This volume contains the proceedings of the conference "Casimir Force, Casimir Operators and the Riemann Hypothesis - Mathematics for Innovation in Industry and Science" held in November 2009 in Fukuoka (Japan). The motive for the conference was the celebration of the 100th birthday of Casimir and the 150th birthday of the Riemann hypothesis. The conference focused on the following topics: Casimir operators in harmonic analysis and representation theory Number theory, in particular zeta functions and cryptography Casimir force in physics and its relation with nano-science Mathematical biology Importance of mathematics for innovation in industry The latter topic was inspired both by the call for innovation in industry worldwide and by the fact that Casimir, who was the director of Philips research for a long time in his career, had an outspoken opinion on the importance of fundamental science for industry. These proceedings are of interest both to research mathematicians and to those interested in the role science, and in particular mathematics, can play in innovation in industry.

"Visions in Mathematics - Towards 2000" was one of the most remarkable mathematical meetings in recent years. It was held in Tel Aviv from August 25th to September 3rd, 1999, and united some of the leading mathematicians worldwide. The goals of the conference were to discuss the importance, the methods, the past and the future of mathematics as we enter the 21st century and to consider the connection between mathematics and related areas.

The aims of the conference are reflected in the present set of survey articles, documenting the state of art and future prospects in many branches of mathematics of current interest.

This is the second part of a two-volume set that will serve any research mathematician or advanced student as an overview and guideline through the multifaceted body of mathematical research in the present and near future.

"Visions in Mathematics - Towards 2000" was one of the most remarkable mathematical meetings in recent years. It was held in Tel Aviv from August 25th to September 3rd, 1999, and united some of the leading mathematicians worldwide. The goals of the conference were to discuss the importance, the methods, the past and the future of mathematics as we enter the 21st century and to consider the connection between mathematics and related areas.

The aims of the conference are reflected in the present set of survey articles, documenting the state of art and future prospects in many branches of mathematics of current interest.

This is the first part of a two-volume set that will serve any research mathematician or advanced student as an overview and guideline through the multifaceted body of mathematical research in the present and near future.

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.

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

Die Mathematiker leben im allgemeinen gut mit den Biologen zusammen. Aber sie reden wenig miteinander. Ihre Erkenntnisse und Beweggriinde sind so weit voneinander entfernt, daB ein Dialog unmoglich scheint. Jedoch kann ein betrachtlicher Gewinn erzielt werden. Niemand wird bestreiten, daB man die Mathematik mit dem Gehirn betreibt. Jedoch ist es bisher keiner yom Menschen konstruierten Maschine gelungen, die verniinftigen und erfinderi- schen Fahigkeiten unserer zerebralen Maschine nachzumachen. Kommt man eines Tages dahin? Kann eine echte kiinstliche Intelligenz aus der unbelebten Materie entstehen? Dies ist die zentrale Frage dieses Buches. Bevor man diese Frage beantworten kann, muB man die Mathematik definieren. Was ist die Natur der mathematischen Objekte? Existieren diese unabhangig yom Gehirn des Menschen, der sie entdeckt? Oder sind sie im Gegenteil nur das Produkt der Gehirnaktivitat, die sie konstruiert? Die jiing- sten Entwicklungen der Neurowissenschaften, der Wissenschaft yom Nerven- system, liefern neue Gesichtspunkte zu einem Fragekomplex, der schon in den Dialogen von Plato behandelt wurde. Die Mathematik ist in Paris, in Moskau und in San Francisco die gleiche.

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 adele 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.

This book gets to the heart of science by asking a fundamental question about its essence: what is the true nature of space and time? Both defy modern physics, and scientists find themselves continually searching for answers. This unique volume brings together world leaders in cosmology, particle physics, quantum gravity, mathematics, philosophy and theology, to provide fresh insights into the deep structure of space and time. In an attempt to understand the question, subjects ranging from dark matter to the philosophical and theological implications of spacetime are covered, ensuring that the issue is thoroughly explored. Interesting and thought-provoking answers provide a well-rounded read.