In 2023 fall semester, we are running a seminar on foliations, focusing on their noncommutative geometry. The noncommutative geometry of foliations was initiated by Alain Connes in the late 1970s, when he realised that the geometry of a foliation can be encoded in a noncommutative C*-algebra. The construction is in two steps: one first builds the holonomy groupoid of the foliation, then takes its reduced groupoid C*-algebra. This C*-algebra might be viewed as the “noncommutative space” of leaves, and is the desired receptacle for the index theory of longitudinal elliptic operators. This was achieved in the seminal work of Connes and Skandalis. We will reach this topic at the end of this seminar.
Another focus of the seminar is to understand certain foliations arising from dynamical systems. Lindsey and Treviño constructed translation surfaces from bi-infinite Bratteli diagrams using combinatorial methods. These were further studied by Putnam and Treviño with operator algebraic techniques in a recent print. Putnam and Treviño built explicit relations between the groupoid C*-algebras constructed from the Bratteli diagram, and the C*-algebras constructed from suitable foliations on the translation surface. This also allows them to compute the K-theory of the aforementioned C*-algebras.
Tuesday 15:15-17:00. Most of the talks will be in Gorlaeus building. The rooms are a little bit hard to hard (see Science Campus map). Instead: you may join us from Snellius building at 15:05.
Yuezhao Li (y.li AT math.leidenuniv.nl)
I. Moerdijk and J. Mrčun, Introduction to foliations and Lie groupoids.
Calvin C. Moore and Claude L. Schochet, Global analysis on foliated spaces.
Ian F. Putnam and Rodrigo Treviño, Bratteli diagrams, translation flows and their C*-algebras. arXiv: 2205.01537
A. Connes and G. Skandalis, The longitudinal index theorem for foliations.
Notes of the talk are available here. Last updated: Nov 19, 2023.
Another version with smaller fonts (10pt by default) and narrower margins (0.8cm) is here (This version is not suggested as I do not concern much with its formatting).
Most of the talks are in Gorlaeus building.
On October 17 we are in Huygens building 2.11-2.14.
On October 31 we are in Gorlaeus lecture hall C2.
|Foliations: motivations, definitions and examples
|Holonomy and stability
|No talk (Liberation Day)
|The C*-algebra of a foliation
|Translation surfaces and bi-infinite Bratteli diagrams
|Lecture hall C2
|The surface associated with a bi-infinite Bratteli diagram
|Translation surfaces, groupoids and C*-algebras
|Fredholm modules and K-theory of Bratteli diagrams