Methods of Modern Mathematical Physics 2024

 

Exam

Date/time: Thursday 9 January 2025 2-4pm.

Venue: Classroom Building 2, Room 505

 

Announcements

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

 

Lecture notes

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 (12 Dec: typos corrected)) Exponential map and cyclic exact sequence

(Part 14 (19 Dec: typos corrected)) 2D topological insulators

(Part 15 ) Pimsner-Voiculescu sequence

(Part 16 ) Noncommutative tori and quantum physics

Miscellaneous notes

(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

 

Assignments

(Assignment 1) Due 10 October, 3pm.

(Assignment 2) Due 7 November, 3pm.

(Assignment 3) Due 28 November, 3pm.

(Assignment 4) Due 19 December, 3pm.

 

Assessment (planned)

60% final exam (Closed book!). 40% take-home assignments (~4)

 

Prerequisites

Basic Functional analysis, Differential geometry and topology, algebra. Some spectral theory/quantum mechanics exposure.

 

References

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