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!

• James Gilliam
• Laurel Langford
• Aaron Lauda
• Alissa Crans
• Valeria Michelle Carrión Álvarez
• Toby Bartels
• Derek Wise
• Jeffrey Morton
• Alex Hoffnung
• Chris Rogers
• John Huerta
• Christopher Walker
• Mike Stay
• Brendan Fong
• Jason Erbele
• Blake S. Pollard
• Brandon Coya
• Daniel Cicala
• Kenny Courser
• Joe Moeller
• Jade Edenstar Master
• Owen Lynch
• Christian Williams

• 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 C. Baez, An algebraic approach to discrete mechanics, Letters in Mathematical Physics 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.

  • 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 C. Baez, 2-Tangles, Letters in Mathematical Physics 43 (1998), 187–197.
  • Laurel Langford and John C. Baez, Higher-dimensional algebra IV: 2-tangles, Advances in Mathematics 180 (2003), 705–764.
  • Laurel Langford, J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, 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 has tenure 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 C. Baez, Higher-dimensional algebra V: 2-groups, Theory and Applications of Categories 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, Theory and Applications of Categories 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 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.
  • 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 C. Baez, Higher-dimensional algebra VI: Lie 2-algebras, Theory and Applications of Categories 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 C. Baez, Danny Stevenson and Urs Schreiber, From loop groups to 2-groups, Homotopy, Homology and Applications, 9 (2007), 101–135.
  • Alissa S. Crans, John C. Baez and Derek K. Wise, Exotic statistics for strings in 4d BF theory, Advances in Theoretical and Mathematical Physics 11 (2007), 707–749.

• Valeria Michelle Carrión Álvarez finished her thesis in 2004. She studied Wilson loops in quantum electromagnetism and Wilson surfaces in the p-form analogue of quantum electromagnetism. You can also see a couple of her talks on this material, as well as a paper she wrote on a generalization of the Gelfand-Naimark theorem.
  • Valeria Michelle 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.
  • Valeria Michelle Carrión Álvarez, Variations on a theme of Gelfand and Naimark.
  • Valeria Michelle Carrión Álvarez, Wilson loop dynamics without regularization, talk.
  • Valeria Michelle Carrión Álvarez, Physics and analysis on noncompact manifolds, talk.

• Toby Bartels finished his thesis in 2006, and now teaches at Southeast Community College. He 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 C. Baez, dialog on Quaternionic functional analysis.
  • Toby Bartels, Expository papers.

• Derek Wise finished his Ph.D. thesis in 2007. He has been working at Lockheed Martin after leaving a tenure-track job at Concordia University St Paul. From 2010 to 2015 he worked in 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, Classical and Quantum Gravity 23 (2006), 5129-5176.
  • Derek K. Wise, John C. Baez and Alissa S. Crans, Exotic statistics for strings in 4d BF theory, Advances in Theoretical and Mathematical Physics 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, Classical and Quantum Gravity 27 (2010), 155010.
  • Derek K. Wise, John C. Baez, Aristide Baratin and Laurent Freidel, Infinite-Dimensional Representations of 2-Groups, Memoirs of the American Mathematical Society 1032, AMS, Providence, Rhode Island, 2012.
  • Derek K. Wise and John C. Baez, Teleparallel gravity as a higher gauge theory, Comm. Math. Phys. 333 (2015), 1537–186.

• Jeffrey Morton did a postdoc at the University of Western Ontario, then worked at Instituto Superior Técnico in Lisbon, and now has tenure at SUNY Buffalo State University. 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, Theory and Applications of Categories 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 works at Ratio Finance. 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 C. Baez, Convenient categories of smooth spaces, Transactions of the American Mathematical Society 363 (2011), 5789–5825.
  • Alexander E. Hoffnung, John C. Baez and Christopher Rogers, Categorified symplectic geometry and the classical string, Communications in Mathematical Physics 293 (2010), 701–715.
  • Alexander E. Hoffnung, John C. Baez and Christopher D. Walker, Higher-dimensional algebra VII: groupoidification, Theory and Applications of Categories 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 finished his thesis in 2011. He now has tenure at the University of Nevada, Reno, and he's had one PhD student of his own. 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 C. Baez and Alexander E. Hoffnung, Categorified symplectic geometry and the classical string, Commun. Math. Phys. 293 (2010), 701–715.
  • Christopher L. Rogers and John C. 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, Letters in Mathematical Physics 100 (2012), 29–50.
  • Christopher L. Rogers, Domenico Fiorenza and Urs Schreiber, A higher Chern-Weil derivation of AKSZ sigma-models, International Journal of Geometric Methods in Modern Physics 10 (2013), 1250078.
  • Christopher L. Rogers, 2-plectic geometry, Courant algebroids, and categorified prequantization, Journal of Symplectic Geometry 11 (2013), 53–91.
  • Christopher L. Rogers, Domenico Fiorenza and Urs Schreiber, L_∞-algebras of local observables from higher prequantum bundles, Homology, Homotopy and Applications 16 (2014), 107–142.

