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Math Graduate Student Seminar

Friday, May 17, 2019
12:00pm to 1:00pm
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Linde Hall 255
A twisted local index formula for curved noncommutative two tori
Jim Tao, Department of Mathematics, Caltech,

We consider the Dirac operator of a general metric in the canonical conformal class on the noncommutative two torus, twisted by an idempotent (representing the K-theory class of a general noncommutative vector bundle), and derive a local formula for the Fredholm index of the twisted Dirac operator. Our approach is based on the McKean-Singer index formula, and explicit heat expansion calculations by making use of Connes' pseudodifferential calculus. As a technical tool, a new rearrangement lemma is proved to handle challenges posed by the noncommutativity of the algebra and the presence of an idempotent in the calculations in addition to a conformal factor.

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