My Students' Theses (and Other Papers)

John Baez

Here are papers by students of mine, especially their theses. You can also see a picture of some of these folks!

• James Gilliam did his thesis on "discrete mechanics", a generalization of classical mechanics in which both the phase space and time are discrete. There's a lot of work on mechanics in which time takes integer values, so the real novelty here was adapting the use of calculus in physics to situations where the phase space is also discrete. It turns out that the Euler-Lagrange equation, Noether's theorem, and the symplectic structure on phase space all generalize to this context! This requires some ideas from algebraic geometry - but don't worry, these are explained from scratch in the thesis. Interesting examples include discrete versions of the harmonic oscillator, the rigid rotating body in n dimensions, and a variety of cellular automata including the Wess-Zumino model. Some but not all of this material was also published as a paper:
  • James Gilliam, Lagrangian and Symplectic Techniques in Discrete Mechanics, PhD thesis, U. C. Riverside, 1996. In PDF and Postscript.

  • John C. Baez and James Giliam, An algebraic approach to discrete mechanics, Lett. Math. Phys. 31 (1994), 205-212. In PDF and Postscript.

• Laurel Langford did her thesis on "2-tangles". Just as a tangle is a bunch of curves embedded in 3-dimensional space, a 2-tangle is a bunch of surfaces embedded in 4-dimensional space. Work on quantum group knot invariants got everyone excited over the fact that tangles form a braided monoidal category. Similarly, 2-tangles form a braided monoidal 2-category! And not just any old braided monoidal 2-category, either: Laurel proved that it's the "free braided monoidal 2-category with duals on an unframed object". The thesis explains these concepts and translates work of Carter, Rieger and Saito into 2-categorical language to prove this result.

  A summary of Laurel's thesis appeared in Letters in Mathematical Physics - but with lots of misprints that aren't in the version below. A much more detailed account appeared as "HDA4" - the fourth of a series of papers on higher-dimensional algebra.

  • Laurel Langford, 2-Tangles as a Free Braided Monoidal 2-Category with Duals, PhD thesis, U. C. Riverside, 1997. Available in Postscript.

  • John C. Baez and Laurel Langford, 2-Tangles, Lett. Math. Phys. 43 (1998), 187-197.

  • John Baez and Laurel Langford, Higher-dimensional algebra IV: 2-tangles, Adv. Math. 180 (2003), 705-764.

  • J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, Laurel Langford and Masahico Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, ERA Amer. Math. Soc. 05 (1999), 146-156.

• Aaron Lauda did his graduate work with Martin Hyland at the University of Cambridge, and now he's at Columbia for a postdoc with Khovanov, but his origins were humble: he wrote a paper on "2-groups" with me while he was getting his masters in physics here at U. C. Riverside. A 2-group, also known as a categorical group, is a category equipped with a multiplication, multiplicative identity and inverses just like a group - but where all the group laws hold up to isomorphism. The idea has been around for a while, but it comes in several slightly different flavors, and it's hard to find a clear exposition of how they're all related, so we decided to write such an exposition. We also explain how 2-groups are classified using group cohomology, and give lots of examples, including examples of "Lie 2-groups".
  • John C. Baez and Aaron D. Lauda, Higher-dimensional algebra V: 2-groups, Th. Appl. Cat. 12 (2004), 423-491.

  • Eugenia Cheng and Aaron D. Lauda, Higher-Dimensional Categories: an Illustrated Guide Book, to be published.

  • Aaron Lauda, Frobenius algebras and ambidextrous adjunctions, Th. Appl. Cat. 16 (2006), 84-122.
  • Aaron Lauda, Frobenius algebras and planar open string topological field theories.
  • Aaron Lauda and Hendryk Pfeiffer, Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras
  • Aaron Lauda, Open-closed Topological Quantum Field Theory and Tangle Homology, PhD thesis, Cambridge University, 2006. Available in Postscript.

• Alissa Crans finished her PhD thesis in August of 2004. This expands on a paper she wrote with me about "Lie 2-algebras". A Lie 2-algebra is a category equipped with algebraic structure much like that of a Lie algebra, but where the laws hold only up to isomorphism. Our paper focuses on a certain class of Lie 2-algebras, the "semistrict" ones, where only the Jacobi identity fails to hold as an equation. It classifies these using Lie algebra cohomology, very much like how 2-groups are classified using group cohomology. Using this classification one can show that any finite-dimensional complex simple Lie algebra admits a one-parameter deformation into a Lie 2-algebra. Her thesis goes on and explores the relationship between groups, Lie algebra, quandles and braids, with an eye towards categorifying this relationship.
  • John C. Baez and Alissa S. Crans, Higher-dimensional algebra VI: Lie 2-algebras, Th. Appl. Cat. 12 (2004), 492-528.

