Index Theory of Hypoelliptic Fredholm Operators

本计画将探讨次椭圆算子的指标理论。主要研究者将应用和发展非交换几何的方法来研究与各种几何结构（最显着的接触结构和叶理）相关的次椭圆Fredholm算子的指数理论。他的信念是，结果将在两个方向上得到回报：亚椭圆算子提供了椭圆算子没有表现出的现象的例子，产生了新的见解，进一步发展指数理论本身，相反，亚椭圆算子的指数公式将导致几何学的新结果。该项目的一个具体目标是发展指数理论在接触几何中的新应用。在20世纪60年代初，英国数学家迈克尔·阿蒂亚和美国数学家伊索·辛格推导出一个重要的公式。“阿蒂亚-辛格指数公式”在数学的两个中心分支之间建立了深刻的联系：分析（微积分的现代版本）和拓扑/几何（高维弯曲空间的研究）。指数公式包含了几个重要的经典结果，并随后导致了许多应用和新的发展，在许多领域的数学以及理论物理。指数理论的持续相关性得到了证明，例如，著名的“韦恩斯坦猜想”的最新证明，该猜想认为（在某些温和的假设下）运动物体的每一种机械排列（“动力系统”）可以配置为这样的方式：系统的集体运动将在有限的时间间隔后精确地重复自己。这个定理的证明在关键的地方使用了阿提亚和辛格的公式。该项目将指数公式的适用性扩展到原始理论未涵盖的一大类新问题。来自非对易几何新兴领域的思想（受量子理论启发，对几何基础概念进行了最近的彻底修订）在这项工作中发挥了核心作用。这个项目在纯数学的不同分支之间建立了大量的新联系，并为指数公式的应用开辟了一个新的领域，指数公式在过去50年中对数学的发展至关重要。

This project will explore the index theory of hypoelliptic operators. The principal investigator will apply and develop methods of noncommutative geometry to study the index theory of hypoelliptic Fredholm operators associated to various geometric structures (most notably contact structures and foliations). His belief is that the results will pay off in two directions: hypoelliptic operators provide examples of phenomena not exhibited by elliptic operators, yielding new insights that further the development of index theory itself, and, conversely, index formulas of hypoelliptic operators will lead to new results in geometry. A specific aim of the project is to develop new applications of index theory to contact geometry.In the early 1960s the British mathematician Michael Atiyah and the American mathematician Isodore Singer derived an important formula. The "Atiyah-Singer index formula" established a deep connection between two central branches of mathematics: analysis (the modern version of differential and integral calculus) and topology/geometry (the study of higher dimensional curved space). The index formula subsumed several important classical results and has subsequently led to numerous applications and new developments in many areas of mathematics as well as in theoretical physics. The continued relevance of index theory is evidenced, for example, by the very recent proof of the celebrated "Weinstein conjecture," which says that (under certain mild hypotheses) every mechanical arrangement of moving objects (a "dynamical system") can be configured in such a way that the collective motion of the system will exactly repeat itself after a finite time-interval. The proof of this theorem makes use, in crucial places, of the formula of Atiyah and Singer. This project extends the applicability of index formulas to a large new class of problems that were not covered by the original theory. Ideas from the emerging area of noncommutative geometry (a recent and radical revision of the foundation concepts of geometry inspired by quantum theory) play a central role in this work. This project establishes substantial new connections between separate branches of pure mathematics, and it opens up a new area of application of the index formula that has been crucial to the development of mathematics for the last fifty years.