Table of contents

CONTENTS Introduction Chapter I. Integrable Lagrange systems with discrete time § 1. Hamiltonian theory of Lagrange systems with discrete time § 2. Discrete analogues of classical integrable systems and factorization of matrix polynomials § 3. Spectral theory of difference operators and discrete systems Chapter II. Integrability in dynamics of general algebraic maps § 1. Commuting polynomial maps and simple Lie groups § 2. Integrable polynomial automorphisms of the plane § 3. Algebraic relations and the Yang-Baxter equation. The growth of the number of images of iterations of multivalued maps References

CONTENTS § 0. Introduction. Basic results § 1. Terminology, concepts, and notation § 2. The class of a 1-form and a Pfaffian equation § 3. Degeneracies of codimension 1 (the first encountered singularities of 1-forms and odd-dimensional Pfaffian equations) § 4. The first encountered singularities of even-dimensional Pfaffian equations (degeneracies of codimension 3) § 5. Odd-dimensional Pfaffian equations: degeneracies of codimension 3 § 6. Differential 1-forms: degeneracies of codimension ≥2 § 7. Odd-dimensional Pfaffian equations: degeneracies of codimension ≥4 § 8. Even-dimensional Pfaffian equations: degeneracies of codimension ≥4 § 9. Singularities connected with the vanishing of a 1-form § 10. The classification of submanifolds of the contact manifold § 11. The classification of first-order partial differential equations § 12. The classification of linear control systems § 13. Holomorphic classification of 1-forms and Pfaffian equations § 14. Comments References

CONTENTS Introduction 1. Singularities of Schubert cells of flag varieties 2. Cascades of projective curves 3. Stabilization of flattenings: projection, trace and duality 4. Infinitesimal transformations that preserve flattenings 5. Simple flattenings 6. Singularities of the bifurcation sets of generic three-parameter families 7. Integrable distributions on finite-dimensional manifolds 8. An integrable distribution in the space of flags of functions 9. The finite determinacy and versality theorems 10. Singularities of the bifurcation sets of the degree of oscillation 11. Bifurcations of Weierstrass points and ramification points of linear series References

CONTENTS Introduction First notebook Second notebook Third notebook Fourth notebook Fifth notebook Sixth notebook Seventh notebook Eighth notebook Ninth notebook Tenth notebook References