Higher Invariants of Elliptic Operators and Applications

流形是模拟宇宙的几何物体。流形的几何结构和拓扑结构通常由一定的自然微分方程决定。数学中最伟大的发现之一是Fredholm指数，它测量了微分方程解空间的大小。弗雷德霍尔姆指数的美妙之处在于，它不会在微小的扰动下发生变化。当流形紧化时，著名的Atiyah-Singer指标定理计算这些微分方程的Fredholm指标。这个定理在几何、拓扑和数学物理中有许多应用。如果考虑到对称性，那么微分方程有一个更精细的不变量叫做高指标。当高指数消失时，一个新的微妙的高次级不变量自然出现。这些新的不变量在数学和物理中有着强大的应用。PI和他的学生计划研究这些不变量及其应用。该项目将通过指导研究生和博士后，为美国劳动力发展做出贡献。微分方程或算子的高不变量存在于非交换几何中某些几何算子代数的k理论中。计算这些较高的不变量仍然是一个主要的挑战。部分潜在的困难来自于缺乏对群体的理解，这在这里扮演着对称性的角色。PI和他的学生打算设计新的方法来计算这些高不变量，并应用这些计算来解决分析、微分几何、拓扑和数学物理中的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Manifolds are geometric objects modeling the universe. Geometry and topology of manifolds are often governed by certain natural differential equations. One of the greatest discovery in mathematics is the Fredholm index, which measures the size of solution space for such differential equations. The beauty of the Fredholm index is that it would not change under small perturbations. The famous Atiyah-Singer index theorem computes the Fredholm index of these differential equations when the manifold is compact. This theorem has numerous applications to geometry, topology, and mathematical physics. If one takes symmetries into consideration, then there is a much more refined invariant of the differential equation called the higher index. When the higher index vanishes, a new subtle higher secondary invariant naturally arises. These new invariants have powerful applications to mathematics and physics. The PI and his students plan to study these invariants and their applications. This project will contribute to US workforce development through mentoring of graduate and post-doctoral students.The higher invariants of differential equations or operators live in K-theory of certain geometric operator algebras in the context of noncommutative geometry. It remains a major challenge to compute these higher invariants. Part of the underlying difficulty comes from the lack of understanding for the universe of groups, which plays the role of symmetries here. The PI and his students intend to devise new methods to compute these higher invariants and apply these computations to solve problems in analysis, differential geometry, topology, and mathematical physics.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.