New Edition of the Diffiety School
The IV Italian School
Forino (Avellino), ITALY, 17 July  29 July 2000
In cooperation with
 Istituto Italiano per gli Studi Filosofici, Naples;
 Diffiety Institute of the
Russian Academy of Natural Sciences;
 Municipality of Forino
and under the scientific direction of
Prof. A. M. Vinogradov
(Universita' di Salerno), we are announcing the
fourth edition of the Diffiety School.
Contents
 On the School (home page)
 On this edition of the School
 Staff of the School
 Beginners' courses
 Veterans' courses
 Our address
On this edition of the School
In this edition the School will offer two series of courses, one for
beginners and one for veterans. Teachers and programs are
listed below. Beginners' course material is the same as first edition
school, and is available here.
We invite interested people to look in our web site or to mail us for information about past editions and
current activities. Here is also a
poster of our
School in .pdf format; it would be of great help for us if you could print
it out and show up in your In
stitution.
Staff of the School
Here follows the list of teachers and tutors in the School:
A. M. Vinogradov, Director (Universita' di Salerno)
V. Chetverikov (Technical University of Moscow)
A. De Paris (Universita' "Federico II" di Napoli)
S. Igonin (Independent University of Moscow)
N. Khorkova (Moscow State Tecnical University "Bauman")
M. Marvan (Silesian University at Opava, Czech Rep.)
A. Verbovetsky (Independent University of Moscow)
M. Vinogradov (Diffiety Institute)
R. Vitolo (Universita' di Lecce)
V. Yumaguzhin (State University of Pereslavl'Zalessky, Russia)
And here's the organising committee:
C. Bove, F. Canfora,
D. Catalano,
S. Cioffi,
A. De Paris
G. Manno,
B. Prinari,
F. Pugliese,
L. Vitagliano,
R. Vitolo.
Beginners' courses
 General Introduction
 Observation mechanism in classical physics
 Commutative algebrae
 States of a classical system
 Spectra of a commutative algebra
 Spectral theorem (manifolds as spectra of commutative algebrae)
 First order differential calculus on man
ifolds
 Tangent vectors and absolute and relative vector fields. D functor
 Behaviour of tangent vectors and vector fields with respect to
smooth
mappings of manifolds
 The flow of a vector field. Lie derivative of vector fields.
Commutators and Lie algebrae
 Tangent covectors and differential forms. Tensors. Main operations
on
tensors. Algebra of differential forms
 Behaviour of differential forms and covariant tensors with respect
to
differentiable mappings of manifolds. Lie derivatives of covariant
tensors.
 Exterior differential and de Rham cohomology
 Cartan's formula and homotopy property of de Rham cohomology
 Integration theory as de Rham cohomology
 Introduction to differential calculus on commutative
algebrae
 Rings and commutative algebrae. Modules and bimodules
 Linear differential operators between modules
 Comparison with the analytical definition of differential operator
 Examples of algebraic differential operators
 Bimodule structure in the set of linear differential operators
 Composition of differential operators
 Symbol of a linear differential operator. Algebra of symbols
 Categories and functors, representative objects
 Functors of the algebraic calculus and their representability
 Derivations and multiderivations
 Differential forms and the de Rham complex
 Interior product and Lie derivative. Comparison with the geometric
approach
Veterans' courses

Contact and symplectic multivalued solutions
of 1st order scalar differential equations
 Elements of symplectic geometry
 Lagrangian Manifolds
 Multivalued solutions of HamiltonJacobi equation
 Elements of contact geometry
 Geometrical interpretation of 1st order scalar differential
operators
 Multivalued generalized solutions of 1
st order equations
 Relations between symplectic and contact geometry
 Rmanifolds and multivalued solutions of
PDE's
 Jet manifolds and partial differential equations (PDE)
 Geometric interpretation of solutions. Rmanifolds
 High order contact transformations
 Extrinsic and intrinsic geometries of PDE
 Singularities of solutions of PDE and associated singularity
equations
 Singularity equations for second order equations in two variables
 Symbols and propagation of singularities
 Rmanifolds and generalized functions (distribution). "Quantization
condition"
 Singularity rays, singular Rpanes and RGrassmannians
 Basic functors of differential calculus over
commutative algebras
 Generalities on functors of differential calculus
 Functors Di and Pi and DiffSpencer complexes
 Algebraic jetspaces and de Rham complexes
 Spencertype complexes associated with a differential operator and
differential equations
 (Spencer type cohomologies and their geometric interpretation
 Higher Spencer and de Rham cohomology
 Green formula
 Algebraic Lagrangian formalism
The deadline has been shifted to 30 June 2000.
As for accommodation, we invite interested people to
read this.
Detailed information on how to reach
Forino is here.
Address
For more details about past edi
tions of the School and current activities,
look in our web site, drop us an email request, fax or phone to us at
 Internet:
http://diffiety.ac.ru
 Email: diffiety @ tiros.dmi.unisa.it
 Fax: +39 089 965226
 Telephone: +39 089 965395
Questions and suggestions should go to
diffiety @ tiros.dmi.unisa.it