DiffietySchoolRussia08_poster_definitivo_thumb VIII Edition of the Russian Winter School.
Kostroma, Russia.
February 1 - 12, 2008.
This page is constantly updated. Please check it frequently for the latest news!

IMPORTANT ANNOUNCEMENT: deadline for applications has been fixed on Sunday, 13th of January, 2008.

The school is organized in cooperation with

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


  1. Courses
  2. Location
  3. Application, Selection and Admission
  4. Prerequisites
  5. Preliminary List of Admitted Participants
  6. Accommodation, Meals and Classes
  7. Organizing Committee
  8. Diffiety School Poster


In this edition of the School, four courses will be proposed.
The general classification of Diffiety School's courses is as follows:

B-COURSE(S): courses for beginners;
A-COURSE(S): advanced courses;
C-COURSE(S): courses for veteran participants.

Course B0.
Title: Introduction to differential calculus over commutative algebras and geometry of smooth manifolds.
Lecturer: Prof. A. M. Vinogradov.
Schedule: 3 hrs per day (the exact timetable will be published later on).

Course B1.
Title: Differential calculus over commutative algebras.
Lecturer: Dr. Christian Di Pietro.
Schedule: 2 hrs per day (the exact timetable will be published later on).

Description: both courses B0 and B1 aim 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. Course B0 covers the same topics of the course B1, but in a more elementary manner. It is recommended, first of all, to undergraduate students of the second and the third years and, more generally, to those whose algebraic background is limited. B0 is designed as a unique course for those who will attend it and will be accompained by numerous exercises one part of which will be discussed in the class and another one will take part of the final exam.

Course B2.
Title: Geometric structures in the theory of PDEs.
Lecturer: Dr. Michael Bächtold.
Schedule: 2 hrs per day (the exact timetable will be published later on).

Description: the course will start with simple examples of non-linear partial differential equations (PDEs) that show in which circumstances symplectic and contact geometries were invented as theories revealing basic structures of first order non-linear PDEs. The general theory of nonlinear PDEs is in a sense a very nontrival generalization of contact geometry and as such is an indispensable first step in understanding the structure of PDEs. Symplectic geometry is a symmetry reduction of contact geometry which is the mathematical basis of most important contemporary physical theories. So, this geometry and the related Hamiltonian formalism are indispensable as the starting point in understandig what mathematics should be developed in order to face basic problems in contemporary physics and mechanics.

Course A1.
Title: Basic differential complexes and cohomology.
Lecturer: Dr. Luca Vitagliano.
Schedule: 3 hrs per day from 1st to 6th of February (the exact timetable will be published later on).

Description: the aim of the course is to introduce the basic complexes in differential calculus over (graded) commutative algebras, among these the de Rham and Spencer complexes, togheter with some cohomology computational techniques. Such differential complexes are of special relevance in Differential (Super-)Geometry, Algebraic Geometry, Commutative Algebra, etc.. Moreover, they are indespensable tools in Secondary Calculus. For instance secondary vector fields are cohomology classes of a certain Spencer C-complex. All the applications (to Berenzin integral on a super-manifold, singular algebraic varieties, geometric theory of PDEs, etc.) of the general constructions will be duly pointed out during the course.

Course A2.
Title: Introduction to Secondary Calculus: symmetries, conservation laws and Lagrangian formalism.
Lecturer: Dr. Giovanni Moreno.
Schedule: 3 hrs per day from 7st to 12th of February (the exact timetable will be published later on).

Description: the course aims to introduce the basic elements of Secondary Calculus for participants already possessing a strong background in the geometry of non-linear PDEs. Partially, the theoretical materials will be proposed in the form of exercises which will be subsequently discussed on the basis of solutions proposed by participants.

Further activities.
In addition, some special lectures and seminars will be organised for veterans.


Kostroma (Russian: Кострома́) 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 (look at the map). Detailed information about the actual location of the school in Kostroma and how to reach it will be duly given later on.

Application, Selection and Admission

To apply, please choose among the registration forms below 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. During the selection phase, applicants may be requested to provide their curriculum and/or scientific background. Selection criteria are mainly based on the following: participants are required to be familiar with fundamentals of Commutative Algebra, Topology and Differential Geometry (see the next section: Prerequisites). Admitted participants will be duly informed by e-mail. The school, including a full board accomodation in double room in a hotel (see the section: Accommodation, Meals and Classes), is free for admitted participants. Travel expenses are on charge of participants.
DEADLINE: Application must be sent not later than Sunday, 13th of January, 2008.

