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## Index theory and noncommutative geometry

### Alan Paterson

University of Mississippi

###
Thursday, May 26, 2005

L02 Carson Hall, 4 pm

Tea 3:30 pm, Math Lounge

**Abstract: ** The Atiyah--Singer index theorem associates with
an elliptic differential operator two numbers, one defined by analytic
means (the analytic index) and the other defined by topological means
(the topological index), and shows (e.g., via *K*-theory) that the two
numbers are the same. The talk will begin by illustrating the index
theorem with the Gauss--Bonnet theorem, which corresponds to the case
of the de Rham operator. The original index theorem works within a
commutative framework (e.g., *C(X)*) but Alain Connes showed that many
naturally ocurring situations require developing index theory in the
non-commutative context (using *C^**-algebras and groupoids). In
particular, he proved an index theorem for measured foliations. A
good example of such a foliation is the Kronecker foliation of the
torus; this involves an irrational rotation *C^**-algebra and the leaf
equivalence relation groupoid. We conclude the talk by discussing
briefly a general index theorem formulated within the framework of
groupoid actions and the *KK*-theory of *C^**-algebras.

This talk will be accessible to graduate students.