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, Letters in Mathematical Physics 31 (1994),
205–212. In PDF and Postscript.
-
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 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 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 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,
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.
-
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'
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:
-
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 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 Baez, Aristide Baratin and Laurent Freidel, Infinite-Dimensional
Representations of 2-Groups, Memoirs of the American Mathematical
Society 1032, AMS, Providence, Rhode Island, 2012.
-
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 Baez,
Convenient categories of
smooth spaces, Transactions of the American Mathematical
Society 363 (2011), 5789–5825.
-
Alexander E. Hoffnung, John Baez and Christopher Rogers,
Categorified symplectic
geometry and the classical string, Communications in
Mathematical Physics 293 (2010), 701–715.
-
Alexander E. Hoffnung, John 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 Baez and Alexander E. Hoffnung,
Categorified symplectic
geometry and the classical string, Commun.
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, 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 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 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 Baez, The algebra of grand unified
theories, with John Huerta, Bulletin of the American Mathematical
Society 47 (2010), 483–552.
-
John Huerta and John Baez,
An invitation to higher
gauge theory, General Relativity and Gravitation 43
(2011), 2335–2392
-
John Huerta and John Baez, The
strangest numbers in string theory, Scientific American,
May 2011, 60–65.
-
John Huerta and John 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 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 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 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 Baez, A
Noether theorem for Markov processes, Journal of Mathematical
Physics 54 (2013), 013301.
-
Brendan Fong and John 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 Baez,
A compositional framework
for passive linear networks, Theory and Applications of Categories
33 (2018), 1158–1222. (Blog article here.)
-
Brendan Fong, John 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 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 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 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 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 Baez, Coarse-graining open Markov
processes, Theory and Applications of Categories 33 (2018),
1223–1268. (Blog article here.)
-
Kenny Courser and John 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 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 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.
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John Baez and Jade Master, Open Petri
nets, Mathematical Structurs in Computer Science 30 (2020)
314–341.
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Jade Master, Petri nets
based on Lawvere theories, Mathematical Structures in Computer
Science 30 (2020), 833–864.
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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.
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John 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.
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Jade Master, Why is homology
so powerful?
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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.
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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.
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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.
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
© 2023 John Baez
baez@math.removethis.ucr.andthis.edu