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X Winter Diffiety School

St. Petersburg
February 1-12, 2010.

Lectures took place in the "Academic Gymnasium" of the St. Peterburg State University, пер. Каховского, 9, 199155, Санкт-Петербург, RUSSIA (tel. +7 (812) 350-10-76).




Courses.

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.

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.

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.

Accommodation.

The attendance to the School is free, but participants are supposed to arrange their own accommodation.
However,  the Organizing Committee can suggest the following lodging solutions, listed by increasing cost for the whole period (12 nights):

Participants interested in one of the flats should immediately inform the Organizing Committee.

Organizing committee.

M. Bächtold, V. Kalnitsky, G. Moreno, M. M. Vinogradov, L. Vitagliano, M.Yu. Zvagelski.

The Organizing Committee can be contacted for any question and suggestion via the e-address:

Prerequisites

Suitable fundamentals for a fruitful participation in the school may be found in the following references:
  • M. F. Atiyah, I. G. MacDonald, - Introduction to Commutative Algebra, - Westview Press, 1969, Chapters 1,2. A beginner participant should be able to solve exercise from these two chapters.
  • John M. Lee, - Introduction to Smooth Manifolds, - Springer-Verlag, Graduate Texts in Mathematics, Vol. 218, 2003. Appendix + Chapters 1-4, 6 (Chapter 1 is also available on the author's web page).
  • Jet Nestruev, - Smooth manifolds and Observables, - Springer-Verlag, Graduate Texts in Mathematics, Vol. 220, 2002. First chapters of this book will introduce you to the spirit of the school. People who have read this book and solved 70% of the exercises will be able to follow the veteran courses.

List of participants.

NAME

SURNAME

AFFILIATION

COUNTRY

Alena

Vasilyeva

Faculty of Mathematics and Mechanics, Saint-Petersburg State University

Russia

Alexandr

Shelekhov

Tver State University

Russia

Anastasiya

Men'shikh

Mathematical Department of Voronezh State University

Russia

Anna

Kondakova

Faculty of Mechanics and Mathematics, Lomonosov Moscow State University

Russia

Denis

Kuznetsov

Physical Department of Voronezh State University

Russia

Elena

Lyapina

Mathematical Department of Voronezh State University

Russia

Grigory

Nesterchuk

Saint-Petersburg State University

Russia

Igor

Kulakovsky

Saint-Petersburg State University

Russia

Iskender

Kamchibekov

Faculty of Mathematics and Mechanics, Saint-Petersburg State University

Russia

Ivan

Gudoshnikov

Mathematical Department of Voronezh State University

Russia

Ivan

Nenashev

Physical Department of Voronezh State University

Russia

Ivan

Kobizev

Saint-Petersburg State University

Russia

Julia

Petrova

Faculty of Mathematics and Mechanics, Saint-Petersburg State University

Russia

Kasatkin

Victor

Physical Department of the Saint-Petersburg State University

Russia

Maria

Sorokina

Faculty of Mathematics and Mechanics, Saint-Petersburg State University

Russia

Mikhail

Hristoforov

Saint-Petersburg State University

Russia

Nikolay

Gorbushin

Faculty of Mathematics and Mechanics, Saint-Petersburg State University

Russia

Olga

Kunakovskaya

Mathematical Department of Voronezh State University

Russia

Sergej

Elfimov

Physical Department of Voronezh State University

Russia

Sergej

Marmo

Physical Department of Voronezh State University

Russia

Svetlana

Azarina

Voronezh State Technical University

Russia

Valeriya

Samoylova

Mathematical Department of Voronezh State University

Russia

Viktor

Konev

Mathematical Department of Voronezh State University

Russia

Proposed talks.

  • "Некоторые топологические вопросы теории криволинейных три-тканей" by prof. Alexandr Shelekhov, Tver State University (Russia).

Poster.

An electronic copy of the official school poster can be downloaded here.

Ċ
Unknown user,
Nov 20, 2009, 12:08 PM
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