My Students' Theses (and Other Papers)

John Baez

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

• James Gilliam
• Laurel Langford
• Aaron Lauda
• Alissa Crans
• Miguel Carrión Álvarez
• Toby Bartels
• Derek Wise
• Jeffrey Morton
• Alex Hoffnung
• Chris Rogers
• John Huerta
• Christopher Walker

• 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, Ph.D. thesis, U. C. Riverside, 1996. Available in PDF and Postscript.
  • James Gilliam and John Baez, An algebraic approach to discrete mechanics, Lett. Math. Phys. 31 (1994), 205-212. In PDF and Postscript.

• Laurel Langford has tenure at the University of Wisconsin River Falls. She 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, Ph.D. thesis, U. C. Riverside, 1997. Available in Postscript.
  • Laurel Langford and John Baez, 2-Tangles, Lett. Math. Phys. 43 (1998), 187-197.
  • Laurel Langford and John Baez, 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, then a postdoc at Columbia for a postdoc with Khovanov, and now he's teaching at the University of Southern California, 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".
  • Aaron D. Lauda, Open-Closed Topological Quantum Field Theory and Tangle Homology, PhD thesis, Cambridge University, 2006. Available in Postscript.
  • Aaron D. Lauda and John Baez, Higher-dimensional algebra V: 2-groups, Th. Appl. Cat. 12 (2004), 423-491.
  • Aaron D. Lauda and Eugenia Cheng, Higher-Dimensional Categories: an Illustrated Guide Book, to be published.
  • Aaron D. Lauda, Frobenius algebras and ambidextrous adjunctions, Th. Appl. Cat. 16 (2006), 84-122.
  • Aaron D. Lauda, Frobenius algebras and planar open string topological field theories.
  • Aaron D. Lauda and Hendryk Pfeiffer, Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras

• Alissa Crans now has tenure at Loyola Marymount University. Her thesis was on "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.
  • Alissa S. Crans, Lie 2-Algebras, Ph.D. thesis, U. C. Riverside, 2004. Available in PDF. Also available in a more user-friendly format on the arXiv.
  • Alissa S. Crans and John Baez, Higher-dimensional algebra VI: Lie 2-algebras, Th. Appl. Cat. 12 (2004), 492-528.
  • Alissa S. Crans, Higher linear algebra, transparencies for a lecture at the Institute of Mathematics and its Applications.
  • Alissa S. Crans, John Baez, Danny Stevenson and Urs Schreiber, From loop groups to 2-groups, Homotopy, Homology and Applications, 9 (2007), 101-135.
  • Alissa S. Crans, John Baez and Derek K. 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, Loop Quantization versus Fock Quantization of p-Form Electromagnetism on Static Spacetimes, Ph.D. thesis, U. C. Riverside, 2004. Available in PDF. Also available in a more user-friendly format on the arXiv.
  • Miguel Carrión Álvarez, Variations on a theme of Gelfand and Naimark.
  • Miguel Carrión Álvarez, Wilson loop dynamics without regularization, talk.
  • Miguel Carrión Álvarez, Physics and analysis on noncompact manifolds, talk.

• 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 , Ph.D. 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 did a postdoc at U. C. Davis and is now at Erlangen, which gave its name to Felix Klein's famous Erlangen Program relating group theory to geometry. This is appropriate, because he 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. More recently, he and I collaborated with Aristide Baratin and Laurent Freidel to write a book on representations of 2-groups.
  • Derek K. Wise, Topological Gauge Theory, Cartan Geometry, and Gravity, Ph.D. Thesis, U. C. Riverside, 2007. Available in PDF.
  • Derek K. Wise, Lattice p-form electromagnetism and chain field theory, Class. Quantum Grav. 23 (2006), 5129-5176.
  • Derek K. Wise, John Baez and Alissa S. Crans, Exotic statistics for strings in 4d BF theory, Adv. Theor. Math. Phys. 11 (2007), 707-749.
  • Derek K. Wise, Exotic statistics and particle types in 3- and 4d BF theory, talk at the Perimeter Institute, Waterloo, Canada, 2006.

  • Derek K. Wise, Symmetric space Cartan connections and gravity in three and four dimensions, SIGMA 5 (2009), 080.
  • Derek K. Wise, MacDowell-Mansouri gravity and Cartan geometry, Class. Quantum Grav. 27 (2010), 155010.
  • Derek K. Wise, John Baez, Aristide Baratin and Laurent Freidel, Infinite-Dimensional Representations of 2-Groups. To appear in Mem. Amer. Math. Soc..

• Jeffrey Morton did a postdoc at the University of Western Ontario and is now at the Instituto Superior Técnico in Lisbon. He 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, Extended TQFT's and Quantum Gravity, Ph.D. thesis, U. C. Riverside, 2007. Available in PDF. Also available in a more user-friendly format on the arXiv.
  • 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, Higher algebra, extended TQFTs, and 3d quantum gravity, talk at the Perimeter Institute, Waterloo, Canada, 2006.
  • Jeffrey Morton, Double bicategories and double cospans, Journal of Homotopy and Related Structures 4 (2009), 389-428
  • Jeffrey Morton, Extended TQFT, gauge theory, and 2-linearization.

