Noncommutative Geometry and Analytic Grothendieck Riemann Roch Theorem

在经典几何中，几何空间上的函数，就像平面上的笛卡尔坐标一样，彼此可以交换。更确切地说，坐标函数的乘法不依赖于阶数。然而，量子力学中的海森堡测不准原理表明，量子系统的两个重要的可观测量位置和动量是不对易的。这种非对易现象不仅在数学和物理中自然出现，而且在生活中无处不在。例如，在游戏中，在B之前点击按钮A可能会产生与在A之前点击B完全不同的结果。非对易几何是一门新兴的数学分支，研究“非对易空间”的几何，在这些空间中，函数，如位置和动量，彼此不对易。这个项目将探索各种有趣的非交换空间，这些空间是由算子理论，奇异空间和数学物理中的问题所激发的。主要研究者提出在非交换几何中研究几个项目。在第一个项目中，首席研究员将从数学的不同分支，包括代数几何，微分几何和微分拓扑学的想法来研究Arveson-Douglas猜想。将Grothendieck Riemann Roch定理推广到（奇异）解析簇，以确定Arveson-Douglas猜想中与本质正规Hilbert模相关的K-同调类。在非对易几何中的次不变量理论将被引入来理解与Brieskorn簇相关的希尔伯特模。第二个项目的目的是发展一个开放的叶状流形的纵向指数理论。特别是，一个真正的李群胚的循环同调将被应用作为一个关键的工具来研究这样的指标理论。第三个项目致力于研究轨道上的量子几何。受到物理学成果的启发，首席研究员打算为étale gerbes的量子几何建立对偶理论。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

In classical geometry, functions on a geometric space, like the Cartesian coordinates on a plane, commute with each other. More precisely, the multiplication of coordinate functions does not depend on the order. However, the Heisenberg uncertainty principle in quantum mechanics indicates that the position and momentum, the two important observables of a quantum system, do not commute. Such noncommutative phenomena turn out to appear naturally not only in mathematics and physics but also everywhere in life. For example, clicking button A before B in a game may produce a completely different outcome from clicking B before A. Noncommutative geometry is an emerging branch of mathematics to study the geometry of "noncommutative spaces", where the functions, like the position and momentum, do not commute with each other. This project will explore various interesting noncommutative spaces motivated by problems in operator theory, singular spaces, and mathematical physics. The principal investigator proposes to investigate several projects in noncommutative geometry. In the first project, the principal investigator will take ideas from different branches of mathematics including algebraic geometry, differential geometry, and differential topology to investigate the Arveson-Douglas conjecture. An analytic generalization of the Grothendieck Riemann Roch theorem for (singular) analytic varieties will be developed to identify the K-homology class associated to the essentially normal Hilbert module in the Arveson-Douglas conjecture. A secondary invariant theory in noncommutative geometry will be introduced to understand the Hilbert modules associated to Brieskorn varieties. The aim of the second project is to develop a longitudinal index theory for open foliated manifolds. In particular, cyclic homology of a proper Lie groupoid will be applied as a key tool to study such an index theory. The third project is devoted to the study of the quantum geometry of étale gerbes on orbifolds. Inspired by results in physics, the principal investigator intends to establish a duality theory for the quantum geometry of étale gerbes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.