Summer School on Algebraic Analysis and Computer Algebra

New Perspectives for Applications

RISC-Linz, Castle of Hagenberg, Austria, July 13-17, 2009

Collocated with the Fourth RISC/SCIEnce Training School

**
**

Franz Winkler (Research Institute for Symbolic Computation, Linz, Austria)

Alban Quadrat (INRIA, Sophia Antipolis, France)

- 13-15 July: Theoretical Module (J.-F. Pommaret)
- 16-17 July: Practical Module (A. Quadrat)

The Theoretical Module takes place in the Hochzeitsraum (Wedding Chamber!), the room to the right
of the Gemeindesaal (Community Hall).

The Practical Module takes place in the Rittersaal
(Seminar Room), and it will include interactive Maple exercises.

There will be a Summer School Dinner in the Hagenberg Schlossrestaurant on Thursday at 7:00pm.

For seeing pictures of the Summer School, you will need the password that was sent out
to the participants.

The pictures are divided into four parts:

- Motivating examples and problems
(control theory, continum mechanics, hydrodynamics, electromagnetism, general relativity, systems depending on parameters, ... )

- Introduction to homological algebra through applications (sequences and diagrams, diagram chasing, controllability indices)

Monday afternoon

- Systems of partial differential equations (jet theory, linear and nonlinear systems, formal linearization, symbols, Spencer cohomology, formal integrability, involution)
- Elementary introduction to the formal theory of Lie pseudogroups through examples

- Linear systems of partial differential equations (Janet and Spencer differential sequences, modified Spencer form, characteristic variety)
- Comparison with Groebner bases on explicit examples

- Rings and modules (homomorphisms and tensor products, short exact sequences, presentation, localization, resolution, extenson modules)

- Rings of differential operators and differential modules
- Formal adjoint and side changing functor

- Solution of the problems
- Classification of modules and systems
- Purity
- Hints towards the future !

- Skew polynomial rings, Ore algebras and Gröbner basis techniques
- Constructive module theory and homological algebra (characterizations of module properties, extension functor, Baer's extensions, basis computation - Quillen-Suslin and Stafford theorems)
- Mathematical systems theory (interpretation of module properties, Monge parametrization)

- Study of the factorization, reduction and decompositions problems
- Study of Serre's reduction
- Study of the purity filtration of differential modules (case n=2)

You can work through the exercise sheets Module Theory I, Module Theory II, Factorization/Reduction/Decomposition, Serre's Reduction.

The Maple worksheets for the exercises are here.

Meanwhile, "commutative algebra", namely the study of modules over rings, was facing a very subtle problem, the resolution of which led to the modern but difficult "homological algebra" with sequences and diagrams. Roughly, one can say that the problem was essentially to study properties of finitely generated modules not depending on the " presentation" of these modules by means of generators and relations. This very hard step is based on homological/cohomological methods like the so-called "extension" modules which cannot therefore be avoided.

As before, using now rings of "differential operators" instead of polynomial rings led to "differential modules" and to the challenge of adding the word "differential" in front of concepts of commutative algebra. Accordingly, not only one needs properties not depending on the presentation as we just explained but also properties not depending on the coordinate system as it becomes clear from any application to mathematical or engineering physics where tensors and exterior forms are always to be met like in the space-time formulation of electromagnetism. Unhappily, no one of the previous techniques for OD or PD equations could work !.

By chance, the intrinsic study of systems of OD or PD equations has been pioneered in a totally independent way by D. C. Spencer and collaborators after 1960, using jet theory and diagram chasing in order to relate differential properties of the equations to algebraic properties of their "symbol", a technique superseding the "leading term" approah of Janet or Grobner but quite poorly known by the mathematical community.

Accordingly, it was another challenge to unify the "purely differential" approach of Spencer with the "purely algebraic" approach of commutative algebra, having in mind the necessity to use the previous homological algebraic results in this new framework. This sophisticated mixture of differential geometry and homological algebra, now called "algebraic analysis", has been achieved after 1970 by V. P. Palamodov for the constant coefficient case, then by M. Kashiwara and B. Malgrange for the variable coefficient case.

The purpose of this intensive course held at RISC is to provide an introduction to "algebraic analysis" in a rather effective way as it is almost impossible to learn about this fashionable though quite difficult domain of pure mathematics today, through books or papers, and no such course is available elsewhere. Computer algbra packages like "OreModules" are very recent and a lot of work is left for the future.

Accordingly, the aim of the course will be to bring students in a self-contained way to a feeling of the general concepts and results that will be illustrated by many academic or engineering examples. By this way, any participant will be able to take a personal decision about a possible way to involve himself into any future use of computer algebra into such a new domain and be ready for further applications.

MAIN REFERENCES:

- J.-F. Pommaret, Partial Differential Control Theory, Kluwer, 2001, 2 vol, 1000
pp (See Zentralblatt review Zbl 1079.93001).

- J.-F. Pommaret, Algebraic Analysis of Control Systems Defined by Partial Differential Equations, in Advanced Topics in Control Systems Theory, chapter 5, Lecture Notes in Control and Information Sciences, LNCIS 311, Springer, 2005, 155-223.

In particular, we shall focus on different aspects of constructive algebra, module theory and homological algebra such as:

- Gröbner basis computations over Ore algebras of functional operators (e.g., differential/shift/time-delay/difference operators).
- Computation of finite free resolutions, dimensions, homomorphisms, tensor products, extension and torsion functors.
- Classification of module properties (e.g., torsion, torsion-free, reflexive, projective, stably free, free, decomposable, simple, pure modules) and their system-theoretic interpretations (e.g., autonomous elements, minimal/successive/injective/Monge parametrizations, Bezout identities, factorization/reduction and decomposition problems).

MAIN REFERENCE:

- A. Quadrat, Systems and Structures, An algebraic analysis approach to mathematical systems theory, soon available here.