XII Edition of the Italian Summer School.
Santo Stefano del Sole (Avellino), Italy. July 16 - 31, 2009. Under the auspices of Levi-Civita Institute |

The school was organized in cooperation with

- Istituto Italiano per gli Studi Filosofici (Italy);
- Municipality of Santo Stefano del Sole (Italy);
- Dipartimento di Matematica e Informatica, Università degli Studi di Salerno (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).

### Contents

### Courses

In this edition of the School, the following courses were proposed.Course B0.

**Title:**Natural Foundations of Differential Calculus.

**Lecturer:**Giovanni Moreno.

**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. The course is especially designed for those undergraduate students who firstly take part in the Diffiety School, much space being dedicated to review the basic algebraic knowledges needed for a fruitful understanding, and constitute the departing point for all subsequent courses.

Course B2.

**Title:**First Order Calculus on Smooth Manifolds II.

**Lecturer:**Alexandre M. Vinogradov.

**Description:**Calculus over smooth manifolds will be developed according to the logic of Differential Calculus over Commutative Algebras. It will be shown that even in basic differential geometry, this approach reveal more details than the standard one. This course is preparatory to more advanced further courses concerning both the geometry of PDEs and Secondary Calculus, and is recommended for those students who had followed the course "First Order Calculus on Smooth Manifolds".

Course A1.

Title: Integrals and Cohomology.

**Lecturer:**Giovanni Moreno and Luca Vitagliano.

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. In the first part of the course will be shown that the integral is actually a cohomological concept. In the simplest case of integration over smooth manifolds, this is an aspect of de Rham cohomology. The main techniques of computation of de Rham cohomology will be introduced on the base of differential calculus by completely 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: Introduction to geometry of PDEs.

**Lecturer:**Michael Bächtold.

Description: The course will first provide a detailed geometrical analysis of the finite jet spaces, which is indispensable for the theory of diffieties. This is the natural environment in which a system of nonlinear PDEs can be inscribed, and where such concepts like transformations, intrinsic/extrinsic and finite/infinitesimal symmetries, generalized Jacobi brakets, integration by characteristics, Cauchy data, first integrals, etc. become part of the geometry of the PDEs. A survey of solution singularity theory will be given, in connection with the quantization problem.

Course C1.

Title: Elementary Diffieties.

**Lecturer:**Luca Vitagliano.

Description: 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.

Course C2.

Title: Functors of Differential Calculus.

**Lecturer:**Alexandre M. Vinogradov.

### List of participants

People who participated to the School are listed below.- Jet Nestruev (Moscow, RUSSIA and Salerno, ITALY) - Levi-Civita Institute;
- Frederic Paugam (Paris, FRANCE) - Jussieus Mathematical Institute;
- Maria Sorokina (St. Petersburg, RUSSIA) - St. Petersburg State University;
- Ekaterina Golikova (St. Petersburg, RUSSIA) - St. Petersburg State University;
- Thomas Leuther (Aubel, BELGIUM) - University of Liege;
- Isabel Cristina Marquez de Mastromartino (Barquisimeto, VENEZUELA) - UCLA;
- Angel Mastromartino (Barquisimeto, VENEZUELA) - UCLA;
- Elizaveta Semenova (Moscow, RUSSIA) - Moscow State University;
- Ivan Kobyzev (St. Petersburg, RUSSIA) - St. Petersburg State University;
- Martin Wen-Yu Lo (Pasadena, U.S.A.) - Jet Propulsion Laboratory, Caltech;
- Irina Gorbunova (St. Petersburg, RUSSIA) - St. Petersburg State University;
- Victor Kasatkin (St. Petersburg, RUSSIA) - St. Petersburg State University;
- Svetlana Azarina (Voronezh, RUSSIA) - Voronezh State University;
- Fan Wu (Beijing, P. R. CHINA) - Capital Normal University;
- Irina Gorodetskaya (St. Petersburg, RUSSIA) - St. Petersburg State University;
- Julia Petrova (St. Petersburg, RUSSIA) - St. Petersburg State University;
- Alyona Vasilyeva (St. Petersburg, RUSSIA) - St. Petersburg State University;
- Denis Tugarev (Tambov, RUSSIA) - Derzhavin Tambov State University;
- Marina Iljina (Tambov, RUSSIA) - Derzhavin Tambov State University;
- Anastasia Nishnevich (St. Petersburg, RUSSIA) - St. Petersburg State University;
- Dmitry Kharkovsky (St. Petersburg, RUSSIA) - St. Petersburg State University;
- Fabian Radoux (Liege, BELGIUM) - University of Liege;
- Ivko Dimitric (Pittsburgh, U.S.A) - Penn State University Fayette;
- Georges Weill;
- Mikhail Khristoforov (St. Petersburg, RUSSIA) - St. Petersburg State University;
- Monika Stypa (Naples, ITALY) - University of Salerno;
- Michael Werman (Jerusalem, ISRAEL) - The Hebrew University;
- Tadeusz Jagodzinski (Warsaw, POLAND) - Mathematics and Information Techniques Faculty;
- Carla Cruz (Aveiro, PORTUGAL) - University of Aveiro.

### Passed exams

Coming soon.