[Apologies if you receive multiple copies] ****************************************************************** * AACA'09 * * * * Summer School on Algebraic Analysis and Computer Algebra * * New Perspectives for Applications * * * * July 13-17, 2009 * * RISC, Castle of Hagenberg, Austria * * * * http://www.risc.uni-linz.ac.at/about/conferences/aaca09/ * ****************************************************************** - Organizers Markus Rosenkranz (Austrian Academy of Sciences, RICAM, Linz, Austria) Franz Winkler (Research Institute for Symbolic Computation, Linz, Austria) - Lecturers Jean-Francois Pommaret (Ecole Nationale des Ponts et Chaussees, France) Alban Quadrat (INRIA, Sophia Antipolis, France) - Schedule The course contains two modules: 13-15 July: Theoretical Module (J.-F. Pommaret) 16-17 July: Practical Module (A. Quadrat) Each day is divided into four blocks: 8:30-10:00 / 10:30-12:00 / 13:30-15:00 / 15:30-17:00 The event is collocated with the Fourth RISC/SCIEnce Training School: http://www.risc.uni-linz.ac.at/projects/science/school/fourth/general.html Possible funding via RISC/SCIEnce Transnational Access Programme: http://www.risc.uni-linz.ac.at/projects/science/access/ In case of interest, please reply to this email. =============================================================================== - Abstract of 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 Groebner 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 Groebner 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 for Theoretical Module 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. The second reference is an elementary introduction coming from a series of European courses. =============================================================================== - Abstract of 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, Quillen-Suslin and Serre, and illustrate them by means of different problems coming from mathematical systems theory, control theory and mathematical physics. In particular, we shall focus on different aspects of constructive algebra, module theory and homological algebra such as: * Groebner 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). 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. - Main Reference for Practical Module A. Quadrat, Systems and Structures, An algebraic analysis approach to mathematical systems theory, soon available ===============================================================================