Mini-course: connections and curvatures in noncommutative geometry

2024 Spring

In 2024 spring semester, Prof. Adam Rennie (Wollongong) will give a mini-course on connections and curvatures in noncommutative geometry. Lectures will take place on Tuesdays from 15.15 to 17.00

Date	Room
Mar 5	Snellius 4.01
Mar 12	Snellius 4.01
Mar 19	New Gorlaeus BM 2.26

Outline

Lecture 1

• 1st-order differential calculus of a spectral triple, connections.
• The abstract picture and the universal differential calculus, its failure.
• The junk forms.
• 2nd-order differential calculus of a spectral triple, the curvature of a connection.
• A non-junk factoid.

Lecture 2

• Dagger bimodules, frames, left/right adjointable operators.
• Index lowering and raising with daggers.
• Connections on dagger bimodules, Hermitian connections, Graßmann connections, conjugate connections.
• Torsion of a connection.
• Torsion-free and Hermitian connections.

Lecture 3

• The two projections problem, concordant submodules.
• Concordant Hermitian differential structures.
• Bram Mesland’s condition for dagger-concordance. Uniqueness of the Levi-Civita connection.
• Uniqueness of the Levi-Civita connection.
• Braidings, “real” forms, dagger-bimodule connections.
• Example: the Podleś sphere.