Index Theory of Hypoelliptic Fredholm Operators

亚椭圆Fredholm算子的指数论

• 批准号: 1244976
• 负责人: Johannes van Erp
• 金额: $ 8.74万
• 依托单位: University of Hawaii
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2014-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1244976&HistoricalAwards=false
• 关键词: Index; Theory; Hypoelliptic; Fredholm; Operators

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.

该项目将探索亚椭圆算子的指数理论。首席研究员将应用和开发非交换几何方法来研究与各种几何结构（最显着的是接触结构和叶状结构）相关的亚椭圆 Fredholm 算子的指数理论。他相信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未表现出的现象的例子，产生了新的见解，进一步发展了指数理论本身；相反，亚椭圆算子的指数公式将带来几何学的新结果。该项目的一个具体目标是开发索引理论在接触几何中的新应用。 20 世纪 60 年代初，英国数学家 Michael Atiyah 和美国数学家 Isodore Singer 推导出了一个重要的公式。 “Atiyah-Singer 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（微分和积分的现代版本）和拓扑/几何（高维弯曲空间的研究）。该指数公式包含了几个重要的经典结果，随后在数学和理论物理的许多领域产生了众多的应用和新的发展。例如，最近著名的“韦恩斯坦猜想”的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）运动物体的每个机械排列（“动力系统”）都可以以这样的方式配置，即系统的集体运动将在有限的时间间隔后精确地重复自身。该定理的证明在关键地方使用了 Atiyah 和 Singer 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。来自非交换几何新兴领域的思想（受量子理论启发对几何基本概念的最新彻底修订）在这项工作中发挥着核心作用。该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，该公式在过去五十年中对数学的发展至关重要。