Index Theory of Hypoelliptic Fredholm Operators

本课题将探索亚椭圆算子的指数理论。主要的研究人员将应用并发展非对易几何的方法来研究与各种几何结构(尤其是接触结构和叶)有关的亚椭圆Fredholm型算子的指数理论。他相信，这些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来的现象的例子，产生了进一步发展指数理论本身的新见解，反过来，亚椭圆算子的指数公式将导致几何上的新结果。该项目的一个具体目标是开发指数理论的新应用来联系几何。20世纪60年代初，英国数学家迈克尔·阿提亚和美国数学家伊索多尔·辛格推导出了一个重要的公式。“Atiyah-Singer指数公式”在数学的两个核心分支之间建立了深刻的联系：分析(微积分学的现代版本)和拓扑学/几何学(研究高维弯曲空间)。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产生了许多应用和新的发展。例如，著名的“温斯坦猜想”的最新证明证明了指数理论的持续相关性，该猜想说(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”)都可以这样配置，即系统的集体运动在有限的时间间隔后准确地重复自己。这个定理的证明在关键的地方使用了Atiyah和Singer的公式。这个项目将指数公式的适用性扩展到了原始理论没有涵盖的一大类新问题。来自新兴的非对易几何领域的想法(受量子理论启发对几何学基础概念进行了最近的激进修订)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实质性的新联系，并开辟了指数公式的新应用领域，在过去50年里，指数公式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.