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14.11.2008. :: XII Summer Diffiety School


School in Geometry of Partial Differential Equations and Secondary Calculus

Santo Stefano del Sole (AV), Italy.
July 16-31, 2009.

The aim of the permanent Diffiety School is to introduce undergraduate and Ph. D. students in Mathematics and Physics as well as post-doctoral researchers in a recently emerged area of Mathematics and Teoretical Physics:


A diffiety is a new geometrical object that properly formalizes the concept of the solution space of a given system of (nonlinear) PDEs, much as an algebraic variety does with respect to solutions of a given system of algebraic equations. Secondary Calculus is a natural diffiety analogue of the standard Calculus on smooth manifolds, and as such leads to a very rich general theory of nonlinear PDEs. It appears that it is this the only natural language of quantum physics, just as the standard Calculus is for classical physics.

The school is 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 Levi-Civita Institute, Italy).


The general classification of Diffiety School`s courses is as follows

B-COURSE(S): courses for beginners;
A-COURSE(S): advanced courses;
C-COURSE(S): courses for veteran participants.

Here is a tentative list of courses that will be activated this year

Course B1.
Title: Observability in Physics and Differential Calculus over Commutative Algebras.
Lecturer: TBA.
Schedule: TBA.

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. 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.

Course B2.
Title: First Order Calculus on Smooth Manifolds.
Lecturer: TBA.
Schedule: TBA.

Description: Calculus over smooth manifolds will be developed according to the philosophy of course B1, i.e., "algebraically". It will be shown that even in basic differential geometry, this approach reveal more details than the standard one. This course is propedeutic for further more advanced courses concerning both the geometry of PDEs and Secondary Calculus.

Course A.
Title: Geometry of Finite Order Jet Spaces.
Lecturer: TBA.
Schedule: TBA.

Description: 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. First of all, the basic geometrical structures on jet spaces will be discussed in the perspective of the subsequent passage to infinite jets. In particular, it will be explained what really PDEs are from the modern point of view, what are their classical symmetries, singularities of their solutions, etc.

Course C.
Title: Secondary Differential Forms and Vector Fields.
Lecturer: TBA.
Schedule: TBA.

Description: 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.

Look at past edition web-pages for more information

Further activities.
For veterans are planned some additional activities (seminars, discussions of problems, etc.)


To apply, please choose among the registration forms you can find on the web-page of the school ( the one that better fits your personal profile, and fill it in all its compulsory fields. Applicants will be selected by the Director of the School, prof. A. M. Vinogradov. During the selection phase, applicants may be requested to provide their curriculum and/or scientific background. Selection criteria are mainly based on the following: participants are required to be familiar with fundamentals of Commutative Algebra, Topology and Differential Geometry (see the next section: Prerequisites).

DEADLINE: Application must be sent not later than Sunday, 21 of June, 2009.
Applicants will be duly informed by e-mail about their (possible) admission
- not later than March 16, 2009, if had applied earlier than March 15, 2009;
- not later than June 21, 2009, if had applied later than March 16, 2009.


The registration fee of the school, including a full board (except for
the meals of the free day) accommodation, amounts to 175 EUR.
Accompanying persons who want to avail themselves of the school
accommodation will be requested to pay 150 EUR. Travel expenses are on
charge of participants.

Registration fee can be paid cash upon arrival or by wire transfer
directly to Diffiety School`s bank account:

IBAN: IT82N0539203404000001274373


Suitable fundamentals for a fruitfull participation in the school may be
found in the following references:

* M. F. Atiyah, I. G. MacDonald, - Introduction to Commutative Algebra, - Westview Press, 1969, Chapters 1,2. A beginner participant should be able to solve exercise from these two chapters.

* John M. Lee, - Introduction to Smooth Manifolds, - Springer-Verlag, Graduate Texts in Mathematics, Vol. 218, 2003. Appendix + Chapters 1-4, 6 (Chapter 1 is also available on the author`s web page).

* Jet Nestruev, - Smooth manifolds and Observables . - Springer-Verlag, Graduate Texts in Mathematics, Vol. 220, 2002. First chapters of this book will introduce you to the spirit of the school. People who have read this book and solved 70% of the exercises will be able to follow the veteran courses.


Men will lodge in very spartan dormitories with shared bathrooms. Women will lodge in some apartment(s) made available by the Municipality of Santo Stefano del Sole. Interested people can contact the Organizing Committee about the availability of more luxurious accommodation, on their charge. The hotel "Il Ciliegio" ( is the only high-Standard lodging solution available in the neighborhoods, located just
1500 meters away from the center of Santo Stefano del Sole. Interested people can contact the hotel to book their own rooms, introducing themselves as participants to the School.
The non-EU participants will be asked, during the first days of the school, for their cooperation to carry out the compulsory registration procedure at the Police`s designated office.

Meals are served to participants in the elementary school`s refectory located on the intersection of "Via Colacurcio" and "Via Pisacreta" (look at the map), one level above the bar. Breakfast is served at 08:00, lunch at 13:30 and supper at 21:30.


More information about excursion(s) will be duly posted herewith.


M. Bächtold, V. Kalnitsky, G. Moreno, C. Ragano, M. M. Vinogradov, L.

The Organizing Committee can be contacted for any question and suggestion
via the e-address:


The Organizing Committee expresses its gratitude to those who spread the
news about the School in their University/Department, by displaying the
attached official poster.


Santo Stefano del Sole is a small ( 2100 inhabitants) village ocated 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. Exact geographical coordinates are: 40°54` N, 14°51` E, 510m a.s.l., planet Earth.

Reaching Santo Stefano del Sole is quite easy. The Organizing Committee warmly advices the participants not to travel in different dates from the official arrival (July 16) and departing (July 31) days, and cannot provide travel information for such different dates. Avellino is 263 km off Rome and 73 km off Naples, the closest cities reached by international flights. A forced stop is the city of Avellino, just 10 kilometers away from Santo Stefano del Sole. Avellino can be easily reached from Rome by bus (there is no direct train connection). More information is avalaibla on the web-page of the school
Institute of Physics Belgrade - Pregrevica 118 - 11080 Belgrade - Serbia, Switchboard: +381 11 3713000, Fax: +381 11 3162190