Harmonic analysis and global invariants

摘要：近年来数学发展的一个明显主题是不同数学学科之间思想的相互交融。作为一个例子，我提到拓扑和分析之间的相互作用在流形的拓扑性质的计算。另一个涉及数论、拓扑和分析的例子，可以在数论中对l函数的主要解析行为的解释中找到，该解释是根据使用调和分析计算的拓扑循环周期的拓扑信息来解释的。类似的跨学科研究一直是近年来PI研究的主要焦点。有几个项目涉及到评估或确定局部对称流形的拓扑/几何性质，或者用调和分析技术，或者通过几何zeta函数的主导行为。这一主题在本提案中继续存在。联合项目已经开始从厄米局部对称空间扩展到它们的“真实形式”，几何zeta函数的构造，以及它们在几何术语中的主导行为的识别。这项工作的一部分涉及开发有关对称空间的各种相容的精细信息，然后将其封装在几何原点的生成函数中。我们期望类扭不变量会出现在这个研究中，但是我们希望能够更好地理解微妙的特征类，比如r类。PI的一个新方向是半单李代数中的幂零轨道。对这些轨道上存在的超kahler度量的代数证明方法正在进行中。本文还开始了用狄拉克算子实现最小轨道的量子化。扭曲理论技术的运用在这些研究中发挥了重要作用。

ABSTRACT:One of the themes visible in recent developments in mathematics is the cross-fertilization of ideas across different mathematical disciplines. As an example I mention interplay between topology and analysis in the computation of topological properties of a manifold. Another example involving number theory and topology and analysis can be found in the explanation of the leading analytic behavior of L-functions from number theory in terms of topological information from periods of topological cycles computed using harmonic analysis.Similar interdisciplinary research has been the main focus of recent research of the PI. Several projects have been concerned with the evaluation or determination of topological/geometrical properties of locally symmetric manifolds either with harmonic analysis techniques or through leading behaviour of geometric zeta functions. This theme continues in the present proposal. Joint projects have been begun to extend from Hermitian locally symmetric spaces to their "real forms" the construction of geometric zeta functions, and the identification of their leading behaviour in geometric terms. Part of this work involves developing refined information concerning various compatifications of symmetric spaces and then encapsulating it in a generating function of geometric origins. It is expected that torsion-like invariants will appear in this investigation but it is hoped that a better understanding of subtle characteristic classes like the R-class may result. A new direction taken by the PI is toward nilpotent orbits in semisimple Lie algebras. An approach to an algebraic proof of the existence of hyperKahler metrics on these orbits is underway. Also a realization of the quantization of the minimal orbit by means of Dirac operators has been started. The use of techniques from twistor theory play an important role in these investigations.