(Assignment 1, Due 26 March 3pm)
Notes will be uploaded shortly before or after the week's lectures. As there will be typos, please keep in mind that the repaired notes will be updated after the lectures.
(Week 1) Introduction, classical geometric laws of motion
Regarding affine/fibre bundle structure of Galilean space-time, see Chapter 2 of this article and Chapter 17 of Penrose's book Road to Reality.
(Week 2) Schrodinger equation, special relativity (optional)
(Week 3) Banach/Hilbert spaces, spectrum of bounded operators
(Week 4) Unbounded operators. Formal self-adjointness and SUSY QM
(Week 5) Genuine self-adjoint operators. Aharonov-Bohm effect.
(Week 6) Projection-valued measures and bounded spectral theorem.
(Week 7) Unbounded Spectral Theorem. Stone's Theorem, time evolution, symmetries
(Week 8) Differentiable manifolds. (SUSY QM (skipped))
(Week 9) Tangent and vector bundles, principal bundles, Lie groups
(Week 10) General fibre bundles, reduction of structure group
(Week 11) HOLIDAY, NO LECTURES
(Week 12) Differential forms
(Week 13) Lie group Geometry; Connections, curvature, parallel transport, covariant derivatives
(Week 14) Spin, spin connections
(Week 15) Atiyah-Singer-Dirac operators, Laplacians, Lichnerowicz identity
(Week 16) Quantum Hall effect and other topics
60% final exam. 40% take-home assignments (~2) and mid-term test (~1)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. 1. Functional Analysis. and Vol. II. Fourier analysis, Self-adjointness, Acad. Press, 1980, 1975
Moretti, V.: Spectral Theory and Quantum Mechanics, Springer, 2013
Strocchi, F.: An Introduction to the Mathematical Structure of Quantum Mechanics, World Scientific, 2005
Arveson, W.: A short course on spectral theory, GTM 209, 2002
Baez, J., Muniain, J.P.: Gauge Fields, Knots, and Gravity, World Scientific, 1994
Bleecker, D., Booss-Bavnbek, B.: Index theory with applications to mathematics and physics. International Press, 2013
Choquet-Bruhat, Y., Dewitt-Morette, C.: Analysis, manifolds, and physics. Part I: Basics. North Holland, 1982
Naber, G.: Topology, Geometry, and Gauge Fields. Foundations. Texts in Appl. Math., Springer, 2011
Lee, J.M.: Introduction to smooth manifolds. GTM 218, Springer, 2013
Friedrich, T.: Dirac operators in Riemannian geometry. GSM 25, AMS, 2000
Folland, G.B.: Quantum field theory. A tourist guide for mathematicians. Math. Surveys and Monogr., 149, 2008
Linear algebra, real analysis, metric and topological spaces
Basic abstract algebra (especially groups)
Idea of differentiable manifolds
Familiarlity with Hilbert spaces (at least semi-rigorously)
Exposure to theoretical physics is helpful, but not compulsory.