Week 0 (Sept. 23) - A history of n-categorical physics from Maxwell's thoughts on relativity to Weyl's introduction of "gauge invariance" to Heisenberg's matrix mechanics.
Week 1 (Sept. 28, 30) - A history of n-categorical physics from Born's probability interpretation of quantum mechanics, to Feynman path integrals, to Mac Lane's introduction of monoidal and symmetric monoidal categories.
Week 2 (Oct. 5, 7) - A history of n-categorical physics from Bénabou's introduction of bicategories to Penrose's spin networks.
Week 3 (Oct. 12, 14) - A history of n-categorical physics from the Ponzano-Regge model of 3d quantum gravity, to Grothendieck's dreams about ω-categories, to string theory.
Week 4 (Oct. 19, 21) - A history of n-categorical physics from Segal and Atiyah's definitions of conformal and topological quantum field theories to Joyal and Street's definition of braided monoidal categories.
Week 5 (Oct. 26) - Conclusion of our history of n-categorical physics: from TQFTs to the periodic table of n-categories.
Week 6 (Nov. 2, 4) - Constructing 2d TQFTs from semisimple algebras: an exposition of the work of Fukuma, Hosono and Kawai.
Week 7 (Nov. 9) - Constructing 2d TQFTs from semisimple algebras, continued.
Week 8 (Nov. 16, 18) - Constructing 2d TQFTs from semisimple algebras, concluded.
Week 9 (Nov. 23) - Computing the vector space for a circle in a 2d TQFT: it's the center of the semisimple algebra we started with!
Week 10 (Nov. 20, Dec. 2) - A final description of the 2d TQFT obtained from a semsimple algebra. A key example of a semisimple algebra, leading ultimately to gauge theory: the group algebra of a finite group. A sneak preview of next quarter, in which we'll build 3d TQFTs by categorifying all these ideas.
Week 2 (Jan. 11, 13) - Iterated index notation for categorified tensors; spin foams as diagrams for categorified tensors.
Week 3 (Jan. 18, 20) - The concept of "extended TQFT", and how to construct 3d extended TQFTs from semisimple 2-algebras. Deriving the 2-3 Pachner move from the pentagon identity. Two equivalent forms of the bubble move.
Week 4 (Jan. 25, 27) - On Tuesday, John gave a talk on quantum gravity to computer scientists in Vancouver. On Thursday, we saw how to derive the 1-4 Pachner move from the 2-3 move and the bubble move, and stated Matveev's theorem describing 3-manifolds in terms of special spines.
Week 5 (Feb. 1, 3) - Gauge theory on a triangulated manifold. Connections and gauge transformations when spacetime is a graph; flat connections when spacetime is a simplicial 2-graph.
Week 6 (Feb. 5, 7) - Computing the partition function in the 2d Dijkgraaf-Witten model, a TQFT built from the semisimple algebra C[G] (the group algebra of a finite group G). In this model, the partition function is a path integral over the space of flat connections mod gauge transformations.
You can learn more about the Dijkgraaf-Witten models from their original paper (especially starting on page 42):
Week 7 (Feb. 15, 17) - Description of the space of flat connections mod gauge transformations on a manifold M in terms of the fundamental group of M. Similar descriptions of other "moduli spaces". A beautiful formula for the partition function of the Dijkgraaf-Witten model in terms of the "groupoid cardinality" of the "moduli stack" of flat connections. (For an explanation of groupoid cardinality, see the Winter 2004 notes, especially starting on page 53 of week 6.)
Week 8 (Feb. 22, 24) - Twisting the multiplication in C[G] by a 2-cocycle; twisting the associator in Vect[G] by a 3-cocycle. Twisted Dijkgraaf-Witten models. Group cohomology and Pachner moves.
Week 9 (Mar. 1, 3) - Group cohomology and TQFTs. The cohomology of the group G as the cohomology of its classifying space BG. Computing group cohomology.
Week 10 (Mar. 8, 10) - Computing group cohomology, continued. Computing the cohomology of Z/2 using the fact that its classifying space is the infinite-dimensional projective space RP∞. Summary of what we've done so far this year.
Week 2 (Apr. 5, 7) - Connections on principal G-bundles. Parallel transport. Connections as Lie-algebra valued 1-forms on the total space of the principal bundle.
Week 3 (Apr. 12, 14) - Trivializations and connections: using a trivialization of a principal bundle to think of connections as Lie-algebra valued 1-forms on the base space. Associated bundles.
Week 4 (Apr. 19, 21) - Parallel transport and covariant derivatives. Calculating covariant derivatives with the help of a trivialization.
Week 5 (Apr. 26, 28) - Covariant derivatives and curvature. Exterior covariant derivatives. Ad(P)-valued differential forms.
Week 6 (May 5, 10) - Gauge transformations: how they act on principal bundles, associated bundles, sections, connections and their curvatures.
Week 7 (May 10, 12) - Towards EF theory (usually called BF theory). Discrete versus smooth connections.
Week 8 (May 17, 19) - The map from smooth to discrete connections; the map from smooth to discrete gauge transformations. The moduli space of flat connections versus the moduli space of flat bundles.
Week 9 (May 24, 26) - The moduli space of flat bundles: examples when the gauge group is U(1), SU(n), and SO(3). From the 2d Dijkgraaf-Witten model to 2d EF theory. The moduli stack of flat bundles.
Week 10 (May 31) - The 2d Dijkgraaf-Witten model and EF theory: measures and Hilbert spaces.