Index Theory of Hypoelliptic Fredholm Operators

本课题将探讨准椭圆算子的指数理论。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（主要是接触结构和叶状）相关的半椭圆Fredholm算子的指标理论。他的信念是，这些结果将在两个方向上得到回报：次椭圆算子提供了椭圆算子没有表现出来的现象的例子，产生了进一步发展指标理论本身的新见解；相反，次椭圆算子的指标公式将在几何上带来新的结果。该项目的一个具体目标是开发指标理论在接触几何中的新应用。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多尔·辛格推导出一个重要公式。“Atiyah-Singer指数公式”在数学的两个中心分支：分析（微分和积分的现代版本）和拓扑/几何（高维弯曲空间的研究）之间建立了深刻的联系。指数公式包含了几个重要的经典结果，并随后在数学和理论物理的许多领域得到了许多应用和新的发展。指数理论的持续相关性得到了证明，例如，最近著名的“温斯坦猜想”的证明，它说（在某些温和的假设下）运动物体的每一个机械排列（一个“动力系统”）都可以以这样一种方式配置，即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Atiyah和Singer的公式。这个项目将指标公式的适用性扩展到原始理论未涵盖的大量新问题。非对易几何（受量子理论启发对几何基础概念的最新激进修正）的新兴领域的思想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实质性的新联系，并为指数公式的应用开辟了一个新的领域，而指数公式在过去50年里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.