Mathematisches Kolloquium

Zusammenfassung: The aim is higher dimensional nonabelian tools for local-to-global problems. Part of the interest is that this turns out to require higher dimensional notions of abstract groups, in order to model not only invertible processes, but also processes of processes, etc. Hence the algebraic methods have also found interest in concurrency in computer science. The testing ground has been algebraic topology. The intuitions involve simple ideas of subdivision and multiple compositions of cubes, where the compositions are partially defined under geometric conditions. This key idea of "higher dimensional algebra" allows for new and powerful algebraic structures for modelling geometry. We generalise paths to "flows". A main achievement is that the gluing of spaces is partially modelled by "gluing" algebraic structures, using a general notion of "colimit", which will be explained. This leads to new calculations. These ideas have yielded new tools in group theory as well as in algebraic topology, and in other fields, such as holonomy. (see "Higher dimensional group theory": www.bangor.ac.uk/~mas010/hdaweb2.html) Recent work was partially supported by a Leverhulme Emeritus Fellowship, 2002-2004.

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