Exam: Thursday 9 January 2025 2pm. Venue to be announced.
Lecture on Thursday 5 December 2024
No lecture on Tuesday 10 December 2024
Lecture on Thursday 12 December 2024
Lecture on Tuesday 17 December 2024
Lecture on Thursday 19 December 2024
Lecture on Tuesday 24 December 2024
No Lecture on Thursday 26 December 2024
I will upload notes before classes so you can read ahead. But they may be updated before class. Typos will be corrected after class. Plan your note-taking methods accordingly.
(Part 1 (19 Sept update: I removed "commutative" from Gelfand transform where possible.)) Spectral view of topology: Gelfand-Naimark duality
(Part 2 (15 Oct: typos corrected)) Vector bundles from NC viewpoint: Serre-Swan duality
(Part 3 (17 Oct: typos corrected)) Key structural aspects of C*-algebras
(Part 4 (24 Oct: typos corrected)) K0 for general rings
(Part 5 (31 Oct: typos corrected)) K0 functor for C*-algebras
(Part 6 (7 Nov: typos corrected)) Stability of K0
(Part 7 (11 Nov: typos corrected) ) Fredholm Index, Toeplitz index theorem
(Part 8 (16 Nov: typos corrected) ) K1 functor
(Part 9 (28 Nov: typos corrected) ) Index connection from K1 to K0
(Part 10 (28 Nov: typos corrected) ) Fredholm index from K-theory viewpoint
(Part 11 (3 Dec: typos corrected) ) Suspensions and Bott element
(Part 12 (5 Dec: typos corrected)) Bott periodicity theorem
(Part 13 ) Exponential map and cyclic exact sequence
(Part 14 ) 2D topological insulators
Possible upcoming topics: Spin geometry and Dirac operators. Spectral view of Riemannian geometry. Noncommutative differential geometry. Cyclic cohomology
(A bit of Banach/Hilbert space basics) If you need a refresher!
Notes on geometry, gauge theory and physics motivated by quantum mechanics (via classical mechanics). This may be helpful if you know physics but need a crash course in differential geometry, or if you want to know "why" differential geometry without knowing anything about modern physics
(Assignment 1) Due 10 October, 3pm.
(Assignment 2) Due 7 November, 3pm.
(Assignment 3) Due 28 November, 3pm.
(Assignment 4) Due 19 December, 3pm.
60% final exam. 40% take-home assignments (~4)
Basic Functional analysis, Differential geometry and topology, algebra. Some spectral theory/quantum mechanics exposure.
K-theory and C*-algebras, N.E. Wegge-Olsen
Introduction to K-theory for C*-algebras, Rordam, Larsen, Lausten
Elements of noncommutative geometry - Gracia-Bondia, Varilly, Figueroa
Connes, Noncommutative Geometry
Khalkali, Very Basic NCG
Murphy, C*-algebras and operator theory
Arveson, A short course on spectral theory