Levi-Civita Institute announces

DS_2009_poster_thumb XII Edition of the Italian Summer School.
Santo Stefano del Sole (Avellino), Italy.
July 16 - 31, 2009.
This page was originally posted on October 23, 2008.
This page is constantly updated. Please check it frequently for the latest news!

The school is organized in cooperation with

and under the scientific direction of Prof. A. M. Vinogradov (Università di Salerno, Italy, and Diffiety Institute, Russia).


  1. Courses
  2. Application, Selection and Admission
  3. Registration fee
  4. Prerequisites
  5. Preliminary List of Admitted Participants
  6. Accommodation, Meals and Classes
  7. Free day
  8. Organizing Committee
  9. Diffiety School Poster
  10. Location


In this edition of the School, the standard courses for beginners will be proposed.
The general classification of Diffiety School's courses is as follows (look at past editions for more information about such courses):

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

Course B0.
Title: Natural Foundations of Differential Calculus.
Lecturer: Giovanni Moreno.
Schedule: 09:00-11:00, room "Biblioteca".

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. The course is especially designed for those undergraduate students who firstly take part in the Diffiety School, much space being dedicated to review the basic algebraic knowledges needed for a fruitful understanding, and constitute the departing point for all subsequent courses.

Course B2.
Title: First Order Calculus on Smooth Manifolds II.
Lecturer: Alexandre M. Vinogradov.
Schedule: 11:00-13:00, room "Istituto Levi-Civita".

Description: 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, and is recommended for those students who had followed the course "First Order Calculus on Smooth Manifolds".

Course A1.
Title: Integrals and Cohomology.
Lecturer: Giovanni Moreno and Luca Vitagliano.
Schedule: 11:00-13:00, room "Sala Consiliare".

Description: 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. In the first part of the course will be shown that the integral is actually a cohomological concept. In the simplest case of integration over smooth manifolds, this is an aspect of de Rham cohomology. The main techniques of computation of de Rham cohomology will be introduced on the base of differential calculus by completely 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.

Course A2.
Title: Introduction to geometry of PDEs.
Lecturer: Michael Bächtold.
Schedule: 09:00-11:00, room "Istituto Levi-Civita" (July 18-22 and 24-29).

Description: The course will first provide a detailed geometrical analysis of the finite jet spaces, which is indispensable for the theory of diffieties. This is the natural environment in which a system of nonlinear PDEs can be inscribed, and where such concepts like transformations, intrinsic/extrinsic and finite/infinitesimal symmetries, generalized Jacobi brakets, integration by characteristics, Cauchy data, first integrals, etc. become part of the geometry of the PDEs. A survey of solution singularity theory will be given, in connection with the quantization problem.

Course C1.
Title: Elementary Diffieties.
Lecturer: Luca Vitagliano.
Schedule: 09:00-11:00, room "Sala Consiliare".

Description: The course introduces the fundamentals of geometry of infinite jets spaces, and specific differential calculus over them. When dealing with this infinite-dimensional objects, differential calculus over commutative algebra allows to overcome every typical difficulty. Will be discussed infinite prolongations of systems of nonlinear PDEs, that are the simplest examples diffieties. There will be constructed secondary vector fields, which are interpreted as higher symmetries of systems of nonlinear PDEs, and is the simplest element of Secondary Calculus over diffieties, and then it will discussed horizontal differential calculus and C-spectral sequence whose first term is interpreted as secondary differential forms. Variational Calculus and the theory of conservation laws for PDEs will be presented as small parts of the C-Spectral Sequence.

Course C2.
Title: Functors of Differential Calculus.
Lecturer: Alexandre M. Vinogradov and TBA.
Schedule: 16:00-18:00, room "Istituto Levi-Civita".

Description: There will be introduced basic functors of Differential Calculus over Commutative Algebras, and their natural transformations, which forms the logic of Differential Calculs.

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

Proposed talks.
The following list is tentative. More precise information and schedule will be provided during the school.
  • Exotic Spacetime and Dark Energy/Matter, by Martin Wen-Yu Lo (Jet Propulsion Laboratory, Caltech).

Application, Selection and Admission

To apply, please choose among the registration forms below the one which 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.

Online registration for Undergraduate Students.
Online registration for Ph.D. Students.
Online registration for Post-Doctoral Students.
Online registration for Researchers and Professors.

Registration fee

The registration fee of the school, including a full board (except for the meals of the free day) accommodation (see the section: Accommodation, Meals and Classes), 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.
Participants coming from Eastern Europe might take advantage of EMS scholarships.
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.

