St. PetersburgFebruary 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): - A 3-rooms flat 1100 meters away from the venue. If a maximum of 7 people lodge there, unit cost is about 4400 RUB (100 EUR, 145 USD).
- Hostel Pilau, 30 min off the venue. The cost of one person in a 4-beds room is 6720 RUB (150 EUR, 220 USD), with the opportunity of a 10% discount.
- One (ore more) 1-room, 4-persons flat(s), within the range of 2km from the venue. The whole flat costs about 22000 RUB.
- Hotel Courtyard (1200 USD per room)
- Hotel park inn (1400 USD for single room, 1720 USD for double room)
- Hotel Sokos (1600 USD per room)
- Hotel Marco Polo (1630 USD per room, 1810 per double room)
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: ## PrerequisitesSuitable 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.
## 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. |

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