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).


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


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.


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.


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.


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.

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

List of participants:

Also among participants:

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