  • Alissa S. Crans, Lie 2-Algebras, PhD thesis, U. C. Riverside, 2004. Available in PDF. Also available in a more user-friendly format on the arXiv.

  • Alissa S. Crans, Higher linear algebra, transparencies for a lecture at the Institute of Mathematics and its Applications.

  • John C. Baez, Alissa S. Crans, Danny Stevenson and Urs Schreiber, From loop groups to 2-groups, Homotopy, Homology and Applications, 9 (2007), 101-135.

  • John C. Baez, Alissa S. Crans and Derek Wise, Exotic statistics for strings in 4d BF theory, Adv. Theor. Math. Phys. 11 (2007), 707-749.

• Miguel Carrión Álvarez wrote his thesis on Wilson loops in quantum electromagnetism and Wilson surfaces in the p-form analogue of quantum electromagnetism. You can also see a couple of his talks on this material, as well as a paper he wrote on a generalization of the Gelfand-Naimark theorem.
  • Miguel Carrión Álvarez, Variations on a theme of Gelfand and Naimark.

  • Miguel Carrión Álvarez, Wilson loop dynamics without regularization.

  • Miguel Carrión Álvarez, Physics and analysis on noncompact manifolds.

  • Miguel Carrión Álvarez, Loop Quantization versus Fock Quantization of p-Form Electromagnetism on Static Spacetimes, PhD thesis, U. C. Riverside, 2004. Available in PDF. Also available in a more user-friendly format on the arXiv.

• Toby Bartels did his thesis on "2-bundles". These play a fundamental role in higher gauge theory, just as bundles underlie ordinary gauge theory. Roughly speaking, a 2-bundle is a bundle where the fiber is a smooth category rather than a smooth manifold. We can build a 2-bundle by pasting together trivial 2-bundles over open sets using "transition functors" g_αβ in place of transition functions, but these only need to satisfy the usual law g_αβ g_βγ = g_αγ up to a specified natural isomorphism, which satisfies a law of its own on quadruple intersections of open sets. Toby defines principal G-2-bundles for any smooth 2-group G, constructs a 2-category of principal G-2-bundles over a given space, and shows that under certain circumstances this is equivalent to a 2-category of nonabelian gerbes:
  • Toby Bartels, Higher Gauge Theory: 2-Bundles , PhD thesis, U. C. Riverside, 2006. Available in PDF. Also available in a more user-friendly format on the arXiv under the title Higher gauge theory I: 2-bundles.
  • Toby Bartels and John Baez, dialog on Quaternionic functional analysis.
  • Toby Bartels, Expository papers.

• Derek Wise wrote his thesis on Cartan geometry and its relation to gravity in 3 and 4 spacetime dimensions - especially the MacDowell-Mansouri formulation of 4d gravity. As a warmup we wrote a paper on statistics for strings coupled to 4d topological gravity. Before that, he wrote a paper about p-form electromagnetism on discrete spacetimes:
  • Derek Wise, Lattice p-form electromagnetism and chain field theory, Class. Quantum Grav. 23 (2006), 5129-5176.
  • John C. Baez, Alissa S. Crans and Derek Wise, Exotic statistics for strings in 4d BF theory, Adv. Theor. Math. Phys. 11 (2007), 707-749.

  • Derek Wise, Exotic statistics and particle types in 3- and 4d BF theory, talk at the Perimeter Institute, Waterloo, Canada, 2006.

  • Derek Wise, MacDowell-Mansouri gravity and Cartan geometry.

  • Derek Wise, Topological Gauge Theory, Cartan Geometry, and Gravity, PhD Thesis, U. C. Riverside, 2007. Available in PDF.

• Jeffrey Morton did his thesis on extended topological quantum field theories and quantum gravity. He gave a precise definition of "extended TQFT", and showed the Dijkgraaf–Witten model gives one of these, in any dimension. As a warmup for this, he wrote a paper on a bicategory nCob₂ where the 2-morphisms are n-dimensional cobordisms between manifolds with boundary. Before this, he wrote a paper about categorifying quantum mechanics, which explains the combinatorics of the quantum harmonic oscillator and Feynman diagrams:
  • Jeffrey Morton, Categorified algebra and quantum mechanics, Th. Appl. Cat. 16 (2006), 785-854.
  • Jeffrey Morton, Categorifying the quantum harmonic oscillator, talk at the International Category Theory Conference (CT06), White Point, Nova Scotia, 2006.
  • Jeffrey Morton, A double bicategory of cobordisms with corners.
  • Jeffrey Morton, Higher algebra, extended TQFTs, and 3d quantum gravity, talk at the Perimeter Institute, Waterloo, Canada, 2006.
  • Jeffrey Morton, Extended TQFT's and Quantum Gravity, PhD thesis, U. C. Riverside, 2007. Available in PDF. Also available in a more user-friendly format on the arXiv.

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