Recent developments in physics related to quantum groups had an immense impact on Algebra, bringing now deep problems and meaning to noncommutative algebra. The geometric background present in most phenomena transpires in the mathematical formalism. Howev er, this geometry remained "virtual" because a suitable geometry connected to the noncommutative algebra was lacking. Developing a geometric counterpart to the noncommutative algebra, often worded in terms of deformations or quantizations of algebras, req uires an abstract foundation allowing the desired generality of its applicability, but should also allow concrete calculations, either in the algebras or in (co)homology groups, reflecting real geometric aspects. The fundamental multidisciplinarity of the mathematics involved is obvious: category theory, algebraic geometry, topology and knot theory, homological methods, Lie theory, representation theory, differential geometry, rings of differential operators, with applications in physics, statistical physics or even Robotics (mobile robots). International and multidisciplinary cooperation is absolutely necessary for the development of this subject. On the European scale a concerted action integrating work-shops, exchange visits, study centers and schools , research in pairs, grants for researchers, intgrating cooperation of top specialists with a training aspect for young researchers in Europe, allow s for interaction at all research levels in many directions.
Project leader: Arnfinn Laudal