• Alex Hoffnung is now at the University of Ottawa. He did his thesis on groupoidified Hecke algebras. Groupoidification is a method of categorifying linear algebra in which vector spaces are replaced by groupoids and linear maps are replaced by spans of groupoids. In this approach, categorified Hecke algebras arise naturally from some groupoids associated to flag varieties of algebraic groups over finite fields. Before this, he wrote a paper with me on various convenient categories of "smooth spaces" (generalizations of smooth manifolds) and also a paper with Chris Rogers and me on Lie 2-algebras arising in multisymplectic geometry. On top of all that, he collaborated with my student Aaron Lauda on a paper about quotients of certain rings that can be used to categorify the positive half of quantum sl(n).
  • Alexander E. Hoffnung, Foundations of Categorified Representation Theory, Ph.D. Thesis, U.C. Riverside, 2010. Available in PDF.
  • Alexander E. Hoffnung and John Baez, Convenient categories of smooth spaces, to appear in Trans. Amer. Math. Soc..
  • Alexander E. Hoffnung, John Baez and Christopher Rogers, Categorified symplectic geometry and the classical string, Comm. Math. Phys. 293 (2010), 701-715.
  • Alexander E. Hoffnung, John Baez and Christopher D. Walker, Higher-dimensional algebra VII: groupoidification, Th. Appl. Cat. 24 (2010), 489-553.
  • Alexander E. Hoffnung, The Hecke bicategory.
  • Alexander E. Hoffnung and Aaron D. Lauda, Nilpotency in type A cyclotomic quotients, Journal of Algebraic Combinatorics 32 (2010), 533.

• Chris Rogers starts a postdoc in Göttingen in 2011. He did his Ph.D. thesis on higher algebraic structures arising from multisymplectic geometry, which is a generalization of symplectic geometry where the symplectic 2-form is replaced by an n-form. Before this, he wrote one paper with Alex Hoffnung and me, another paper with just me, and two papers all on his own, all dealing with this general subject.

  It's a big subject, since it shows up naturally when you generalize the classical mechanics of point particles to strings and higher-dimensional membranes! One recurrent theme is the appearance of Lie n-algebras as generalizations of the usual Poisson algebra of observables for a symplectic manifold. In particular, a "2-plectic" manifold has a closed nondegenerate 3-form, and gives rise to a Lie 2-algebra of observables.

  • Christopher L. Rogers, Higher Symplectic Geometry, Ph.D. thesis, U. C. Riverside, 2011. Available in PDF.
  • Christopher L. Rogers, John Baez and Alexander E. Hoffnung, Categorified symplectic geometry and the classical string, Comm. Math. Phys. 293 (2010), 701-715.
  • Christopher L. Rogers and John Baez, Categorified symplectic geometry and the string Lie 2-algebra, Homotopy, Homology and Applications 12 (2010), 221-236.
  • Christopher L. Rogers, L_∞-algebras from multisymplectic geometry, to appear in Lett. Math. Phys..
  • Christopher L. Rogers, 2-plectic geometry, Courant algebroids, and categorified prequantization.

• John Huerta starts a postdoc postdoctoral position at Australian National University in Canberra in 2011. He did his thesis on using normed division algebras to construct the higher algebraic structures—Lie 2-supergroups and 3-supergroups—used in theories of supersymmetric strings and 2-branes. Before that he wrote papers with me on grand unified theories and higher gauge theory. We also wrote a paper on octonions for Scientific American!

  • John Huerta, Division Algebras, Supersymmetry and Higher Gauge Theory, Ph.D. thesis, U.C. Riverside, 2011. Available in PDF. Also available in a more user-friendly format on the arXiv.
  • John Huerta, L_∞ superalgebras for superstring and M-theory, talk at the AMS special session on Topology, Geometry and Physics, November 2010.
  • John Huerta, A categorified supergroup for string theory, talk at the Workshop and School on Higher Gauge Theory, TQFT and Quantum Gravity in Lisbon, February 2011.
  • John Huerta and John Baez, The algebra of grand unified theories, with John Huerta, Bull. Amer. Math. Soc. 47 (2010), 483-552.
  • An invitation to higher gauge theory, with John Huerta, Gen. Rel. Grav. 43 (2011), 2335-2392
  • John Huerta and John Baez, Division algebras and supersymmetry I, in Superstrings, Geometry, Topology, and C*-Algebras, eds. Robert Doran, Greg Friedman and Jonathan Rosenberg, Proc. Symp. Pure Math. 81, AMS, Providence, 2010, pp. 65-80.
  • John Huerta and John Baez, Division algebras and supersymmetry II.
  • John Huerta and John Baez, The strangest numbers in string theory, Scientific American, May 2011, 60-65.

• Christopher Walker finished his Ph.D. thesis in 2011. His thesis is on groupoidified Hall algebras. Starting from a simply-laced Dynkin diagram, and labelling the edges with arrows, one gets a "quiver". The groupoid of representations of this quiver comes with a structure that's a groupoidified version of the positive half of the quantum group associated to this Dynkin diagram. Before writing his thesis he wrote a paper on groupoidification with Alex Hoffnung and me, and also a paper on how to see Hall algebras as Hopf algebras in a certain braided monoidal category. Both these play important roles in his thesis work.
  • Christopher D. Walker, A Categorification of Hall Algebras, Ph.D. thesis, U. C. Riverside, 2011. Available in PDF.
  • Christopher D. Walker, John Baez and Alexander E. Hoffnung, Higher-dimensional algebra VII: groupoidification, Th. Appl. Cat. 24 (2010), 489-553.
  • Christopher D. Walker, Hall algebras as Hopf objects.
  • Christopher D. Walker, Groupoidified linear algebra, talk at Groupoidfest 2008.
  • Christopher D. Walker, A categorification of Hall algebras, talk at the AMS Fall Western Section Meeting, November 2009.

© 2011 John Baez
baez@math.removethis.ucr.andthis.edu

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