Table of contents

Contents Introduction §1. The general Maupertuis principle §2. The Maupertuis principle in the dynamics of a massive rigid body §3. The Maupertuis principle and the explicit form of the metric generated on the sphere by a quadratic Hamiltonian on the Lie algebra of the group of motions of R³ §4. Classical cases of integrability in rigid body dynamics and the corresponding geodesic flows on the sphere §5. Integrable metrics on the torus and on the sphere §6. Conjectures §7. The complexity of integrable geodesic flows of on the sphere and on the torus §8. A rougher conjecture: the complexities of non-singularly integrable metrics on the sphere or on the torus coincide with those of the known integrable §9. The geodesic flow on an ellipsoid is topologically orbitally equivalent to the Euler integral case in the dynamics of a rigid body

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Contents Introduction §1. Preliminaries 1.1. The main results 1.2. Wiles' strategy 1.3. Representability of and the ring 1.4. Construction of a Hecke ring 1.5. The map §2. Wiles' isomorphism criterion 2.1. The invariants and 2.2. Basic properties of and 2.3. Complete intersections 2.4. Isomorphism theorems 2.5. A resolution theorem 2.6. A criterion for complete intersections 2.7. Proof of Wiles' isomorphism criterion 2.8. The relative invariant 2.9. Interpretation of the Gorenstein case §3. Interpretation of and 3.1. Interpretation of 3.2. A formula for 3.3. Connection with the Bloch-Kato conjecture §4. Proof of the inequality 4.1. Reduction to the case 4.2. Proof of the inequality when §5. Wiles' general strategy

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Contents Introduction §1. Monads and algebras §2. Functors in the category of compacta §3. Monads induced by functors in the category Comp §4. Lifting of functors to a category of -algebras and extensions of functors to the Kleisli category §5. Geometry of structure maps of -algebras

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Contents Introduction §1. Strong cohomology of dual complexes §2. Hyperhomology §3. Examples §4. Typical limit relations for Steenrod-Sitnikov homology §5. The strong homology of topological spaces §6. On the special position held by singular theory

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