• John Huerta John Huerta finished his thesis in 2011, took a postdoctoral position at Australian National University in Canberra, and is now at the Instituto Superior Técnico, part of the University of Lisbon. 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, and won a prize, the 2013 Levi L. Conant Prize, for the best paper in the Bulletin of the American Mathematical Society, namely our overview of grand unified theories.

  • 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 and John C. Baez, Division algebras and supersymmetry I, in Superstrings, Geometry, Topology, and C*-Algebras, eds. Robert Doran, Greg Friedman and Jonathan Rosenberg, Proceedings of Symposia in Pure Mathematics 81, AMS, Providence, Rhode Island, 2010, pp. 65–80.
  • John Huerta and John C. Baez, Division algebras and supersymmetry II, Advances in Theoretical and Mathematical Physics 15 (2011), 1373–1410.
  • John Huerta, Division algebras and supersymmetry III, Advances in Theoretical and Mathematical Physics 16 (2012), 1485–1589.
  • John Huerta, Division algebras and supersymmetry IV.
  • 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 C. Baez, The algebra of grand unified theories, with John Huerta, Bulletin of the American Mathematical Society 47 (2010), 483–552.
  • John Huerta and John C. Baez, An invitation to higher gauge theory, General Relativity and Gravitation 43 (2011), 2335–2392
  • John Huerta and John C. Baez, The strangest numbers in string theory, Scientific American, May 2011, 60–65.
  • John Huerta and John C. Baez, G₂ and the rolling ball, Transactions of the American Mathematical Society 366 (2014), 5257–5293.

• Christopher Walker finished his Ph.D. thesis in 2011, and now teaches at the College of San Mateo. 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.
  • Christopher D. Walker, A Categorification of Hall Algebras, Ph.D. thesis, U. C. Riverside, 2011. Available in PDF.
  • Christopher D. Walker, John C. Baez and Alexander E. Hoffnung, Higher-dimensional algebra VII: groupoidification, Theory and Applications of Categories 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.

• Mike Stay is the co-founder and CTO of Pyrofex Corp. He 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.
  • Mike Stay, Physics and Computation, Ph.D. thesis, Department of Computer Science, University of Auckland, 2015. Available in PDF.
  • John C. Baez and Mike Stay, Physics, topology, logic and computation: a Rosetta Stone, in New Structures for Physics, ed. Bob Coecke, Lecture Notes in Physics 813, Spinger, Berlin, 2011, 95–172.
  • John C. Baez and Mike Stay, Algorithmic thermodynamics, Mathematical Structures in Computer Science 22 (2012), 771–787.
  • Mike Stay and Jamie Vicary, Bicategorical semantics for nondeterministic computation, Proceedings of Mathematical Foundations of Programming Semantics 29 (2013), 345–359.
  • Mike Stay, Compact closed bicategories, Theory and Applications of Categories 31 (2016), 755–798.

• Brendan Fong is the chief executive of the Topos Institute, a research institute in applied category theory. He finished his Ph.D. thesis in 2016; he was a graduate student in the Department of Computer Science at the University of Oxford under Bob Coecke, but I was his advisor for most practical purposes. He worked with me on networks such as electrical circuits, Markov processes, and developed the formalism of decorated cospan categories and decorated corelation categories to study these. He subsequently applied them to control theory. He spent some time at the University of Pennsylvania and then got a postdoc at M.I.T. working with David Spivak.
  • Brendan Fong, The Algebra of Open and Interconnected Systems, Ph.D. thesis, University of Oxford, 2016. Available in PDF and on the arXiv.
  • Brendan Fong, Causal theories: a categorical perspective on Bayesian networks.
  • Brendan Fong and John C. Baez, A Noether theorem for Markov processes, Journal of Mathematical Physics 54 (2013), 013301.
  • Brendan Fong and John C. Baez, Quantum techniques for studying equilibrium in reaction networks, Journal of Complex Networks 3 (2014), 22–34.
  • Brendan Fong, Decorated cospans, Theory and Applications of Categories 30 (2015), 1096–1120. (Blog article here.)
  • Brendan Fong, Decorated corelations, Theory and Applications of Categories 33 (2018), 608–643.

