Harmonic analysis and global invariants

摘要：在数学的最新发展中，一个显而易见的主题是不同数学学科之间的思想交叉。作为一个例子，我提到拓扑和分析之间的相互作用，在计算拓扑性质的流形。另一个涉及数论、拓扑学和分析的例子可以在从数论中解释L-函数的主要分析行为中找到，这是根据使用调和分析计算的拓扑循环周期的拓扑信息。类似的跨学科研究一直是PI最近研究的主要焦点。几个项目已经关注的评价或局部对称流形的拓扑/几何性质的测定无论是与调和分析技术或通过几何zeta函数的领先行为。本提案继续讨论这一主题。联合项目已经开始从厄米特局部对称空间扩展到其“真实的形式”的几何zeta函数的建设，并确定其领先的行为在几何方面。这项工作的一部分涉及开发有关对称空间的各种兼容性的精细信息，然后将其封装在几何起源的生成函数中。预计扭样不变量将出现在这次调查中，但它希望能更好地了解微妙的特征类，如R-类可能会导致。 PI所采取的一个新方向是半单李代数中的幂零轨道。一种方法的代数证明存在的hyperKahler度量这些轨道正在进行中。此外，实现量子化的最小轨道的狄拉克运营商的手段已经开始。扭量理论的技术在这些研究中起着重要的作用。

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.