherds. Based on this, the main goal is to establish (further) generalizations of the Borcherds lift with applications to arithmetic algebraic geometry and Riemannian geometry.This proposal deepens the relationship between several different areas of mathematics related to number theory, the most classical discipline in mathematics. More precisely, it involves an interaction between representation theory (the study of symmetries) on one hand and geometry on the other. Interesting in their own right, these subjects have contributed to advances in cryptography and physics, among others. Specifically, parts of this project are concerned with certain aspects of the work of Fields medalist R. Borcherds that turned out to be important in string theory. It is therefore not unreasonable to expect applications of the proposed work to theoretical physics. The collaborations outlined in this project are part of a larger network of researchers working in these kind of mathematical problems nationally and internationally. Through these collaborations, the proposed work strengthens this infrastructure, which has been effective in facilitating research progress and dissemination of results.