Pictures of XIII Diffiety School may be viewed via the Facebook group "The Diffiety School Experience".
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. From mathematical side this allows to reveal the "logic" of differential calculus. 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 concepts 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.
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.
The course will start with simple examples of non-linear partial differential equations (PDEs) that show in which circumstances symplectic and contact geometries were invented as theories revealing basic structures of first order non-linear PDEs. The general theory of nonlinear PDEs is in a sense a very non-trival generalization of contact geometry and as such is an indispensable first step in understanding the structure of PDEs. Symplectic geometry is a symmetry reduction of contact geometry which is the mathematical basis of most important contemporary physical theories. So, this geometry and the related Hamiltonian formalism are indispensable as a starting point in understanding which type of mathematics could be developed in order to face basic problems in contemporary physics and mechanics.
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 the C-spectral sequence theory.
The Organizing Committee can be contacted via the e-address: Santo Stefano del Sole is a small (≈ 2100 inhabitants) village located in the heart of a mountainy region called "Irpinia". Its territory begins 328 meters above the sea level and stretches over the shoulders of the hills untill it reaches 1146 meters, covering more than ten square kilometers, half of which is forest. Section "Pictures" of the X Edition of Summer Diffiety School displays many shots of the School location.
Exact geographical coordinates are: 40°54' N, 14°51' E, 510m a.s.l., planet Earth.
Reaching Santo Stefano del Sole is quite easy.