Please join us for the Coxeter Lecture Series with Professor Alain Connes.

Lecture Series: From rings of operators to noncommutative geometry

Abstract: I will first explain the origin of noncommutative geometry starting form Rings of Operators and the classification of factors in my thesis in 1972 where I reduced type III factors to type II and automorphisms and the invariants deduced from the uniqueness of the modular time evolution. The construction of the von Neumann algebra of a foliation (1976) showed the relevance of operator algebras in differential geometry and begged for an extension of classical geometric concepts to the new spaces such as the space of leaves of foliations, the key concepts such as cyclic homology were developed from the beginning of the 80's.

After this general introduction the lectures will focus on two fundamental examples of geometric spaces on which the new theory (NCG) brings a new light. The first is space-time and I will explain briefly the impact of our joint works with A. Chamseddine, S. Mukhanov and W. van Suijlekom. The new geometric paradigm of spectral triples, allows one to express the very elaborate Lagrangian given by gravity coupled with the Standard Model, with all its subtleties (V-A, BEH, seesaw, etc etc...) as pure gravity on a geometric space-time whose texture is slightly more elaborate than the 4-dimensional continuum and is dictated by the economy of presentation provided by a small amount of noncommutativity.

The action functional is given by the entropy of the Fermions and this singles out the test function to be used in the spectral action. This specific test function is intimately related to the Riemann zeta function. In turns, this leads us to the second  fundamental space for which NCG gives an unexpected new light: the space Spec Z of prime numbers. The key role of the BC system as exhibiting the thermodynamics of noncommutative spaces, will lead to the Hasse-Weil interpretation of the Riemann zeta function as a counting function of the fixed points of the Frobenius action on the adele class space found in our joint work with K. Consani. The second lecture will explain how this leads to absolute algebraic geometry based on Segal's Gamma rings and will then be illustrated in the talk of K. Consani on our Riemann-Roch formula for the ring Z of integers. The third lecture will concentrate on the recent  advances in the program of the operator theoretic approach to the Riemann Hypothesis.

I will present two recent discoveries on the spectral realization of the zeros of the Riemann zeta function:

On the one hand, joint results with K. Consani show that the semilocal trace formula gives a conceptual explanation of Weil's positivity for the archimedean place, and shows that one can access to the infrared part of the zeros of the Riemann zeta function using  the scaling operator with periodic boundary conditions, restricted to the orthogonal of the radical of the Weil quadratic form.

On the other hand, the unusual ultraviolet behavior of the zeros of the Riemann zeta function has found an unexpected spectral incarnation in our joint work with H. Moscovici in terms of the negative eigenvalues of the  classical prolate wave operator extended to the whole real line.

These presentations will be a preparation to the talk of H. Moscovici on our latest paper (in collaboration with him and with K. Consani) which provides a unified framework which integrates the above recent discoveries.

You can find all lecture recordings here: https://www.youtube.com/playlist?list=PLArBKNfJxuunVAOYrHhO5ZQDKMEo6wNBX