Mini-course: connections and curvatures in noncommutative geometry

2024 Spring

In March 2024, Prof. Adam Rennie (Wollongong) gave a mini-course on connections and curvatures in noncommutative geometry, based on joint work of him and Bram Mesland:

• B. Mesland and A. Rennie, Existence and uniqueness of the Levi-Civita connection on noncommutative differential forms.
• B. Mesland and A. Rennie, Curvature and Weitzenbock formula for spectral triples.
• B. Mesland and A. Rennie, Curvature and Weitzenbock formula for the Podleś quantum sphere.

Lectures took 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. The universal differential calculus and its failure.
• 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 lowerings and raisings with daggers.
• Connections on dagger bimodules. Hermitian connections. Graßmann connections. Conjugate connections.
• The torsion of a connection.
• Torsion-freeness and Hermitian connections.

Lecture 3

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