B-COURSES: they are intended as courses for beginners.
B1 - Smooth Manifolds and Observables (by G. Moreno): 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 - Symplectic, Contact Geometry and Jet Spaces (by C. Di Pietro):
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.
A-COURSES: they are intended as advanced course.
A1 - Cohomological Theory of Integration and the Leray-Serre Spectral Sequence (by M. Bächtold): 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. (For detail programm see here.)
A2 - Geometry of Jet Spaces and Symmetries of PDEs (by L. Vitagliano):
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.
(For detail programm see here.)
A3 - Differential Calculus over Commutative Algebras (by A. M. Vinogradov): 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).
C-COURSE: it is intended as a course for veteran participants.
C1 - Introduction to Secondary Calculus (by A. M. Vinogradov): 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.
The following lecture schedule is preliminary and may be subject to change:
Daily Schedule
Diffiety School scientific activities consist of two blocks of 7 full days.
First block goes from July 19 to July 25. Second block goes from July 27 to
August 2.
During these days lectures are given from 9:00 to 11:00 a.m. and from 11:15 to
1:15 p.m. Exercise correction sessions take from 4:00 to 6:00 p.m. The hour
from 6:00 to 7:00 is reserved for seminars.
On the arrival day (Wednesday, July 18) only dinner is served to participants.
On the departure day (Friday, August 3) only breakfast and lunch is served to
participants. During the free day (Thursday, July 26) no meals are served.
During the 14 days of scientific activities, breakfast, lunch and dinner are
served. Breakfast is given at 8:30 a.m., lunch at 1:30 p.m., and dinner at 9:30
p.m.
Preliminary list of admitted participants:
Ansalone Domenico Patrizio, Politecnico di Torino (Italy);
Ali Sajid, National University of Sciences and Technology (Pakistan);
Bahayou Mohamed Amine, University of Ouargla (Algeria);
