Higher Invariants of Elliptic Operators and Applications

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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.