  • Brendan Fong and John C. Baez, A compositional framework for passive linear networks, Theory and Applications of Categories 33 (2018), 1158–1222. (Blog article here.)
  • Brendan Fong, John C. Baez and Blake S. Pollard, A compositional framework for Markov processes, Journal of Mathematical Physics 57 (2016), 033301. (Blog article here.)
  • Brendan Fong, Paolo Rapisarda and Paweł Sobociński, A categorical approach to open and interconnected dynamical systems, to appear in Proceedings of Logic in Computer Science 2016, LICS 16.
  • Brandon Coya and Brendan Fong, Corelations are the prop for extraspecial commutative Frobenius monoids, Theory and Applications of Categories 32 (2017), 380–395. (Blog article here.)
  • Brendan Fong, Modelling interconnected systems with decorated corelations, talk at the Simons Institute for the Theory of Computing, December 6, 2016.

• Jason Erbele finished his Ph.D. thesis in 2016 and now teaches at Victor Valley College. 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 C. Baez and Jason Erbele, Categories in control, Theory and Applications of Categories 30 (2015), 836–881. (Blog article here.)
• Jason Erbele, Categories in control, video of talk at QPL 2015, University of Oxford, 2016.

• Blake S. 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 at NIST working with Eswaran Subrahanian and Spencer Breiner on an NSF-funded project called A Categorical Approach to Systems Modeling for Systems Engineering. He is now a Senior Scientist at CrossnoKaye, which builds advanced control systems for industrial cold storage facilities.
• Blake S. Pollard, Open Markov Processes and Reaction Networks, Ph.D. thesis, U. C. Riverside, 2017. Available in PDF. Also available in a more user-friendly format on the arXiv.
• Blake S. Pollard, Open Markov processes and reaction networks, thesis defense slides, June 1, 2017.
• Blake S. Pollard, A Second Law for open Markov processes, Open Systems and Information Dynamics 23 (2016), 1650006. (Blog article here.)
• Blake S. Pollard, John C. Baez and Brendan Fong, A compositional framework for Markov processes, Journal of Mathematical Physics 57 (2016), 033301. (Blog article here.)
• Blake S. Pollard, Open Markov processes: A compositional perspective on non-equilibrium steady states in biology, Entropy 18 (2016), 140. (Blog article here.)
• Blake S. Pollard and John C. Baez, A compositional framework for reaction networks, Reviews of Mathematical Physics 29 (2017), 1750028. (Blog article here.)

• Brandon Coya finished his thesis in 2018, and now teaches at Whittier College. He began his research by studying corelations with my student Brendan Fong. These are a way of describing electrical circuits made solely of ideal conductive wires. He then expanded his research by working with me and Franciscus Rebro on other electrical circuit diagrams, and then wrote a paper by himself on "bond graphs", another form of diagram used by engineers to describe not only electrical circuits but a large class of other systems.
• Brandon Coya, Circuits, Bond Graphs, and Signal-Flow Diagrams: A Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2018. Available in PDF. Also available in a more user-friendly format on the arXiv.
• Brandon Coya, Circuits, bond graphs, and signal-flow diagrams: a categorical perspective, thesis defense slides, May 15, 2018.
• Brandon Coya and Brendan Fong, Corelations are the prop for extraspecial commutative Frobenius monoids, Theory and Applications of Categories 32 (2017), 380–395. (Blog article here.)
• John C. Baez, Brandon Coya and Franciscus Rebro, Props in network theory. (Blog article here.)
• Brandon Coya, A compositional framework for bond graphs.
• Brandon Coya, Frobenius monoids, weak bimonoids, and corelations, talk at Applied Category Theory 2017, November 5, 2017. Video: part 1 and part 2.

• Daniel Cicala finished his thesis in 2019 and now teaches at Southern Connecticut State University. His thesis was inspired by graph rewriting, with a strong focus on on 'open' graphs, which can be glued together to form larger graphs. But he worked at a high level of generality, considering not just graphs but objects in any topos. He developed two different approaches to rewriting open objects, both using the idea of structured cospans.
• Daniel Cicala, Rewriting Structured Cospans: A Syntax For Open Systems, Ph.D. thesis, U. C. Riverside, 2019. Available in PDF. Also available in a more user-friendly format on the arXiv.
• Daniel Cicala, Spans of cospans, Theory and Applications of Categories 33 (2018), 131–147.
• Daniel Cicala and Kenny Courser, Spans of cospans in a topos, Theory and Applications of Categories 33 (2018), 1–22.
• Daniel Cicala, Categorifying the ZX-calculus, in Proceedings 14th International Conference on Quantum Physics and Logic, eds. Bob Coecke and Aleks Kissinger.
• Daniel Cicala, Rewriting structured cospans.

