The practical task consists in the development of a program library which aims to represent a tool for tackling selected parametric polynomial and algebro-differential equation systems occuring in natural science applications and engineering. So there is the claim to produce some useful software at an advanced stage of the TERA project.
The theoretical task however, which can be even misunderstood as counter-productive to any generalistic approach, is the search for a satisfactory definitive proof (i.e. a proof independent of any open conjectures in mathematics or theoretical computer science) of our hypothesis, that only very exceptional, ``well suited", equation systems allow an efficient treatment.
This amounts to giving a rigorous proof of the impossibility to construct a general purpose software solving any generic polynomial or algebro-differential equation system in a reasonable amount of time and memory space.
Let us mention - by the way - that solving any one of these tasks is also subject of an award, the ``Jacques Morgenstern Challenge", set up by the University of Cantabria with the collaboration of a large number of other scientific organizations. For more information, consult http://matsun1.matesco.unican.es/CAG/
The resolution of this double task requires a well organized interdisciplinary work between so distant (or so close) fields as: algebraic, semialgebraic and diophantine geometry, singularity theory, differential topology, commutative and differential algebra, theory of invariants, diophantine approximation and transcendence theory representing the aspect of pure mathematics, numerical analysis and approximation theory representing the aspect of applied mathematics and complexity theory, computer algebra, seminumerical algorithms, data structures, programming languages and software engineering representing the aspect of theoretical and practical computer science.
These fields, at first glance unrelated, are subject to a unified view through our project oriented approach. Most of them are represented by at least one specialist in the working group.
The Project TERA
Complexity theory and TERA