Online registration for Undergraduate Students.
Online registration for Ph.D. Students.
Online registration for Post-Doctoral Students.
Online registration for Researchers and Professors.


Suitable fundamentals for a fruitfull 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.

Preliminary list of admitted participants

Admitted participants are listed below accordingly to the time of their application. Such a list is regularly refreshed (last update: December 17, 2007).

  1. Pavel Vladimirovich Shubin (Nizhni Novgorod, RUSSIA) - N. I. Lobachevsky State University of Nizhni Novgorod;
  2. Aleksey Kozlov (Nizhni Novgorod, RUSSIA)- N. I. Lobachevsky State University of Nizhni Novgorod;
  3. Mohamed Amine Bahayou (Gharda-a, ALGERIA) - GGTM;
  4. Andrey Olegovich Krutov (Shuya, RUSSIA) - Ivanovo State Power University;
  5. Mohammed Tayeb Benmoussa (Sidikhuiled Ouargla, ALGERIA);
  6. Olga Veniaminovna Kunakovskaya (Voronezh, RUSSIA) - Voronezh State University;
  7. Ivan Mikhailovich Gudoshnikov (Voronezh, RUSSIA) - Voronezh State University;
  8. Julia Yurievna Popova (Voronezh, RUSSIA) - Voronezh State University;
  9. Ekaterina Vladimirovna Trepacheva (Voronezh, RUSSIA) - Voronezh State University;
  10. Darya Alexandrovna Nikishina (Voronezh, RUSSIA) - Voronezh State University;
  11. Marina Evgenievna Zalygaeva (Voronezh, RUSSIA) - Voronezh State University;
  12. Margarita Alexandrovna Kalashnikova (Voronezh, RUSSIA) - Voronezh State University;
  13. Valeria Vitalievna Samoilova (Voronezh, RUSSIA) - Voronezh State University;
  14. Mikhail Yurievich Kuzmin (Voronezh, RUSSIA) - Voronezh State University;
  15. Svetlana Azarina (Voronezh, RUSSIA) - Voronezh State University;
  16. Lev Yurievich Yakovlev (Voronezh, RUSSIA) - Voronezh State University;
  17. Antuan Leonidovich Zemlyanukhin (Voronezh, RUSSIA) - Voronezh State University;
  18. Stanislav Dubrovskiy (Beer-Sheva, ISRAEL) - Ben-Gurion University;
  19. Leonid Sudov (St. Petersburg, RUSSIA) - St. Petersburg State University;
  20. Victor Erikovich Kasatkin (St. Petersburg, RUSSIA) - St. Petersburg State University;
  21. Svetlana Valerievna Bächtold (Zurich, SWITZERLAND);
  22. Nastya Lashova (St. Petersburg, RUSSIA) - St. Petersburg State University;
  23. Andrey Borozdin (St. Petersburg, RUSSIA) - St. Petersburg State University;
  24. Longla Martial (Moscow, RUSSIA) - Peoples' Friendship University of Russia;
  25. Evgeniya Nikolaevna Moreva (Ivanovo, RUSSIA) - Ivanovo State Power University;
  26. Elizaveta Gennadievna Vishnyakova (Tver, RUSSIA) - Tver State University;
  27. Alexander Mikhailovic Shelekhov (Dzerzhinskii, RUSSIA).

Accommodation, Meals and Classes

Participants will lodge in double room and have meals in the

Sanatorij "Kostromskoj" (Russian: Санаторий Костромской),
micro-rajon Malyshkovo,
156011 Kostroma, RUSSIA.
Look at the map ("micro-rajon Malyshkovo" is not indicated: it is the left branch at the end of ул. Малышковская).
Exact geographical coordinates are: 57°43'45" N, 40°55'27" E, 117m a.s.l., planet Earth.
You may also be interested at the Russian web site.

Classes and seminars will take place in the Sanatorij itself. A detailed schedule of the school activities will be published later.

Organizing committee

M. Bächtold, C. Di Pietro, V. Kalnitsky, G. Moreno, R. Piscopo, M. M. Vinogradov, L. Vitagliano, M. Y. Zvagelsky.

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


Diffiety School poster

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