• Istituto Italiano per gli Studi Filosofici, Naples;
• Diffiety Institute of the Russian Academy of Natural Sciences;
• Municipality of Forino

Contents

1. On the School (home page)
2. On this edition of the School
3. Staff of the School
4. Beginners' courses
5. Veterans' courses
6. Our address

On this edition of the School

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

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

1. 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)
2. 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
3. Introduction to differential calculus on commutative algebrae
   • Rings and commutative algebrae. Modules and bi-modules
   • Linear differential operators between modules
   • Comparison with the analytical definition of differential operator
   • Examples of algebraic differential operators
   • Bi-module 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 multi-derivations
   • Differential forms and the de Rham complex
   • Interior product and Lie derivative. Comparison with the geometric approach

Veterans' courses

1. Contact and symplectic multi-valued solutions of 1st order scalar differential equations
   • Elements of symplectic geometry
   • Lagrangian Manifolds
   • Multi-valued solutions of Hamilton-Jacobi equation
   • Elements of contact geometry
   • Geometrical interpretation of 1st order scalar differential operators
   • Multi-valued generalized solutions of 1 st order equations
   • Relations between symplectic and contact geometry
2. R-manifolds and multi-valued solutions of PDE's
   • Jet manifolds and partial differential equations (PDE)
   • Geometric interpretation of solutions. R-manifolds
   • 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
   • R-manifolds and generalized functions (distribution). "Quantization condition"
   • Singularity rays, singular R-panes and R-Grassmannians
3. Basic functors of differential calculus over commutative algebras
   • Generalities on functors of differential calculus
   • Functors Di and Pi and Diff-Spencer complexes
   • Algebraic jet-spaces and de Rham complexes
   • Spencer-type 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

• Internet: http://diffiety.ac.ru
• E-mail: diffiety @ tiros.dmi.unisa.it
• Fax: +39 089 965226
• Telephone: +39 089 965395