Harmonic analysis and global invariants

摘要：在数学的最新发展中，一个明显的主题是不同数学学科之间的思想交流。作为一个例子，我提到了在计算流形的拓扑性质时拓扑学和分析之间的相互作用。另一个涉及数论、拓扑学和分析的例子可以用调和分析计算的拓扑圈周期的拓扑信息从数论的角度解释L函数的主要解析行为。类似的跨学科研究一直是近年来PI研究的主要焦点。已有几个项目涉及用调和分析技术或通过几何Zeta函数的引导行为来评估或确定局部对称流形的拓扑/几何性质。这一主题在本提案中继续存在。联合项目已经开始从厄米特局部对称空间扩展到它们的“实形式”，几何Zeta函数的构造，以及用几何术语识别它们的主导行为。这项工作的一部分涉及开发关于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。预计在这项研究中将出现类似挠率的不变量，但希望能更好地理解像R-类这样的微妙特征类。PI的一个新方向是指向半单李代数中的幂零轨道。一种证明这些轨道上存在超Kahler度量的代数证明的方法正在进行中。并利用狄拉克算子实现了最小轨道的量子化。在这些研究中，来自扭曲理论的技术的使用起着重要的作用。

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.