Roger Penrose

John Polkinghorne

Alain Connes

Michael Heller

Gets to the heart of science by asking a fundamental question: what is the true nature of space and time?

On Space and Time

This English version of the path-breaking French book on this subject gives the definitive treatment of the revolutionary approach to measure theory, geometry, and mathematical physics developed by Alain Connes. Profusely illustrated and invitingly written, this book is ideal for anyone who wants to know what noncommutative geometry is, what it can do, or how it can be used in various areas of mathematics, quantization, and elementary particles and fields. Key Features * First full treatment of the subject and its applications * Written by the pioneer of this field * Broad applications in mathematics * Of interest across most fields * Ideal as an introduction and survey * Examples treated include: @subbul* the space of Penrose tilings * the space of leaves of a foliation * the space of irreducible unitary representations of a discrete group * the phase space in quantum mechanics * the Brillouin zone in the quantum Hall effect * A model of space time

Noncommutative Geometry

Matilde Marcolli

The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory. The two main objects of investigation are spaces where both the noncommutative and the motivic aspects come to play a role: space-time, where the guiding principle is the problem of developing a quantum theory of gravity, and the space of primes, where one can regard the Riemann Hypothesis as a long-standing problem motivating the development of new geometric tools. The book stresses the relevance of noncommutative geometry in dealing with these two spaces. The first part of the book deals with quantum field theory and the geometric structure of renormalization as a Riemann-Hilbert correspondence. It also presents a model of elementary particle physics based on noncommutative geometry. The main result is a complete derivation of the full Standard Model Lagrangian from a very simple mathematical input. Other topics covered in the first part of the book are a noncommutative geometry model of dimensional regularization and its role in anomaly computations, and a brief introduction to motives and their conjectural relation to quantum field theory. The second part of the book gives an interpretation of the Weil explicit formula as a trace formula and a spectral realization of the zeros of the Riemann zeta function. This is based on the noncommutative geometry of the adèle class space, which is also described as the space of commensurability classes of Q-lattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the space of Q-lattices, in one and two dimensions, exhibits spontaneous symmetry breaking. In the low-temperature regime, the equilibrium states of the corresponding systems are related to points of classical moduli spaces and the symmetries to the class field theory of the field of rational numbers and of imaginary quadratic fields, as well as to the automorphisms of the field of modular functions. The book ends with a set of analogies between the noncommutative geometries underlying the mathematical formulation of the Standard Model minimally coupled to gravity and the moduli spaces of Q-lattices used in the study of the zeta function.

Noncommutative Geometry, Quantum Fields and Motives

Caterina Consani

Mathematics Institute, these essays collectively provide mathematicians and physicists with a comprehensive resource on the topic.

Proceedings of the Twenty-First Meeting of the Japan-U.S. Mathematics Institute

Noncommutative Geometry, Arithmetic, and Related Topics

Joachim Cuntz

Erik G. Guentner

Nigel Higson

Jerome Kaminker

John E. Roberts

Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeper insights for instance in Foliations, K-Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subject and closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists.

Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 3-9, 2000

Institut Henri Poincaré

Institut des hautes études scientifiques (Paris, France)

Institut de mathématiques de Jussieu

The work of Alain Connes has cut a wide swath across several areas of mathematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics. Specific themes covered by the articles are as follows: entropy in operator algebras, regular $C^*$-algebras of integral domains, properly infinite $C^*$-algebras, representations of free groups and 1-cohomology, Leibniz seminorms and quantum metric spaces; von Neumann algebras, fundamental Group of $\mathrm{II}_1$ factors, subfactors and planar algebras; Baum-Connes conjecture and property T, equivariant K-homology, Hermitian K-theory; cyclic cohomology, local index formula and twisted spectral triples, tangent groupoid and the index theorem; noncommutative geometry and space-time, spectral action principle, quantum gravity, noncommutative ADHM and instantons, non-compact spectral triples of finite volume, noncommutative coordinate algebras; Hopf algebras, Vinberg algebras, renormalization and combinatorics, motivic renormalization and singularities; cyclotomy and analytic geometry over $F_1$, quantum modular forms; differential K-theory, cyclic theory and S-cohomology.

Quanta of Maths

En 1637, Descartes révolutionne la manière que l'on a de faire de la géométrie : en associant à chaque point de l'espace trois coordonnées, il pose les bases de la géométrie algébrique. Cette géométrie est dite " commutative " : le produit de deux quantités ne dépend pas de l'ordre des termes, et A × B = B × A. Cette propriété est fondamentale, l'ensemble de l'édifice mathématique en dépend. Mais au début du XXe siècle, la découverte du monde quantique vient tout bouleverser. L'espace géométrique des états d'un système microscopique, un atome par exemple, s'enrichit de nouvelles propriétés, qui ne commutent plus. Il faut donc adapter l'ensemble des outils mathématiques. Cette nouvelle géométrie, dite " non commutative ", devenue essentielle à la recherche en physique, a été développée par Alain Connes. En un texte court, vif et fascinant, ce grand mathématicien nous introduit à la poésie de sa discipline.

La géométrie et le quantique

Marcel Paul Schützenberger

André Lichnerowicz

« Les grandes découvertes scientifiques du XXe siècle telles que la relativité générale, la mécanique quantique ou le théorème de Gödel modifient en profondeur notre perception de la réalité. Les physiciens sont arrivés en un siècle à un modèle extraordinairement efficace de la réalité matérielle. Aux échelles microscopiques, la mécanique quantique résiste avec succès aux tentatives les plus ingénieuses pour la prendre en défaut. Aux échelles des galaxies, l’observation des pulsars binaires donne des confirmations éclatantes de la relativité générale d’Einstein. Le but de cet ouvrage est de permettre de franchir le décalage croissant entre les subtilités de ces modifications, appréciées des seuls spécialistes, et l’image souvent incroyablement déformée qu’en reçoit le public. Chacun des trois auteurs est l’un des sommets de ce triangle de pensées au sein duquel le lecteur est invité à trouver sa place. » Alain ConnesAlain Connes, mathématicien est professeur au Collège de France sur la chaire d’Analyse et Géométrie, membre de l’Académie des sciences et de plusieurs académies étrangères dont la National Academy of Science des États-Unis. Il a obtenu la médaille Fields en 1982. André Lichnerowicz, mathématicien, grand géomètre et physicien théoricien, spécialiste de la relativité générale était Professeur au Collège de France, membre de l’Académie des sciences depuis 1963 et Docteur honoris causa de plusieurs académies et universités étrangères. Marcel Paul Schützenberger, membre de l’Académie des sciences, a contribué de manière éclatante à la combinatoire et à la théorie des graphes. Il était un esprit pluridisciplinaire, à la fois médecin, biologiste, psychiatre, linguiste, et algébriste.

Alain Connes

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