Preliminary list of admitted participants

List below is populated with the names of the admitted participants, accordingly to the time of their application.
Last update: 9th of July, 2009.
Admitted participants are listed below accordingly to the time of their application. Such a list is regularly refreshed.
  1. Jet Nestruev (Moscow, RUSSIA and Salerno, ITALY) - Levi-Civita Institute;
  2. Farnaz Farzi (Tehran, IRAN) - Khaie Nasir Toosi University of Technology;
  3. Sajid Ali (Rawalpindi, PAKISTAN) - National University of Sciences and Technology;
  4. Reza Aghayan (Tehran, IRAN) - Iran University of Science and Technology;
  5. Filiz Kanbay (Istanbul, TURKEY) - Yildiz Technical University;
  6. Frederic Paugam (Paris, FRANCE) - Jussieus Mathematical Institute;
  7. Ekaterina Kutafina (Krakow, POLAND) - AGH University of Sciences and Technology;
  8. Maria Sorokina (St. Petersburg, RUSSIA) - St. Petersburg State University;
  9. Ekaterina Golikova (St. Petersburg, RUSSIA) - St. Petersburg State University;
  10. Thomas Leuther (Aubel, BELGIUM) - University of Liege;
  11. prof. Yves Nogier (Barquisimeto, VENEZUELA);
  12. Isabel Cristina Marquez de Mastromartino (Barquisimeto, VENEZUELA) - UCLA;
  13. Angel Mastromartino (Barquisimeto, VENEZUELA) - UCLA;
  14. Erik Caseres (Barquisimeto, VENEZUELA) - UCLA;
  15. Elizaveta Semenova (Moscow, RUSSIA) - Moscow State University;
  16. Ivan Kobyzev (St. Petersburg, RUSSIA) - St. Petersburg State University;
  17. Martin Wen-Yu Lo (Pasadena, U.S.A.) - Jet Propulsion Laboratory, Caltech;
  18. Irina Gorbunova (St. Petersburg, RUSSIA) - St. Petersburg State University;
  19. Jasmina Djordjevic (Nis, SERBIA);
  20. Victor Kasatkin (St. Petersburg, RUSSIA) - St. Petersburg State University;
  21. Svetlana Azarina (Voronezh, RUSSIA) - Voronezh State University;
  22. Fan Wu (Beijing, P. R. CHINA) - Capital Normal University;
  23. Irina Gorodetskaya (St. Petersburg, RUSSIA) - St. Petersburg State University;
  24. Julia Petrova (St. Petersburg, RUSSIA) - St. Petersburg State University;
  25. Alyona Vasilyeva (St. Petersburg, RUSSIA) - St. Petersburg State University;
  26. Denis Tugarev (Tambov, RUSSIA) - Derzhavin Tambov State University;
  27. Elena Vodolazhskaya (Tambov, RUSSIA) - Derzhavin Tambov State University;
  28. Marina Iljina (Tambov, RUSSIA) - Derzhavin Tambov State University;
  29. Anastasia Nishnevich (St. Petersburg, RUSSIA) - St. Petersburg State University;
  30. Dmitry Kharkovsky (St. Petersburg, RUSSIA) - St. Petersburg State University;
  31. Fabian Radoux (Liege, BELGIUM) - University of Liege;
  32. Ivko Dimitric (Pittsburgh, U.S.A) - Penn State University Fayette;
  33. Georges Weill;
  34. Mikhail Khristoforov (St. Petersburg, RUSSIA) - St. Petersburg State University;
  35. Monika Stypa (Naples, ITALY) - University of Salerno;
  36. Michael Werman (Jerusalem, ISRAEL) - The Hebrew University;
  37. Tadeusz Jagodzinski (Warsaw, POLAND) - Mathematics and Information Techniques Faculty;
  38. Carla Cruz (Aveiro, PORTUGAL) - University of Aveiro.

Accommodation, Meals and Classes

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 (look at the map). 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.

On the arrival day (July 16) only supper is served.
On the departure day (July 31) only breakfast is served
On the free day (July 23) no meals are served.

Classes take place every morning from July 17 to July 22 and from July 24 to July 29 in the headquarters of the Levi-Civita Institute (look at the map). Seminars, talks and other scientific activities will take place there in the afternoons, and diplomas delivery and closing ceremony will be organized there on the last day (July 30).

Free day

On Thursday, July 23, the School activity is suspended. Participants are free to visit the neighborhoods, to remain in Santo Stefano del Sole, or to take part to one of the guided excursions which will be proposed by the Organizing Committee. More information about excursion(s) will be duly posted herewith.
24px-Feature_suburban_buses.svg People who intend to go away from Santo Stefano del Sole during the free day and plan their own itinerary, should keep in mind that the buses from Avellino to Santo Stefano del Sole leave at 07:50, 10:10, 11:10, 12:10, 13:15, 14:00, 14:30, 16:00, 17:30, 18:30 and 20:00 (LAST BUS!).

Free day

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

Organizing committee

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

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

Diffiety School poster

The Organizing Committee expresses its gratitude to those who spread the news about the School in their University/Department, by displaying the official poster. A PDF electronic copy of it is available here for download.


LogoSantoStefano 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.
Upon their arrival in Santo Stefano del Sole, participants are kindly requested to present themselves at the registration desk, to be welcomed by the Organizing Committee. Registration desk will be open from 10:30 to 13:30 in the morning, from 15:00 to 21:30 in the evening and from 22:30 to 23:00 in the night of the arrival day (Thursday, July 16). Participants will be asked for registration fee, and given the designated accommodation. Registration desk is located in the headquarters of the Levi-Civita Institute (look at the map).Download the picture below: Levi-Civita_Location_LT2.
Levi-Civita_Location_LT2 copy

Reaching Santo Stefano del Sole is quite easy.
Foreign participants are kindly suggested to print and/or save an electronic copy of this page and to keep it for the whole duration of their stay in Italy.
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)