# Mathematics Colloquia and Seminars

We describe basic concepts of noncommutative geometry and a general construction which extends the familiar duality between ordinary spaces and commutative algebras to a duality between Quotient spaces and Noncommutative algebras. The basic tools of the theory, K-theory, Cyclic cohomology, Morita equivalence, Operator theoretic index theorems, Hopf algebra symmetry are reviewed. They cover the global aspects of noncommutative spaces, such as the transformation $\theta \rightarrow 1/\theta$ for the noncommutative torus $\Tb_{\theta}^2$ which are unseen in perturbative expansions in $\theta$ such as star or Moyal products. We discuss the foundational problem of "what is a manifold in NCG" and explain the fundamental role of Poincare duality in K-homology which is the basic reason for the spectral point of view. We show that any compact Riemannian spin manifold whose isometry group has rank $r \geq 2$ admits isospectral deformations to noncommutative geometries. We give a survey of other recent developments. First our work with H. Moscovici on the transverse geometry of foliations which yields a diffeomorphism invariant geometry on the bundle of metrics on a manifold and a natural extension of cyclic cohomology to Hopf algebras. Second, our work with D. Kreimer on renormalization and the Riemann-Hilbert problem. FROM "NOTICES of the American Mathematical Society": The Royal Swedish Academy of Sciences will award the 2001 CrafoordPrizeinmath- ematics to A1AIN CONNES of the Institut des Hautes Etudes Sdentifiques and the College de France, Paris, for his penetrating work on the theory of operator algebras and for having been a founder of noncommutative geometry. Alain Connes is counted among the world's foremost mathematicians. For his work in operator algebras, Connes received the Fields Medal in 1983. Noncom- AI .C mutative geometry is a new aln onnes field of mathematics, and Connes played a decisive role in its creation. His work has also provided pow- erful new methods for treating renormalization the- ory and the standard model of quantum and par- ticle physics. He has demonstrated that these new mathematical tools can be used for understanding and attacking the Riemann Hypothesis. Alain Connes was born in Draguignan, France, on April1, 1947. He attended the Ecole Normale Superieure in Paris during 1966- 70. Since 1979 he has held the Leon Motchane Professorship at the Institut des Hautes Etudes Sdentifiques in Bures- sur-Yvette, outside Paris, and since 1984 he has also held a professorship in analysis and geometry at the College de France in Paris. He is a member of many scientific academies, including the Academie des Sciences de Paris and the National Academy of Sciences of the U.S. The 2001 Crafoord Prize will be presented by the King of Sweden on September 26, 2001, at a ceremony at the Royal Swedish Academy of Sciences in Stockholm. The prize consists of a gold medal and US\$500,000. The Anna-Greta and Holger Crafoord Foundation was established in 1980 for promoting basic research in mathematics, astronomy, the bio- sciences (particularly ecology), the geosciences, and polyarthritis (joint rheumatism). Previous laureates in mathematics are Vladimir I. Arnold and Louis Nirenbeig (1982), Pierre Deligne and Alexandre Grothendieck (1988) (Grothendieck declined the prize), and Simon Donaldson and Shing- Tung Yau (1994). -From a Royal SWedish Academy news release