Graph complexes are chain complexes populated by formal linear combinations of finite graphs, with a differential defined by collapsing edges. They were introduced by Maxim Kontsevich in the early 90's as a tool to compute the homology of certain infinite-dimensional Lie algebras which he related to various invariants in low-dimensional topology and group theory. Ever since then these complexes have been under heavy investigation, however their overall structure is still mysterious: We have learned many things about them, and found connections to various other fields of mathematics (deformation quantization, deformation theory of operads, GRT, invariants of manifolds/embedding spaces, (topological) quantum field theory etc.), but a complete understanding is still far out of sight.
This course aims to give an introduction to graph complexes (background and definition; 1st half of the course), with special emphasis on a geometric point of view (2nd half): We will realize them as cellular chain complexes of certain topological spaces and discuss what can be seen from this point of view (and what not). In particular, we will review the recent work of Francis Brown who used Feynman-esque integrals to define "differential forms on graph complexes".
- Kursverantwortliche/r: Marko Berghoff