Jonathan Rosenberg

Abstract:   Part of Alain Connes' philosophy of "non-commutative geometry" involves modeling certain topological or geometric structures by non-commutative algebras. For example, given a suitable topological space X with an equivalence relation ~, instead of just forming the quotient space X/~, one can sometimes construct a non-commutative algebra (with the quotient space as its space of primitive ideals) that better captures the structure of the equivalence relation, and that can be viewed as the algebra of "functions on the quotient".

We give an illustration of this philosophy in the case of a "Z/k manifold", which is a manifold with an identification of k identical boundary components. The associated non-commutative algebra captures more of the structure than does the commutative algebra of continuous functions on the quotient space. We use this algebra to give a new proof of the Z/k index theorem of Freed and Melrose. Our proof is parallel to the proofs of many other index theorems.

Here are some directions to Northeastern University. Lake Hall and Nightingale Hall can be best accessed from the entrance on the corner of Greenleaf Street and Leon Street. The two halls are connected, with no well-defined boundary in between. In particular, 509 Lake Hall is on the same corridor as 544 Nightingale Hall.

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