Paris Seminar Notice: http://www.di.ens.fr/~longo/geocogni.html Groupe de Travail GÃOMÃTRIE ET COGNITION (Giuseppe LONGO*, Jean PETITOT°, Bernard TEISSIER*) 30 Mai : R. Brown (Math., Univ. of Wales) "The intuitions of higher dimensional algebra for the study of structured space." 16h, SALLE U ou V, étage -2 , Dépts. de Math. et d'Informatique, ENS, 45, Rue D'Ulm, 75005 Paris: ABSTRACT One definition of Higher Dimensional Algebra is that it is about algebraic structures whose operations are partially defined under geometric conditions, and whose axioms are defined by the geometry - so for example one considers composing squares and cubes in various directions. This allows for a more detailed transition from geometry to algebra. One overall aim is to provide new tools for local-to-global problems by the procedure `algebraic inverses to subdivision'. The pursuit of this aim has led to a range of new algebraic structures, new understanding and computations in algebraic topology, as well as new tools for concurrency problems. Its potential relevance to this seminar is the speculative idea that a mathematics `on a line' cannot be adequate for describing the complex modular and hierarchical interactions of the brain. So Higher Dimensional Algebra can perhaps open out new modelling possibilities for future development. For more background, see http://www.bangor.ac.uk/~mas010/hdaweb2.htm The structure of the talk will be 1. Origins and intuitions 2. Landmark theorems: these are of the type (a) equivalent descriptions of the algebraic structures (b) gluing (colimit) theorems (c) realisation of algebra by spaces 3. Computational issues 4. The future? (The aim is to give an impression of this area for a more general audience, and so consideration of `weak structures' and many other applications will be omitted.) Ronnie Brown