Teaching

Here are the slides and some references for the courses I am currently teaching.


Stone Dualities

Lecture 1: Neighbourhoods

Lecture 2: Topology

Lecture 3: Lattices

Lecture 4: Filters

References:

1. Non-Hausdorff Topology and Domain Theory by Jean Goubault-Larrecq

A very nice introduction to both topology and order theory.

2. Topology and Groupoids by Ronald Brown

Another nice introduction to topology, including (topological) neighbourhood systems.

3. Convergence Foundations of Topology by Frédéric Mynard and Szymon Dolecki

A very interesting approach to topology via convergence.

(Note: the general neighbourhood systems in lecture 1 are the "pretopologies" found here).

4. Continuous Lattices and Domains by G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott

The canonical reference for order theory and (locally compact) topology.

5. Frames and Locales: Topology without points by Jorge Picado and Aleš Pultr

An excellent modern reference for point-free topology.

6. Stone spaces by Peter T. Johnstone

The classic reference for point-free topology.

7. Lattice Theory by Garrett Birkhoff

The classic reference for lattice theory.

8. General Lattice Theory by George Grätzer

Another classic reference for lattice theory, including the author's generalisation of Stone's original duality to distributive join-semilattices. The 2nd edition also includes some interesting appendices by various people.

9. Introduction to Lattices and Order by B. A. Davey and H. A. Priestley

Another great lattice theory reference, including Priestley duality.


Introduction to C*-Algebras

Lecture 1: Semigroups

Lecture 2: Involutions and Vector Spaces

Lecture 3: Algebra and Norms

Lecture 4: Hilbert Spaces

References:

1. Lecture Notes on C*-algebras by Ian F. Putnam

Excellent introductory notes on C*-algebras, including groupoid C*-algebras.

2. C*-Algebras and Operator Theory by Gerard J. Murphy.

Probably the best introductory C*-algebra book around.

3. C*-algebras and Their Automorphism Groups by Gert K. Pedersen

My personal favorite but a little tough for the beginner.

4. Operator Algebras: Theory of C*-Algebras and von Neumann Algebras by Bruce Blackadar

A very good encyclopedic up-to-date reference.

5. Hausdorff étale groupoids and their C*‡-algebras by Aidan Sims

Great introduction to groupoids. The C*-algebra material is more advanced though.


Once we have reviewed the necessary prerequisites, the course will roughly follow Putnam's notes.