Matrix factorisations are to isolated hypersurface singularities as bounded complexes of coherent sheaves are to compact Calabi-Yau varieties, and they are organised by various kinds of categories: the matrix factorisations of an individual singularity form a triangulated category, which may be refined to a differential graded category (or even better/worse an A-infinity category), while the collection of all isolated hypersurface singularities may be organised into a bicategory with matrix factorisations as 1-morphisms between singularities. I will begin with a general introduction to this story. 

One of the reasons to like matrix factorisations is that one can get one’s hands dirty and compute with them! I will explain some of the interesting structures (Clifford representations and Atiyah classes) that arise in the concrete details of working with bicategories and A-infinity categories of matrix factorisations.