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The Atiyah-Singer index formula for subelliptic operators on contact manifolds /

by 1962- Van Erp, Johannes

Abstract (Summary)
The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the naturally arising geometric operators are not elliptic, but subelliptic. A filtration on the algebra of differential operators that is adapted to these geometric structures, naturally leads to a symbolic calculus that is noncommutative, and a corresponding subelliptic theory can be developed. For such subelliptic operators we construct a symbol class in the K-theory of a noncommutative C?-algebra naturally associated to the algebra of symbols. There is a canonical map from this noncommutative K-theory to the ordinary cohomology of the manifold, which gives a class to which the Atiyah-Singer formula can be applied. In this way we define the topological index of a subelliptic operator, and we prove that it is equal to its analytic index. iii
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School:Pennsylvania State University

School Location:USA - Pennsylvania

Source Type:Master's Thesis

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