Ronnie Brown Seminar at Liverpool University School of Mathematical Sciences, 4 December, 2009 What is and what should be `Higher dimensional group theory'? ABSTRACT: The presentation will show, including some knot demos, some of the problems and intuitions which have led to this question, and how certain cubical algebraic structures with partial operations whose domains are given by geometric conditions have been found quite natural for expressing modes of higher dimensional subdivision and composition which are related to long term concerns in algebraic topology. pdf file (2MB) of beamer presentation available from http://www.bangor.ac.uk/r.brown/brownpr.html Disclaimer: The seminar could not in one hour answer the questions posed in the title, or reference the literature! More information can be obtained by a web search on "Higher dimensional algebra". [For admin and other information see: http://www.mta.ca/~cat-dist/ ]