Universitą degli studi di Modena e Reggio Emilia

Crystallization theory is a graph-theoretical representation method for compact PL-manifolds of arbitrary dimension, with or without boundary, which makes use of a particular class of edge-coloured graphs, which are dual to coloured (pseudo-) triangulations. These graphs are usually called gems, i.e. Graph Encoding Manifolds. One of the principal features of crystallization theory relies on the purely combinatorial nature of the representing objects, which makes them particularly suitable for computer manipulation.

Recent research by the local group focuses on:

• relationships between crystallization theory and coloured tensor models in high dimensional quantum gravity
• generation of catalogues of PL-manifolds for increasing values of the vertex number of the representing graphs, both in dimension three and four
• definition and/or computation of invariants for PL-manifolds, directly from the representing graphs, in dimension n ? 3
• trisections of PL 4-manifolds with boundary