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!
Here they are:

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 EulerLagrange 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 WessZumino 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), 205212. 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's teaching at the University of Southern California,
but his origins were humble: he wrote a paper on "2groups"
with me while he was getting his masters in physics here at
U. C. Riverside. A 2group, 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 2groups are classified using group
cohomology, and give lots of examples, including examples of "Lie
2groups".

Aaron D. Lauda, OpenClosed Topological Quantum Field Theory and
Tangle Homology, PhD thesis, Cambridge University, 2006.
Available in Postscript.

Aaron D. Lauda and John Baez,
Higherdimensional algebra
V: 2groups, Th. Appl. Cat.
12 (2004), 423491.

Aaron D. Lauda and Eugenia Cheng,
HigherDimensional Categories: an Illustrated Guide Book,
to be published.

Aaron D. Lauda,
Frobenius algebras and ambidextrous adjunctions,
Th.
Appl. Cat. 16 (2006), 84122.

Aaron D. Lauda,
Frobenius algebras and planar open string topological field theories.

Aaron D. Lauda and Hendryk Pfeiffer,
Openclosed
strings: twodimensional extended TQFTs and Frobenius algebras

Alissa Crans now has
tenure at Loyola Marymount University. Her thesis was on "Lie
2algebras". A Lie 2algebra 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 2algebras, 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 2groups are
classified using group cohomology. Using this classification one can
show that any finitedimensional complex simple Lie algebra admits a
oneparameter deformation into a Lie 2algebra. 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 2Algebras, Ph.D. thesis, U. C. Riverside,
2004. Available in PDF. Also
available in a more userfriendly format on the arXiv.

Alissa S. Crans and John Baez,
Higherdimensional
algebra VI: Lie 2algebras,
Th.
Appl. Cat. 12 (2004), 492528.

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 2groups,
Homotopy,
Homology and Applications, 9 (2007), 101135.

Alissa S. Crans, John Baez and Derek K. Wise,
Exotic
statistics for strings in 4d BF theory,
Adv. Theor. Math. Phys. 11 (2007), 707749.

Miguel Carrión
Álvarez wrote his thesis on Wilson loops in
quantum electromagnetism and Wilson surfaces in the pform
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 GelfandNaimark theorem.

Toby Bartels
did his thesis on "2bundles". These play a fundamental
role in higher gauge theory, just as
bundles underlie ordinary gauge theory. Roughly speaking, a 2bundle
is a bundle where the fiber is a smooth category rather than a smooth
manifold. We can build a 2bundle by pasting together trivial
2bundles 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 G2bundles for any smooth 2group G,
constructs a 2category of principal G2bundles over a given
space, and shows that under certain circumstances this is equivalent
to a 2category of nonabelian gerbes:

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 MacDowellMansouri
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 pform electromagnetism on discrete spacetimes. More
recently, he and I collaborated with Aristide Baratin and Laurent Freidel
to write a book on representations of 2groups.

Derek K. Wise, Topological Gauge Theory, Cartan Geometry, and
Gravity, Ph.D. Thesis, U. C. Riverside, 2007. Available in PDF.

Derek K. Wise, Lattice
pform electromagnetism and chain field theory,
Class. Quantum Grav. 23 (2006), 51295176.

Derek K. Wise, John Baez and Alissa S. Crans,
Exotic
statistics for strings in 4d BF theory,
Adv. Theor. Math. Phys. 11 (2007), 707749.

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, MacDowellMansouri gravity and
Cartan geometry, Class. Quantum Grav. 27 (2010),
155010.

Derek K. Wise, John Baez, Aristide Baratin and Laurent Freidel, InfiniteDimensional
Representations of 2Groups. 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 nCob_{2} where the 2morphisms are
ndimensional 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 userfriendly
format on the arXiv.

Jeffrey Morton, Categorified algebra and
quantum mechanics, Th.
Appl. Cat. 16 (2006), 785854.

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), 389428

Jeffrey Morton, Extended TQFT,
gauge theory, and 2linearization.

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 2algebras
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), 701715.

Alexander E. Hoffnung, John Baez and
Christopher D. Walker, Higherdimensional algebra VII:
groupoidification, Th.
Appl. Cat. 24 (2010), 489553.

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 2form is replaced by an nform. 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
higherdimensional membranes! One recurrent theme is the appearance
of Lie nalgebras as generalizations of the usual Poisson algebra of
observables for a symplectic manifold. In particular, a
"2plectic" manifold has a closed nondegenerate 3form, and
gives rise to a Lie 2algebra 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), 701715.

Christopher L. Rogers and John Baez,
Categorified symplectic
geometry and the string Lie 2algebra,
Homotopy, Homology
and Applications 12 (2010), 221236.

Christopher L. Rogers, L_{∞}algebras from multisymplectic geometry, to appear in Lett.
Math. Phys..

Christopher L. Rogers, 2plectic 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 2supergroups and 3supergroups—used
in theories of supersymmetric strings and 2branes. 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 userfriendly format on the arXiv.

John Huerta, L_{∞} superalgebras for superstring and Mtheory, 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), 483552.

An
invitation to higher gauge theory, with John Huerta, Gen.
Rel. Grav. 43 (2011), 23352392

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. 6580.

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, 6065.

Christopher
Walker finished his Ph.D. thesis in 2011. His thesis is on
groupoidified Hall algebras.
Starting from a simplylaced 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, Higherdimensional algebra VII:
groupoidification, Th.
Appl. Cat. 24 (2010), 489553.

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