Topology of integrable Hamiltonian systems - учебный курс | ИСТИНА – Интеллектуальная Система Тематического Исследования НАукометрических данных

Автор: Ошемков А.А.
Год создания: 2024
Организация: МГУ имени М.В. Ломоносова
Курс на иностранном языке
Описание: This course is devoted to the presentation of the basic concepts, ideas and methods necessary to study the topology of integrable systems, in particular singularities of integrable Hamiltonian systems. By Liouville theorem, it is the structure of singularities of an integrable Hamiltonian system determines its topological properties and makes it possible to investigate its topological invariants. We will discuss the local classification of singularities of integrable Hamiltonian systems, the semi-local structure of non-degenerate singularities (i.e., their structure in a neighborhood of a singular fiber), as well as some modern results on the global properties of integrable systems reflecting their behavior in the large (i.e., on the whole phase space). Course Plan: 1. Symplectic manifolds. Poisson brackets. Liouville integrable systems. 2. Moment map. Liouville foliation. Non-degenerate singularities. Williamson theorem. 3. Local classification of non-degenerate singularities. Eliasson theorem. 4. Basic semilocal singularities. Notion of atoms. 5. Almost direct products. Zung theorem on semilocal description of singularities. 6. Singularities of corank 1. The case of two degrees of freedom. 3-Atoms. 7. Rough Liouville equivalence. Fomenko-Zieschang theorem. 8. Examples of singularities in classical integrable systems. 9. Hamiltonian torus action. Atiyah-Guillemin-Sternberg theorem. Delzant polytopes. 10. Homological properties of the complex of singularities of an integrable Hamiltonian system. 11. Smooth and symplectic invariants of singularities.
Добавил в систему: Ошемков Андрей Александрович

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Topology of integrable Hamiltonian systemsучебный курс

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