Alain Connes' research
"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
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
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
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):
"Noncommutative geometry and the Riemann zeta function"
This is Connes' expository paper on the subject matter of the course,
describing his motivations and some background.
"Trace formula in noncommutative geometry and the zeros of
the Riemann zeta function"
This is another of Connes' papers, on the subject matter of
The scientific life of mathematicians can be pictured as a trip
inside the geography of the "mathematical reality" which
they unveil gradually in their own private mental frame.
It often begins by an act of rebellion with respect to the existing
dogmatic description of that reality that one will find in existing
books. The young "to be mathematician" realize in their own
mind that their perception of the mathematical world captures some
features which do not fit with the existing dogma. This first act is
often due in most cases to ignorance but it allows one to free
oneself from the reverence to authority by relying on one's intuition
provided it is backed by actual proofs. Once mathematicians get to
really know, in an original and "personal" manner, a small
part of the mathematical world, as esoteric as it can look at first,
their trip can really start. It is of course vital not to break the
"fil d'arianne" which allows one to constantly keep a fresh
eye on whatever one will encounter along the way, and also to go back
to the source if one feels lost at times...
It is also vital to always keep moving. The risk otherwise is to confine
oneself in a relatively small area of extreme technical specialization, thus
shrinking one's perception of the mathematical world and its bewildering
The really fundamental point in that respect is that while so many
mathematicians have been spending their entire life exploring that
world they all agree on its contours and on its connexity: whatever
the origin of one's itinerary, one day or another if one walks long
enough, one is bound to reach a well known town i.e. for instance to
meet elliptic functions, modular forms, zeta functions. "All
roads lead to Rome" and the mathematical world is
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.
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