Table of contents

Contents § 1. The A-, B-, and C-cohomologies for dynamical systems § 2. The B-cohomology ring for smooth maps § 3. B-cohomology for twist maps § 4. An invariant meaning of the non-degeneracy condition

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Contents § 1. Introduction 1.1. Statement of results § 2. The algebraic density of closed trajectories and the Euler characteristic 2.1. Types of trajectories. Degree of irrationality 2.2. Stability of closed trajectories 2.3. The algebraic density of closed trajectories 2.4. The heights of critical points 2.5. The heights of cylinders of closed trajectories 2.6. The density of the Euler characteristic 2.7. The critical cycle and the mean density of the Euler characteristic § 3. Stability regions 3.1. Minimal generic components 3.2. The set U(H) is a closed interval 3.3. Essential components 3.4. k-subordination § 4. Degeneration 4.1. The simplest types of degeneration. The boundary of a stability region 4.2. The general case 4.3. The sizes of the stability regions 4.4. Rotational symmetry

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Contents Introduction § 1. Fuchsian groups and their sequential generators § 2. Geometry of Fuchsian groups § 3. Free Fuchsian groups of rank two § 4. Spaces of Fricke-Klein-Teichmüller type § 5. Moduli of Riemann surfaces § 6. The space of holomorphic morphisms of a Riemann surface § 7. Lifting Fuchsian groups to § 8. Topological classification of Arf functions and of pairs of Arf functions § 9. Topological classification of independent sets of Arf functions on compact surfaces § 10. The moduli space of spinor bundles of rank one § 11. Super Fuchsian groups, super Riemann surfaces, and their topological types § 12. Moduli of super Riemann surfaces § 13. super Fuchsian groups, super Riemann surfaces, and their topological invariants § 14. Moduli of super Riemann surfaces § 15. Superholomorphic morphisms of super Riemann surfaces

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Contents § 1. Introduction and statement of results § 2. Morse-Novikov theory § 3. Brief survey of some results of [2] § 4. Condition and homological gradient descent operator § 5. Morse-type filtrations of cobordisms § 6. Algebraic lemmas § 7. Condition , Morse-type filtrations for Morse maps , and -generic rationality of Novikov incidence coefficients § 8. Proof of the main theorem

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Contents 1. Introduction 2. Rigidity and arithmeticity problems for semisimple groups 3. Rigidity and arithmeticity problems for groups with radical 4. Conjugacy classes of finite subgroups 5. Finitely generated integer matrix groups

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Contents § 1. Introduction § 2. Proof of Theorem 1 § 3. Anderson localization of the operator H_α § 4. Exponential decay of the approximate eigenfunctions and other asymptotic relations

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Contents § 1. Introduction § 2. Algebras of Krichever-Novikov type (a) General set-up (b) Almost-graded structure (c) Central extensions and affine algebras of higher genus § 3. Representations of multi-point Krichever-Novikov algebras § 4. Moduli of curves with marked points and the general form of the KZ equations (a) Moduli spaces (b) Krichever-Novikov algebras and tangent vectors of moduli spaces (c) The formal KZ equations § 5. The case of small genus: § 6. The case of small genus: § 7. Appendix: The KN basis in the elliptic case

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