• Kenny Courser finished his Ph.D. thesis in 2020. He did his thesis on double categories as a formalism for describing open systems. Most of his thesis is about 'structured cospans', a way to build double categories of networks — but a large chunk is about open Markov processes, which require some other techniques.
• Kenny Courser, Open Systems: A Double Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2020. Available in PDF. Also available in a more user-friendly format on the arXiv. (Blog articles here.)
• Kenny Courser, A bicategory of decorated cospans, Theory and Applications of Categories 32 (2017), 995–1027.
• Kenny Courser and Daniel Cicala, Spans of cospans in a topos, Theory and Applications of Categories 33 (2018), 1–22.
• Kenny Courser and John C. Baez, Coarse-graining open Markov processes, Theory and Applications of Categories 33 (2018), 1223–1268. (Blog article here.)
• Kenny Courser and John C. Baez, Structured cospans, Theory and Applications of Categories 35 (2020), 1771–1822. (Blog article here.)
• Kenny Courser, Structured cospans, talk at the 4th Symposium on Compositional Structures, May 22, 2019.
• Kenny Courser, Coarse-graining open Markov processes, talk at Quantum Physics and Logic 2019, June 12, 2019.

• Joe Moeller finished his Ph.D. thesis in 2020. He did his thesis on network models and the monoidal Grothendieck construction. In 2021 he got a postdoc at NIST working with Eswaran Subrahanian and Spencer Breiner.
• Joe Moeller, The Grothendieck Construction in Categorical Network Theory, Ph.D. thesis, U. C. Riverside, 2020. Available in PDF. Also available in a more user-friendly format on the arXiv.
• Joe Moeller, John C. Baez, John Foley and Blake Pollard, Network models, Theory and Applications of Categories 35 (2020), 700–744.
• Joe Moeller and Christina Vasilakopoulou, Monoidal Grothendieck construction, Theory and Applications of Categories 35 (2020), 1159–1207.
• Joe Moeller, Noncommutative network models, Mathematical Structures in Computer Science 30 (2020), 14—32.
• Joe Moeller, John C. Baez and John Foley, Network models from Petri nets with catalysts, Mathematical Structures in Computer Science 30 (2020), 14–32.
• Joe Moeller, Monoidal Grothendieck construction, talk at the 4th Symposium on Compositional Structures, May 22, 2019.
• Joe Moeller, Network models from Petri nets with catalysts, talk at Quantum Physics and Logic 2019, June 12, 2019.

• Jade Edenstar Master finished her Ph.D. thesis in 2021 and now has a postdoc in computer and information sciences at the University of Strathclyde. She started by working with me on 'open' Petri nets, that is, Petri nets with some places specified as inputs and some as outputs. She then generalized these to Q-nets, which are specified by an arbitrary Lawvere theory Q: when this is the Lawvere theory for commutative monoids, we get Petri nets. She also worked on the 'algebraic path problem', a generalization of the shortest path problem from graphs to R-matrices, meaning matrices valued in an arbitrary quantale Q: when this is the booleans, such matrices are just graphs. She did her thesis on open Q-nets, open R-matrices, and their operational semantics.
• Jade Master, Composing Behaviors of Networks, Ph.D. thesis, U.C. Riverside, 2021. Available in PDF. Also available in a more user-friendly format on the arXiv.
• John C. Baez and Jade Master, Open Petri nets, Mathematical Structurs in Computer Science 30 (2020) 314–341.
• Jade Master, Petri nets based on Lawvere theories, Mathematical Structures in Computer Science 30 (2020), 833–864.
• Tai-Danae Bradley, Martha Lewis, Jade Master and Brad Theilman, Translating and evolving: towards a model of language change in DisCoCat, Electronic Notes Theor. Comput. Sci. 283 (2018), 50–61.
• John C. Baez, Fabrizio Genovese, Jade Master and Michael Shulman, Categories of nets, in 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), IEEE, Rome, Italy, 2021, pp. 1–13.
• Jade Master, Why is homology so powerful?
• Jade Master, The open algebraic path problem, in 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021), eds. Fabio Gadducchi and Alexandra Silva, Schloss Dagstuhl – Leibniz-Zentrum, Dagstuhl, Germany, 2021, pp. 20:1–20:20.

• Owen Lynch finished his masters thesis in 2022 at the mathematics department of the Universiteit Utrecht with Wioletta Ruszel and me. His thesis was on compositional thermodynamics. He is now a research software engineer at the Topos Institute.
  • Owen Lynch, Relational Composition of Physical Systems: A Categorical Approach, M.Sc. Thesis, Universiteit Utrecht, 2022, corrected and improved version at the arXiv.
  • John C. Baez, Joe Moeller and Owen Lynch, Compositional thermostatics, Journal of Mathematical Physics 64 (2023), 023304.

• Christian Williams finished his thesis on categorical logic in 2023. He has since taken a job with Conexus, a company that uses category theory to create database software.
  • Christian Williams, The Metalanguage of Category Theory, Ph.D. Thesis, U. C. Riverside, 2023.
  • John C. Baez and Christian Williams, Enriched Lawvere theories for operational semantics, Proceedings ACT 2019, Electronic Proceedings Theoretical Computer Science 323 (2020), 106-135.

© 2023 John Baez
baez@math.removethis.ucr.andthis.edu

home