University of Edinburgh, 2011-2012.
Notes (mostly) by Teruji Thomas.
Note:
These notes are undoubtedly filled with errors with a variety of origins. Please let me know if you notice any.
Geometric Quantization
Teruji Thomas. August-September 2011.
Lecture 1
Symplectic geometry. Hamiltonian mechanics.
Lecture 2
Prequantization. The Heisenberg group.
Lecture 3
More Heisenberg group. Change of Lagrangian. Moyal product. Half-forms. The metaplectic gerbe.
Lecture 4
Polarizations. Half-forms again. Quantization. Metalinear and metaplectic structures.
Quantization of Elementary Dynamical Systems
Will Donovan. September-October 2011.
Lecture 1
Group actions. Coadjoint orbits.
Lecture 2
Lie theory. Flags. Nilpotent cone. Borel-Weil Theorem for complex groups.
Introduction to Quantum Field Theory
Teruji Thomas. October-November 2011.
Lecture 1
Generalizing quantum mechanics. The Wightman axioms. Symmetry.
Lecture 2
Free field theories. (In preparation...)
Topological Quantum Field Theory
January-April 2012.
1. Intro to topological quantum field theory
(Will Donovan)
Definition of TQFTs. Duality. 0+1 dimension.
2. The path integral formalism and Freed's finite gauge theory
(Teruji Thomas)
1+1 dimensions. The Path Integral Formalism. Group algebras and finite gauge theory (following Freed).
3. Gauge theory
(Yoshi Hashimoto)
Introduction to Gauge Theory. Connections. Classical Yang-Mills theory. Chern-Weil theory. A little about Seiberg-Witten.
4. The cobordism hypothesis
(Teruji Thomas)
The Cobordism Hypothesis. Infinity categories.
5. The Mumford conjecture
(Carmen Rovi)
Thom: cobordism groups as homotopy groups. Galatius-Madsen-Tillmann-Weiss:
homotopy types of cobordism categories.
6. Chern-Simons theory and knot invariants
(Teruji Thomas)
Witten: QFT and the Jones polynomial. Chern-Simons gauge theory.
Algebraic Quantum Field Theory
March-April 2012.
Lecture 1
(Teruji Thomas)
C
*-algebras. The commutative case. Gelfand, Gelfand-Naimark, Gelfand-Naimark-Segal. The geometry of the space of
states. 2x2 matrices.