X Edition of the Italian Summer School.

Santo Stefano del Sole (AV), Italy.

July 18 - August 3, 2007.

### Contents

CoursesIn this edition of the School, six courses have been given.

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 B1.

**Title:**Smooth Manifolds and Observables.

**Lecturer:**G. Moreno.

**Duration:**14 lessons (from 19.07 to 25.07 and from 27.07 to 02.08) of 1h30' each (from 08:30 to 10:00 a.m.), 21h overall.

**Description:**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.

**Lecture Notes**:

*soon available*.

**Video:**

*coming soon*.

**List of Participants**(with attendance to lectures):

- Prof. Belkhelfa Mohamed (Algeria), 5/14;
- Prof. Benaissa Abbes (Algeria), 9/14;
- Prof. Jagodzinski Tadeusz (Poland), 14/14;
- Matak Peter (Slovak Republic), 14/14;
- Prof. Ntumba Patrice (South Africa), 11/14;
- Zhelezov Dmitry (Russia), 12/14;
- Prof. Kashtanov Arseny (Russia), 1/14;
- Kashtanova Stanislava (Russia), 2/14.

**Diplomas delivered to**:

*N.A.*

Course B2.

**Title:**Symplectic, Contact Geometry and Jet Spaces.

**Lectures:**C. Di Pietro.

**Description:**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.

Course A1.

**Title:**Cohomological Theory of Integration and the Leray-Serre Spectral Sequence.

**Lecturer:**M. Bächtold.

**Description:**In mathematical applications to some fundamental problems in Physics and Mechanics one needs to perform integration over the "solution space" of a given non-linear PDE (Feynman path integral, etc.). It seems that this goal cannot be reached by standard measure theory methods. The first part of the course aims to show that the integral is actually a cohomological concept and, in the simplest case (integral over a smooth manifold), an aspect of the theory of de Rham cohomology. The main techniques of computation of de Rham cohomology will be introduced on the base of differential calculus thus avoiding the standard use of algebraic topology. Among these techniques a central role is played by the differential version of the Leray-Serre spectral sequence. Such a sequence is not only important in its own but it is also the most simple example of a C-spectral sequence, which is a key notion in Secondary Calculus.

Course A2.

**Title:**Geometry of Jet Spaces and Symmetries of PDEs.

**Lecturer:**L. Vitagliano.

**Description:**The functional analytic approach to differential equations hides the basic natural structures of PDEs. On the other side they clearly appears in the geometric approach based on the theory of jet spaces. In the first part of the course basic geometrical structures on jet spaces will be discussed. In particular, it will be explained what really PDEs are from the modern point of view. The second part of the course is introductory to the geometry of the infinite jet space. The passage to infinite jets is not only natural but also has a lot of practical advantages and applications, such as the possibility to introduce the concept of higher symmetries of systems of PDEs, that are nothing but secondary vector fields.

Course A3.

**Title:**Differential Calculus over Commutative Algebras.

**Lecturer:**A. M. Vinogradov.

**Decription:**The "logic" of differential calculus as an aspect of commutative algebra will be presented. Indeed, differential calculus is the study of certain functors, their representative objects and natural transformations in suitable categories of modules over commutative algebras. On the base of this study all differential geometric concept may be formalized over an arbitrary commutative algebra. Differential calculus over commutative algebras is not only the "mathematical grammar" of classical nature but it is an indispensable tool in Secondary Calculus. Its knowledge is also very useful to better understand basic aspects of classical and quantum physics (as an example, calculus on super-manifolds is just a particular case of differential calculus over graded-commutative algebras).

Course C1.

**Title:**Introduction to Secondary Calculus.

**Lecturer:**A. M. Vinogradov.

**Description:**The aim of the course is to introduce the category of diffieties and the fundamentals of Secondary Calculus on the base of the C-Spectral Sequence whose first term is naturally interpreted as the space of differential forms on the "solution manifold" of a system of (nonlinear) PDEs. It will be proved that calculus of variations, conservation laws theory, etc., are just small aspects of C-spectral sequence theory.

### Pictures

*The photo albums presented below contain 237 pictures overall*.

- The School is usually given an official opening
ceremony. The chief authorities of the municipality
welcome the participants and the Director introduces them
to the philosophy of the School. Look at the album
The Opening Ceremony.
- The heart of the School life is made by the lectures,
which take place at morning. Also exercises corrections
and scientific seminars, though intended as a side
activity, play an important role, and are held during the
afternoon. Look at the album
Lectures &
Seminars.
- For any participant who had worked hard during the
whole period of the School, the highest moment of
satisfaction is when he/she receives the diploma. An
informal ceremony is usually held during one of the last
days of the school, and the "winners" receive the
diplomas directly from the Director, together with his
handshake. Look at the album Diplomas Delivery.
- Social life is an essential aspect of the School.
Without it, working on mathematics would be an unbearable
challenge for anyone. Look at the album Fun & Friendship.
- There are many ways to enjoy social life and have fun
together. But some of them have become "tradition" for
the School. Look at the album
Fiesta,
Discoteque, Mafia & Karaoke.
- Undoubtedly the most traditional social event is the
excursion to the mount "Terminio", the highest peak in
the neighborhoods (1783 m). It became so important during
the years that the free day (July 26, in this year's
edition) is customarily devoted to the excursion, and
most of the participants take part to it. Look at the
album Excursion to Mount
"Terminio".
- Sport activity is the best complement for the mind
activity. Participants from various part of the world
organized an "international" football match. Look at the
album The Football Match.
- Due to the affordability of huge memory cards,
thousands of picture are usually shot during the School,
by many participants. And not unlikely some people try to
ease the pressure of the School by making some (harmless)
craziness, or by displaying a weird behavior. These two
ingredients, combined together, give rise to some
"artistic" pictures. Look at the album Genre-pieces.
- The School is an international meeting. Students,
lecturers, professors, researchers come here from all
around the world. Look at the album Portraits.
- "Santo Stefano del Sole" means precisely "Saint
Stephan of the Sun". So it is true that
*nomen est omen*: sun, light and colors can be found everywhere in this small village, and in the nature which surrounds it, and in the people who live there. Look at the album Santo Stefano del Sole & its Neighborhoods.

*The original shots displayed in the above section were taken mostly by Stan Dubrovskiy. Our gratitude goes to his unmatched skills with digital camera and ability to freeze instantaneous emotions. We thank also Ewa Falkiewicz, Ilya Kachkovskiy, Mikhail Zvagelskiy, Michal Cukrowski and Peter Matak for their contributions. Giovanni Moreno took care of the digital development of the originals and added some "artistic touches" to them. He apologizes if his manipulations hurt the photographers' feelings about their own pictures, and hopes that whoever walks in this site will enjoy visiting the gallery. If we had more time, we would have made shorter albums.*