Mathematical Sciences: K-Theory, C*-Algebras and Index Theory for Elliptic Operators

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The central mathematical concept in this project is that of index. In its most basic manifestation, index has to do with the number of extra conditions that must be imposed on a solution to a differential equation in order to specify it unambiguously. A famous theorem of Atiyah and Singer asserts that this sort of index is given in the simplest case by a winding number (counting the number of times a path in the plane winds around the origin) and more generally by an analogous higher-dimensional topological invariant. The Atiyah-Singer Theorem and its numerous progeny all have the form: lefthand side (index of a differential operator) equals righthand side (something you can compute from the ingredients that go into the operator). Progress consists in achieving a more and more sophisticated understanding of what the righthand side really is. Nowadays, this involves geometry, topology, and functional analysis. The most advanced point of view, represented by Professors Kaminker and Ji among others, computes index in the K-theory of an appropriate algebra of operators. Classically, this would be commutative, an algebra of functions on the underlying manifold, but in the presence of additional structure a more complicated algebra is called for. The investigators will seek to understand the algebra and its K-theory better, and to extract new invariants for differential operators from elements of the K- groups.