Summer 2006 Italian School

THE ITALIAN SUMMER 2006 SCHOOL

Santo Stefano del Sole (Avellino), Italy,

July 14 - 31, 2006

Last updated September 13, 2006

This school is organized in cooperation with

Istituto Italiano per gli Studi Filosofici, Italy;
Municipality of Santo Stefano del Sole (AV), Italy;
Comunità Montana Serinese-Solofrana, Italy;
Ente Morale S. Vito Martire, Italy,

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

Secondary Calculus
Courses
Prerequisites
List of participants
Organizing committee
School photos. (last updated September 13, 2006.)
Color School poster (A3, 854 K, and A4, 320 K).
How to reach S. Stefano del Sole

Courses

In this edition of the school three series of courses, one for beginners, one for the veterans, and one for all interested participants, were given. Courses started on Saturday , July 15 at 9:00 a.m. and ended on Sunday, July 30. Diplomas was delivered on Sunday, July 30 afternoon. Saturday , July 22 was free of scientific activities.

For BEGINNERS

B1 - Smooth Manifolds and Observables (by A. M. Vinogradov): 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 - - Introduction to Geometry of Jet Spaces (by V. Chetverikov): the role of jet bundles in the theory of nonlinear partial differential equations is very similar to that of space-time in physics. It will be explained why and basic geometrical structures on jet bundles will be discussed. In particular, it will be explained what really partial differential equations are from the modern point of view.

For VETERANS

A1 - Differential Cohomology (by L. Vitagliano): the aim of the course is twofold. First, to introduce de Rham cohomology and, in particular, to show that integration over manifolds is really a cohomological concept. On the other side, to introduce algebraic theory of linear connections (which is a ground-stone for the future construction of field theory) and the differential Leray-Serre spectral sequence (which is an embryonic form of the C-spectral sequence, one of the central notions in Secondary Calculus). The approach is nontraditional, based on philosophy of B1.

A2 - Coverings and Non-Local Theory of PDEs (by S. Igonin): the course aims to introduce the concept of a covering in the category of PDEs. Coverings are the mean that allows to formalize properly the so-called non-local quantities in the theory of PDEs, i.e., quantities arising when inverting differential operators. There will be discussed fundamentals of the theory of coverings with a particular emphasis on non-local symmetries, non-local conservation laws of a PDE and Bæcklund transformations connecting PDEs.

For ALL INTERESTED PARTICIPANTS

C1 - Symplectic and Contact Geometry (by A. M. Vinogradov): the aim of the course is to introduce the general theory of (geometric) distributions and then focus on contact structures and their relations with symplectic geometry. Contact geometry helps much to understand better many constructions and ideas of the "higher" geometry of PDEs.

A scientific session aiming to involve interested participants to our research programs is also planned.

Seminars
Daily seminar sessions were done by dr. C. Di Pietro, dr. G. Moreno, dr. R. Piscopo, dr. V. Golovko.

List of participants:

Dominique Aberlin (Swiss Federal Institute of Technology, Zürich, Switzerland),
Svetlana Azarina (Voronezh State University, Russia),
Konrad Bartkowski (Warsaw University of Technology, Poland),
Denis Biryukov (Bauman Moscow State Technical University, Russia),
Irina Bobkova (St.-Petersburg State University, Russia),
Andrew Borozdin (St.-Petersburg State University, Russia),
Michal Cukrowski (Warsaw Technical University, Poland),
Stanislav Dubrovskiy (Keio University, Yokohama, Japan),
Ewa Falkiewicz (University of Technology in Radom, Poland),
Valentina Golovko (Moscow State University, Russia),
Adrian Mihai Ionescu ("Politehnica" University, Bucharest, Romania),
Sergey Kolesnikov (St.-Petersburg State University, Russia),
Alexander Kuklev (Göttingen University, Germany),
Marek Multarzynski (Warsaw, Poland),
Patrice Pungu Ntumba (University of Pretoria, South Africa),
Elena Rumyantseva (Bauman Moscow State Technical University, Russia),
Davide G. M. Salvetti (University of Trieste, Italy),
Amandip Sangha (University of Oslo, Norway),
Stefan Sakalos (Central European University, Budapest, Hungary),
Andrew Shevliakov (Bauman Moscow State Technical University, Russia),
Sergey Slizovskiy (St.-Petersburg Nuclear Physics Institute, Russia),
Svetlana Vangesova (St. Petersburg Pedagogical University Herzen, Russia).

Also among participants:

Prof. Augustin Batubenge (University of the Witwatersrand, South Africa),
Prof. Piotr Multarzynski (Warsaw University of Technology, Poland),
Prof. Wieslaw Sasin (Warsaw University of Technology, Poland),
Prof. Robert Stanisław Stępnicki (John Paul II Catholic University of Lublin, Poland).

Diplomas of participation in the School were handed to all participants. Moreover, there were examinations in all courses, which were organized as follows. For each course, students received a list of examination problems. To pass an examination, one had to solve a reasonably large number of problems. Students having passed an examination received diplomas certifying this fact.

The following students have passed examinations in Smooth Manifolds and Observables:

Konrad Bartkowski (Warsaw University of Technology, Poland).

The following students have passed examinations in Introduction to Geometry of Jet Spaces:

Irina Bobkova (St.-Petersburg State University, Russia),
Andrew Borozdin (St.-Petersburg State University, Russia),
Stanislav Dubrovskiy (Keio University, Yokohama, Japan),
Sergey Kolesnikov (St.-Petersburg State University, Russia),
Amandip Sangha (University of Oslo, Norway),
Sergey Slizovskiy (St.-Petersburg Nuclear Physics Institute, Russia).

And Svetlana Azarina (Voronezh State University, Russia) has passed examinations in Contact and Symplectic Geometry.

Organizing committee:

M. Bächtold, C. Di Pietro, V. Fiore, M. Langastro, G. Moreno, R. Piscopo, V. Vingo, M. M. Vinogradov, L. Vitagliano.

Contents

Courses

List of participants:

Organizing committee: