THE ITALIAN SUMMER 2002 SCHOOL
Santo Stefano del Sole (Avellino), Italy,
July 19 - 31, 2002
Last updated October 1, 2002

This school was organized also in cooperation with
- Istituto Italiano per gli Studi Filosofici
- Municipality of Santo Stefano del Sole (AV), Italy
and under the scientific direction of Prof. A. M. Vinogradov (Universita' di Salerno, Italy, and Diffiety Institute, Russia). The aim of the school was to introduce undergraduate and graduate students in Mathematics and Physics, as well as post-doctoral researchers and other mathematicians into a recently emerged area of mathematics and theoretical physics:

Secondary Calculus

Secondary calculus is the result of a natural evolution of the classical geometrical theory of partial differential equations (PDE) originated by Sophus Lie. In particular, it allows the construction of a general theory of PDE, in the same manner as algebraic geometry does with respect to algebraic equations. It became also clear that secondary calculus is a natural language for quantum field theory, just in the same way as standard calculus is for classical physics. From the mathematical point of view secondary calculus is a complex mathematical construction putting into a natural interrelation many parts of modern mathematics such as commutative and homological algebra, algebraic and differential topology, differential geometry, etc.
The strategic goal of the school is to involve interested participants into a series of large scale research programs the Diffiety Institute is launching.

A first idea of the area can be got from the books:

  1. Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. - AMS, Mathematical Monographs, volume 182, 1999.
  2. A. M. Vinogradov: Cohomological Analysis of Partial Differential Equations and Secondary Calculus. - AMS, Mathematical Monographs, volume 204, 2001.
  3. J. Nestruev: Smooth manifolds and Observables -. MCCME, Moscow, 2000 (in Russian); Springer-Verlag, New York, 2002 (in English), Graduate Texts of Mathematics series.

In this edition of the school the following two courses was given:

  1. Differential Calculus over Commutative Algebras and Smooth Manifolds (suggested to beginners).
  2. Infinite Jets and Diffieties (suggested to more experienced participants).
Diplomas of participation in the School were handed to all participants. Moreover, there were examinations for 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 reasonably large number of problems. Students having passed an examination received (or will receive) a diploma certifying this fact. Students having not passed all examinations yet, can do this by email. Click here for school pictures.

During the School there was discussed the following starting programs, intended to involve interested participants into our research projects:

  1. "Einstein";
  2. Singular Lagrangians and Hamiltonian formalism;
  3. Differential Invariants: theory and applications;
  4. Towards Differential Algebraic Geometry.

Also there was organized a scientific session on the base of research interests of the participants.

Lecturers and tutors in the School:

Organizing committee:

D. Catalano Ferraioli, A. De Paris, C. Di Pietro, G. Manno, R. Piscopo, F. Pugliese, G. Rotondaro, F. Strazzullo, R. Vitolo.

List of participants:

Questions and suggestions should go to school @ diffiety.org