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

主要研究人员：Jens Funke命题编号：0305448机构：新墨西哥州立大学题目：局部对称空间中的圈和模形式摘要：利用对偶理论和theta对应来研究某些局部对称空间中几何定义的圈与自同构形式之间的关系。该建议的一个主要主题是将Kudla和Millson引入的theta提升推广到局部对称空间的完全上同调，以及具有非平凡系数的上同调。另一个主要主题是基于PI与Bruinier合作的最新工作，在Kudla-Millson Lift和BorCherds最先引入的奇异theta Lift之间建立了二元性结果。在此基础上，主要目的是建立(进一步)推广的Borcherds提升及其在算术代数几何和黎曼几何中的应用。这一建议加深了与数论有关的几个不同数学领域之间的关系，数论是数学中最经典的学科。更准确地说，它涉及到表示理论(对称性研究)和几何学之间的相互作用。这些学科本身很有趣，为密码学和物理学等领域的进步做出了贡献。具体地说，这个项目的一部分与菲尔兹奖牌获得者R.博尔切尔德的工作的某些方面有关，这些方面在弦理论中被证明是重要的。因此，期望将拟议的工作应用于理论物理并不是不合理的。这个项目中概述的合作是一个更大的研究网络的一部分，这些研究人员在国内和国际上研究这类数学问题。通过这些合作，拟议的工作加强了这一基础设施，有效地促进了研究进展和成果的传播。

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.