Number Theory and Physics

1. Front Matter

   Pages I-XIII

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2. Conformally Invariant Field Theories, Integrability, Quantum Groups

   1. Front Matter

      Pages 1-1

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   2. Z/NZ Conformal Field Theories

      • P. Degiovanni

      Pages 2-19

   3. Affine Characters and Modular Transformations

      • M. Bauer, C. Itzykson

      Pages 20-32

   4. Conformal Field Theory on a Riemann Surface

      • H. Ooguri

      Pages 33-43

   5. Modular Invariance of Field Theories and String Compactifications

      • A. Shapere

      Pages 44-55

   6. Yang-Baxter Algebras and Quantum Groups

      • H. J. de Vega

      Pages 56-67

   7. Representations of Uq sl(2) for q a Root of Unity

      • H. Saleur

      Pages 68-76

3. Quasicrystals and Related Geometrical Structures

   1. Front Matter

      Pages 77-77

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   2. Some Quasiperiodic Tilings as Modulated Lattices

      • M. Duneau, C. Oguey

      Pages 78-85

   3. Types of Order and Diffraction Spectra for Tilings of the Line

      • C. Godrèche

      Pages 86-99

   4. A Topological Constraint on the Atomic Structure of Quasicrystals

      • A. Katz

      Pages 100-110

   5. From Approximants to Quasicrystals: A Non-standard Approach

      • R. Mosseri, F. Bailly

      Pages 111-117

   6. The Topological Structure of Grain Boundaries

      • N. Rivier

      Pages 118-127

   7. Calculation of 6D Atomic Surfaces from a Given Approximant Crystalline Structure Using Approximate Icosahedral Periodic Tilings

      • J.-L. Verger-Gaugry

      Pages 128-137

4. Spectral Problems, Automata and Substitutions

   1. Front Matter

      Pages 139-139

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   2. Spectral Properties of Schrödinger’s Operator with a Thue-Morse Potential

      • J. Bellissard

      Pages 140-150

   3. On the Non-commutative Torus of Real Dimension Two

      • P. Cohen

      Pages 151-156

   4. Topological Invariants Associated with Quasi-Periodic Quantum Hamiltonians

      • H. Kunz

      Pages 157-161

   5. Spectra of Some Almost Periodic Operators

      • A. Sütö

      Pages 162-169

7 Les Houches Number theory, or arithmetic, sometimes referred to as the queen of mathematics, is often considered as the purest branch of mathematics. It also has the false repu­ tation of being without any application to other areas of knowledge. Nevertheless, throughout their history, physical and natural sciences have experienced numerous unexpected relationships to number theory. The book entitled Number Theory in Science and Communication, by M.R. Schroeder (Springer Series in Information Sciences, Vol. 7, 1984) provides plenty of examples of cross-fertilization between number theory and a large variety of scientific topics. The most recent developments of theoretical physics have involved more and more questions related to number theory, and in an increasingly direct way. This new trend is especially visible in two broad families of physical problems. The first class, dynamical systems and quasiperiodicity, includes classical and quantum chaos, the stability of orbits in dynamical systems, K.A.M. theory, and problems with "small denominators", as well as the study of incommensurate structures, aperiodic tilings, and quasicrystals. The second class, which includes the string theory of fundamental interactions, completely integrable models, and conformally invariant two-dimensional field theories, seems to involve modular forms and p­ adic numbers in a remarkable way.

• Riemann zeta function
• algebraic number theory
• arithmetic
• crystal
• dynamical systems
• number theory
• quasicrystal
• theoretical physics
• zeta function

• Service de Physique Théorique, C.E.N Saclay, Gif-sur-Yvette Cedex, France

  Jean-Marc Luck, Pierre Moussa

• Institut Henri-Poincaré, Problèmes Diophantiens, Université de Paris 6, Paris Cedex 05, France

  Michel Waldschmidt

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