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
(Austrian Academy of Sciences,
RICAM, Linz, Austria)
(Research Institute for Symbolic Computation,
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.
- 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:
Outline of Theoretical Module
- 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)
- 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
- Hints towards the future !
Outline of Practical Module:
Lectures, Exercise Classes and Exercises with Maple
- 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,
A summary of commands can be found
here for the package "OreModules"
and here for the package "OreMorphisms".
- 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,
The Maple worksheets for the exercises are
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
The second reference is an elementary introduction coming from a series of European courses; see
here for a preprint version.
- 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,
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.
- Gröbner basis computations over Ore algebras of functional operators (e.g.,
- 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).
The slides for the lecture notes are here:
constructive algebraic analysis and algebraic systems theory,
Factorization, Reduction and Decomosition,
Stafford and Quillen-Suslin theorems
- A. Quadrat, Systems and Structures, An algebraic analysis approach to mathematical systems theory, soon available here.