Distinguished Lecture Series Alain Connes, Collège de France, IHES & Vanderbilt University Location Map of Campus-see BA

I will show that the first five axioms I had given in 96 on spectral triples suffice in the commutative case to characterize smooth compact manifolds. I will also define a new invariant in Riemannian geometry, which when combined with the spectrum of the Dirac operator is a complete invariant of the geometry. It is an analogue of the CKM mixing matrix of the Standard model. In the last lecture I shall describe joint work with C. Consani and M. Marcolli which shows how the inductive structure of the algebraic closure of the mysterious "field with one element" gives rise to the quantum statistical mechanical system known to give, after passing to the dual system, a spectral realization of the zeros of the Riemann zeta function, as well as a trace formula interpretation of the Riemann-Weil explicit formulas."

Born in France in 1947, Connes entered the École Normale Supérieure in Paris in 1966, and graduated in 1970. He became a researcher at the Centre National de la Recherche Scientifique where he completed his thesis on von Neumann algebras under Jacques Dixmier in 1973. He was elected to the Académie des Sciences in 1982 and he was awarded the Fields Medal in 1982. He has contributed to the theory of operator algebras, including the general classification and structure theorem of factors of type III and applications of the theory of C*-algebras to foliations and differential geometry which led to the development of the new field of non-commutative geometry. He was awarded the Crafoord Prize in 2001