X Зимняя диффеотопическая школа (Diffiety School)

1–12 февраля 2010 года, Санкт-Петербург, Академическая гимназия СПбГУ.

Организаторы:
Санкт-Петербургское отделение математического института им. В. А. Стеклова РАН,
Международный математический институт им. Эйлера.

Диффеотопия – новый динамично развивающийся раздел математики, который представляет собой своеобразный и естественный синтез многих современных математических дисциплин, таких как геометрическая теория нелинейных дифференциальных уравнений в частных производных, коммутативная и гомологическая алгебра, алгебраическая топология, алгебраическая и дифференциальная геометрия, дифференциальное исчисление в коммутативных алгебрах и других.

Курсы:

• A1. The Structure of Hamiltonian Mechanics (lecturer: A. Vinogradov, in Russian).
  We present the foundation of Hamiltonian Mechanics and its mathematical language, from the point of view given in course B1.
• A2. Geometry of Infinite-Order Jet Spaces (lecturer: L.  Vitagliano, in English).
  The course introduces the fundamentals of geometry of infinite jets spaces, and specific differential calculus over them. When dealing with this infinite-dimensional objects, differential calculus over commutative algebra allows to overcome every typical difficulty. Will be discussed infinite prolongations of systems of nonlinear PDEs, that are the simplest examples diffieties. There will be constructed secondary vector fields, which are interpreted as higher symmetries of systems of nonlinear PDEs, and is the simplest element of Secondary Calculus over diffieties, and then it will discussed horizontal differential calculus and C-spectral sequence whose first term is interpreted as secondary differential forms. Variational Calculus and the theory of conservation laws for PDEs will be presented as small parts of the C-Spectral Sequence.
• B1. Smooth Manifolds and Observables (lecturer: A. Vinogradov, in Russian).
  The course aims to show that the natural language of classical physics is differential calculus over commutative algebras and that this fact is a consequence of the classical observability mechanism. As a key example, calculus over smooth manifolds will be developed according to this philosophy, i.e., «algebraically». Hence it will be shown that differential geometry can be developed over an arbitrary commutative algebra as well.
• B2. Cohomological Theory of Integration, Distributions and Lie Algebras (lecturers: G. Moreno & M. Bächtold, in English).
  The first part of the course aims to re-establish the «old alliance» between integrands and differential forms, whose breaking led to the common belief that integration cannot be performed without measure theory. It will be shown that integration is a feature of the differential cohomology of smooth manifolds, needing only one analytical result, the Isaac Barrow's fundamental formula.
  The second part of the course begins with the algebraic (i.e., in the philosophy of course B1) definition of distributions, their geometric definition (historically appeared first) being used only to visualize properties and theorems. Then the calculus on distributions is developed. The fundamentals of symmetries, infinitesimal symmetries, and characteristics will be introduced, the Frobenius theorem, the Morse and Darboux lemmas will be proved, and the Cartan distribution in simple cases will be analyzed. The theory of Lie algebras is presented as the result of a deep insight of Sophus Lue, who was struggling to study the symmetries of PDEs, and the Lie's third theorem is proved.

Рабочие языки: английский, русский.

Окончание приема заявок на участие: 15 января 2010 г.

Контакты:
E-mail: diffiety(a)gmail.com.

Сайт школы