AACA '09

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


Markus Rosenkranz (Austrian Academy of Sciences, RICAM, Linz, Austria)
Franz Winkler (Research Institute for Symbolic Computation, Linz, Austria)


Jean-Francois Pommaret (Ecole Nationale des Ponts et Chaussées, France)
Alban Quadrat (INRIA, Sophia Antipolis, France)


The course contains two modules: Each day is divided into four blocks: 8:30-10:00 / 10:30-12:00 / 13:30-15:00 / 15:30-17:00.

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:

Outline of Theoretical Module

Monday morning

Monday afternoon

Tuesday morning Tuesday afternoon Wednesday morning Wednesday afternoon

Outline of Practical Module: Lectures, Exercise Classes and Exercises with Maple

Thursday Friday A summary of commands can be found here for the package "OreModules" and here for the package "OreMorphisms".
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.

Theoretical Module

With only a slight abuse of language, one can say that the birth of the " formal theory" of systems of ordinary differential (OD) equations or partial differential (PD) equations is coming from the work of M. Janet in 1920 along algebraic ideas brought by D. Hilbert at the same time in his study of sygyzies for finitely generated modules over polynomial rings. The work of Janet has then been used (without any quotation !) by J.F. Ritt when he created "differential algebra" around 1930, namely when he became able to add the word "differential" in front of most of the classical concepts concerned with algebraic equations, successively passing from OD algebraic equations to PD algebraic equations. In 1965 B. Buchberger invented Groebner bases, named in honor of his PhD advisor W. Groebner, whose earlier 1940 work on polynomial ideals and PD equations with constant coefficients provided a source of inspiration. However, Janet and Grobner approaches suffer from the same lack of "intrinsicness" as they both highly depend on the ordering of the n independent variables and derivatives of the m unknowns.

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.


The second reference is an elementary introduction coming from a series of European courses; see here for a preprint version.

Practical Module

The purpose of the practical part of the lectures is to give deeper insights into constructive issues of algebraic analysis, present their implementations in the symbolic packages OreModules, OreMorphisms, Stafford, QuillenSuslin and Serre, and illustrate them by means of different problems coming from engineering sciences and physics.

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

The different results and constructive algorithms will be illustrated by examples coming from mathematical systems theory, control theory and mathematical physics. Finally, the attendees will have to study explicit problems by means of the packages OreModules, OreMorphisms, Stafford, QuillenSuslin and Serre.


The slides for the lecture notes are here: constructive algebraic analysis and algebraic systems theory, Factorization, Reduction and Decomosition, Serre Reduction, Stafford and Quillen-Suslin theorems