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24.10.2007. :: VIII Edition of the Russian Winter Diffiety School

VIII Edition of the Russian Winter Diffiety School
Kostroma, Russia.
February 1 - 12, 2008.
(web page:

The aim of this permanent School is to introduce undergraduate and Ph. D. students in Mathematics and Physics as well as post-doctoral researchers in a recently emerged area of Mathematics and Teoretical Physics:


A diffiety is a new geometrical object that properly formalizes the concept of the solution space of a given system of (nonlinear) PDEs, much as an algebraic variety does with respect to solutions of a given system of algebraic equations. Secondary Calculus is a natural diffiety analogue of the standard calculus on smooth manifolds, and as such leads to a very rich general theory of nonlinear PDEs. Moreover, it appears to be a natural language for quantum physics, just as the standard calculus is the natural language for classical physics.

The school is organized in cooperation with
- the "Bol`shaya peremena" program of Michael Batin (Kostroma, Russia);
- Istituto Italiano per gli Studi Filosofici, Italy;

and under the scientific direction of Prof. A. M. Vinogradov (Universita` di Salerno, Italy, and Diffiety Institute, Russia).


In this edition two series of courses will be given.

B-COURSES: they are intended as courses for beginners.

(B1) Smooth Manifolds and Observables: 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) Symplectic, Contact Geometry and Jet Spaces: historically, symplectic and contact geometries were first studied as geometric theories of first order scalar differential equations. In particular the basic geometry of non-linear partial differential equations (PDEs) appears in its simplest form in contact geometry. Contact geometry is therefore an indispensable tool in understanding the structure of PDEs.
The aim of the course is to present this non-standard point of view by first introducing the geometry of symplectic and contact structures and, finally, the analogous structures in the geometric theory of PDEs.

A-COURSES: they are intended as advanced course for veterans and will be programmed later on according to the interests and backgrounds of actual participants. To get an idea of the may be contents check on the pages of the last editions of the school (the 10th Italian Edition is here on this site, while the other Editions can be found on the Diffiety Institute site).


Kostroma is a historic city in central Russia, administrative centre of the Kostroma Oblast. A part of the Golden ring of the Russian towns, it is located at the confluence of the rivers Volga and Kostroma, 65 km east of Yaroslavl. Detailed information about the actual location of the school in Kostroma and how to reach it will be duly given later on.


To apply, please choose among the registration forms on the web page of the school the one which better fits your personal profile, and fill it in all its compulsory fields. Applicants will be selected by the Director of the School, prof. A. M. Vinogradov. Admitted participants will be informed by e-mail. The school, including a full board accomodation in hotel, is free for admitted participants. Travel expenses are on charge of participants.


M. Bchtold, C. Di Pietro, G. Moreno, R. Piscopo, M. M. Vinogradov, L. Vitagliano.

Questions and suggestions should be sent to the e-address:


The Organizing Committee expresses its gratitude to those who spread the news about the School in their Institution/University/Department, by displaying the official poster. A PDF electronic copy of it is available for download on the web page of the school.
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