Higher Invariants of Elliptic Operators and Applications

流形是模拟宇宙的几何对象。流形的几何学和拓扑学通常由某些自然微分方程所支配。数学中最伟大的发现之一是Fredholm指数，它测量了此类微分方程的解空间的大小。Fredholm指数的美妙之处在于它不会在小扰动下发生变化。著名的Atiyah-Singer指标定理计算这些微分方程的Fredholm指标时，流形是紧的。这个定理在几何学、拓扑学和数学物理学中有许多应用。如果考虑对称性，则微分方程有一个更精细的不变量，称为高阶指数。当高阶指数消失时，一个新的微妙的高阶次不变量自然产生。这些新的不变量在数学和物理学中有很强的应用。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学生为美国劳动力发展做出贡献。微分方程或算子的高级不变量存在于非交换几何背景下某些几何算子代数的K-理论中。计算这些更高的不变量仍然是一个重大挑战。部分潜在的困难来自于缺乏对群体宇宙的理解，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更高的不变量，并将这些计算应用于解决分析，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。

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.