"By any standards, Alain Connes is one of the world's cleverest mathematicians. Certainly many of his peers think so. Among the mathematicians working on the Riemann Hypothesis, there were few who would claim to understand Connes's particular specialism, but many who felt that if anyone was going to prove the RH, it will be him. For a man who might be considered to have, in Douglas Adam's words, 'a brain the size of a planet', he is remarkably unintimidating."

K. Sabbagh, Dr. Riemann's Zeros (Atlantic, 2002), p.203

Alain Connes' homepage

A. Connes and M. Marcolli, "From Physics to Number Theory via Noncommutative Geometry. Part I: Quantum Statistical Mechanics of Q-lattices" (preprint 04/04)

[abstract:] "This is the first installment of a paper in three parts, where we use noncommutative geometry to study the space of commensurability classes of Q-lattices and we show that the arithmetic properties of KMS states in the corresponding quantum statistical mechanical system, the theory of modular Hecke algebras, and the spectral realization of zeros of L-functions are part of a unique general picture. In this first chapter we give a complete description of the multiple phase transitions and arithmetic spontaneous symmetry breaking in dimension two. The system at zero temperature settles onto a classical Shimura variety, which parameterizes the pure phases of the system. The noncommutative space has an arithmetic structure provided by a rational subalgebra closely related to the modular Hecke algebra. The action of the symmetry group involves the formalism of superselection sectors and the full noncommutative system at positive temperature. It acts on values of the ground states at the rational elements via the Galois group of the modular field."

M. Planat explains:

"[This paper] is quite remarkable: it still generalizes the 1995 Bost and Connes paper to a more general Hamiltonian (proposition 1.17), which is the logarithmic determinant of 2 × 2 matrices associated to an integer lattice. Instead of the Riemann zeta function at temperature β, the partition function becomes a product of 2 Riemann zeta functions at β and β–1. This product appears also as a Mellin transform of the logarithm for the number of unrestricted partitions p(n), a function I used in a recent paper, which generalizes Planck theory of radiation"

The recent paper was this:

M. Planat, "Quantum 1/f noise in equilibrium: from Planck to Ramanujan", Physica A 318 (2003) 371
 

M. Marcolli and A. Connes, "From physics to number theory via noncommutative geometry. Part II: Renormalization, the Riemann–Hilbert correspondence, and motivic Galois theory", from Frontiers in Number Theory, Physics, and Geometry: On Random Matrices, Zeta Functions, and Dynamical Systems (Springer, 2006)

J. Baez, This Week's Finds in Mathematical Physics week 218 contains similarly illuminating discussion of the work of Connes and Marcolli, also framing certain issues concerning the bewildering array of zeta and L-functions in terms of category theory.

In Autumn 1998 Alain Connes offered a course at Ohio State University entitled "Noncommutative Geometry, Trace Formulas and the Zeros of the Riemann Zeta Function"

Video of lectures 2–9 is available here (in .rm format): 2 3 4 5 6 7 8 9

This is Connes' expository paper on the subject matter of the course, describing his motivations and some background.

This is another of Connes' papers, on the subject matter of this course.

In other words there is just "one" mathematical world, whose exploration is the task of all mathematicians, and they are all in the same boat somehow.

Alain Connes