By Nigel Higson and John Roe, Nigel Higson; John Roe

In June 2000, the Clay arithmetic Institute equipped an educational Symposium on Noncommutative Geometry together with the AMS-IMS-SIAM Joint summer time examine convention. those occasions have been held at Mount Holyoke university in Massachusetts from June 18 to 29, 2000. the educational Symposium consisted of a number of sequence of expository lectures which have been meant to introduce key issues in noncommutative geometry to mathematicians unusual with the subject.Those expository lectures were edited and are reproduced during this quantity. The lectures of Rosenberg and Weinberger talk about a variety of purposes of noncommutative geometry to difficulties in 'ordinary' geometry and topology. The lectures of Lagarias and Tretkoff speak about the Riemann speculation and the potential program of the equipment of noncommutative geometry in quantity idea. Higson provides an account of the 'residue index theorem' of Connes and Moscovici. Noncommutative geometry is to an strange volume the production of a unmarried mathematician, Alain Connes. the current quantity provides a longer creation to a number of elements of Connes' paintings during this attention-grabbing region.

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**Additional resources for Surveys in noncommutative geometry: proceedings from the Clay Mathematics Institute Instructional Symposium, held in conjuction with the AMS-IMS-SIAM Joint Summer Research Conference on Noncommutative Geometry, June 18-29, 2000, Mount Holyoke College, Sou**

**Example text**

Then Ind D ∈ K0 (C ∗ (M, F )) agrees with a “topological index” Indtop (D) computed from the characteristic classes of σ(D), just as in the usual Atiyah-Singer index theorem. 5. 19. 19, can also be viewed as a case of a Mishchenko-Fomenko index. 15). Let (M, F ) be a compact foliated manifold and let D be the Euler characteristic operator along the leaves. ) 7Think of Z as a manifold transverse to the leaves of F, and take the “push-forward” of the class of the trivial vector bundle over Z. 4 and its corollaries is that we don’t need to assume the existence of an invariant transverse measure, which is quite a strong hypothesis.

Math. Soc. ) 6 7. (1982), no. 1, 87–89. MR 83f:58076 8. Lawrence G. Brown, Philip Green, and Marc A. Rieﬀel, Stable isomorphism and strong Morita equivalence of C ∗ -algebras, Paciﬁc J. Math. 71 (1977), no. 2, 349–363. MR 57 #3866 9. Marc Burger and Alain Valette, Idempotents in complex group rings: theorems of Zalesskii and Bass revisited, J. Lie Theory 8 (1998), no. 2, 219–228. MR 99j:16016 ´ 10. Alberto Candel, Uniformization of surface laminations, Ann. Sci. Ecole Norm. Sup. (4) 26 (1993), no.

Suppose Γ is a discrete group and e = e2 ∈ C[Γ]. Show that τ (e) must lie in Q, the algebraic closure of Q. Hint ([9]): Consider the action of Gal(C/Q) on C[Γ], as well as the positivity of τ . In fact, it is even proved in [9] that τ (e) ∈ Q, but this is much harder. 18. Let π = Z/2, a cyclic group of order 2, so that the real group C ∗ -algebra R[π] of π is isomorphic to R ⊕ R and the classifying space Bπ = RP∞ . Show that the assembly map A : KO1 (Bπ) → KO1 (R[π]) ∼ = Z/2⊕Z/2 is surjective. Hints: For the summand corresponding to the trivial representation, you don’t have 24 JONATHAN ROSENBERG to do any work, because of the commutative diagram KO∗ (pt) KO∗ (Bπ) A{1} ∼ = / KO∗ (R) Aπ / KO∗ (R ⊕ R).