## My Students' Theses (and Other Papers)

#### John Baez

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

Here they are:

• James Gilliam finished his thesis in 1996. He worked 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 finished her thesis in 1997. She now 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.

• Alissa Crans finished her Ph.D. thesis in 2004. She is now a full professor 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.

• 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" gαβ in place of transition functions, but these only need to satisfy the usual law gαβ gβγ = gαγ 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:

• 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).

• 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 Walker finished his Ph.D. thesis in 2011. His thesis was 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.

• Mike Stay started his PhD work at U. C. Riverside but wound up taking a job at Google in 2007 and getting a Ph.D in computer science at the University of Auckland 2015. That's where he had previously gotten his masters degree in computer science under Cristian Calude, and Calude and I served as his co-advisors for his Ph.D. Apart from a paper connecting thermodynamics to algorithmic entropy, Mike and I worked on applications of symmetric monoidal categories and bicategories to computation. He has continued developing these ideas ever since, and in 2016 he began working for a startup called Pyrofex, which will try to apply them in practical ways.

• Jason Erbele finished his Ph.D. thesis in 2016. He did his thesis on the use of symmetric monoidal categories, and specifically PROPs, to study control theory. He started by writing a paper with me that gives a presentation of the symmetric monoidal category of finite-dimensional vector spaces and linear relations. Starting from here, he constructed a symmetric monoidal category whose morphisms are the 'signal-flow' diagrams used in control theory. The all-important properties of 'observability' and 'controllability' of a linear time-invariant system can be nicely understood in this framework.
• Jason Michael Erbele, Categories in Control: Applied PROPs, Ph.D. thesis, U. C. Riverside, 2016. Available in PDF. Also available in a more user-friendly format on the arXiv.
• John Baez and Jason Erbele, Categories in control, Th. Appl. Cat. 30 (2015), 836–881. (Blog article here.)
• Jason Erbele, Categories in control, video of talk at QPL 2015, University of Oxford, 2016.

• Blake Pollard finished his Ph.D. thesis in 2017. He did his thesis on open systems, particularly open versions of Markov processes and chemical reaction networks. He studied the change in relative entropy in open Markov processes, and nonequilibrium steady states for open Markov processes and reaction networks. He described a 'black-boxing' functor sending any such open system to the the relation between input and output concentrations and flows that holds in a steady state. He and I also worked with Metron Scientific Solutions on their Complex Adaptive System Composition and Design Environment project, funded by DARPA. In the last summer of his thesis work he did an internship with Siemens at Princeton working with Arquimedes Canedo on the project Next-Generation Engineering with Category Theory and Sheaves. Then he got a postdoc with Eswaran Subrahanian at Carnegie Mellon and Spencer Breiner at NIST working on an NSF-funded project called A Categorical Approach to Systems Modeling for Systems Engineering.

It is important that students bring a certain ragamuffin, barefoot irreverence to their studies; they are not here to worship what is known, but to question it. - Jacob Bronowski

© 2018 John Baez
baez@math.removethis.ucr.andthis.edu