Table of contents

CONTENTS Introduction §1. Definition of a conformal structure on a differential manifold and the construction of the invariant conformal connection §2. Conformal structure on a tangentially non-degenerate hypersurface of projective space §3. Four-dimensional pseudoconformal structure of signature (2, 2) and its completely isotropic two-dimensional submanifolds §4. Asymptotic pseudoconformal structure on a four-dimensional hypersurface and its completely isotropic two-dimensional submanifolds References

CONTENTS Introduction Chapter I. Group approach §1. Two-dimensional quasicrystallographic groups 1.1. Finite generation of two-dimensional quasicrystallographic groups 1.2. Examples of two-dimensional quasicrystallographic groups with infinite point group. Classification of admissible rotation angles 1.3. Classification of two-dimensional quasicrystallographic groups with finite point group §2. Three-dimensional and multidimensional quasicrystallographic groups with finite point group 2.1. The abstract construction of quasicrystallographic groups with finite point group 2.2. Classification of admissible representations in certain special cases §3. Connection with other approaches and definitions 3.1. Connection between quasiperiodic tilings and quasicrystallographic groups 3.2. Connection with projections of multidimensional crystals 3.3. Connection with the Mermin-Rokhsar-Wright phase multipliers 3.4. Connection with the generalized Mermin-Rokhsar-Wright phase multipliers. The case of an infinite point group §4. Quasicrystallographic groups with infinite point group 4.1. Complete reducibility and pseudo-orthogonality of the action of the point group on a quasilattice 4.2. Classification of admissible rotation angles in the case of a quadratic quasilattice 4.3. Some examples 4.4. Questions connected with finite generation Chapter II. The geometry of local rules §1. Definitions and notation §2. Quasiperiodicity and local rules 2.1. Projection method and quasiperiodicity 2.2. An example of quasiperiodic local rules 2.3. The section method 2.4. The construction of local rules §3. The geometry of the forbidden set 3.1. The topology of complements of 2-planes §4. Multidimensional case §5. An example of local rules Acknowledgements Appendices 1. Tilings with -symmetry 2. Generalization to the case References

CONTENTS Introduction § 1. The space of partial maps § 2. Basic properties of solution spaces § 3. Convergent sequences of solution spaces § 4. R(U) as a topological space § 5. The equicontinuity condition § 6. Continuous dependence of solutions on the parameters in the right-hand side § 7. The position of the superior limit of a sequence of spaces § 8. On condition (n) § 9. Gluing spaces together §10. Additional remarks §11. Autonomous spaces and spaces close to them §12. Periodic spaces §13. First note on asymptotic integration §14. Second note on asymptotic integration References