Research
My main research interests are mathematical physics, topology, and
category theory, and in particular, how "higheralgebraic" ideas from
category theory can illuminate questions in quantum gravity and the
foundations of quantum mechanics. This has connections to Wittentype
pathintegral invariants for manifolds, combinatorics, representation
theory, and more.
Published Research Papers
2Group Actions and Moduli Spaces of Higher Gauge
Theory (with Roger
Picken, arXiv:1904.10865)
Describes a double category of 2group connections on a
discretized manifold, which plays the role of the moduli space in
higher gauge theory. This is based on the transformation double
category (see below) derived from the action of a global symmetry
2group on a category. (Accepted by Journal of Geometry and
Physics, in press)
Transformation Double Categories Associated to 2Group
Actions (with Roger
Picken, arXiv:1401.0149)
Categorifies the relationship between global and
local symmetry which is expressed in a transformation groupoid
associated to a group action. (Theory and Applications of
Categories, vol. 30, no. 43, pp. 14291468,
2015,)
The Categorified Heisenberg Algebra I: A Combinatorial
Representation (with Jamie
Vicary, arXiv:1207.2054)
Demonstrates that groupoidification of the Heisenberg
algebra gives a concrete model of Khovanov's diagrammatic calculus,
and suggests how this might work for other cases. First part of two 
second forthcoming. (Journal of Pure and Applied Algebra vol. 222, no. 3, pp. 703745, 2018,
DOI:10.1016/j.jpaa.2017.05.004)
Cohomological Twisting of 2Linearization and Extended TQFT
(arXiv:1003.5603)
Uses 2linearization (see "2Vector Spaces and Groupoids")
to build an Extended TQFT associated to a discrete gauge group. A
special case, for closed manifolds, yields the untwisted
DijkgraafWitten model. (Journal of
Homotopy and Related Structures, vol. 10, no. 3, pp. 127187,2013,
DOI:10.1007/s4006201300472.)
2Vector Spaces and Groupoids
(arXiv:0810.2361; Applied
Categorical Structures; DOI: 10.1007/s1048501092250)
Constructs a 2functor from the bicategory of spans of
groupoids into the bicategory of 2Vector spaces. I
interpret this as a kind of categorified version of quantization of a
physical system. (Applied Categorical Structures, vol. 19, no. 4,
pp. 659707, 2011,
https://doi.org/10.1007/s1048501092250.)
Double Bicategories and Double Cospans
(arXiv:math.CT/0611930;
Journal of Homotopy and Related Structures, Vol. 4(2009), No. 1,
pp. 389428)
Describes a kind of weak cubical 2category, how a
construction using cospans (or spans) gives examples, and how they
relate to more familiar 2categories. (Journal of Homotopy and Related
Structures, vol. 4, no. 1, pp. 389428, 2009)
Categorified Algebra and Quantum Mechanics
(arXiv:math.QA/0601458,
Theory and Applications of Categories, Vol. 16, 2006, No. 29, pp
785854.)
Describes the quantum harmonic oscillator in terms of "stuff
types", related to combinatorial species, and gives a combinatorial
interpretation of Feynman diagrams. (Theory and Applications of
Categories, vol. 16, pp. 785854, 2006, doi:10.1.1.69.5789.)
Expository Papers
These are some shorter papers explaining some of the same material as
in the above, but for different audiences.
Cubical nCategories and Finite Limits Theories (arXiv:1001.2628; unpublished)
An expository piece explaining how the double cospan
construction in math.CT/0611930 may be seen in terms of taking models
of finite limits theories. Higher cubical ncategories can be
constructed by iteratively taking models of a theory of
categories/bicategories etc.)
Groupoidification and Categorification in Physics (hosted here)
Originally written for a philosophy of science proceedings,
but not used. Talks about the significance of various forms of
categorification as explanations (i.e. descriptive accounts) of
mathematical structures in physics. Makes (possibly naive)
philosophical remarks about how this relates to structural
realism.
Articles in Preparation
Transport Functors and 2Group Actions in Higher Gauge
Theory (with Roger Picken)
Proves an equivalence of two constructions of a double
groupoid playing the role of moduli space in higher gauge theory:
one based on transport 2functors assigning holonomies to edges,
and one based on the transformation structure for a global
symmetry action of a 2group of gauge transformations on an
underlying category of connections.
The Categorified Heisenberg Algebra II:
Representations from Free Constructions (with Jamie Vicary)
Derives a categorification of Vicary's Fock monad for an
important class of 2categories, and shows how this gives rise
both to the groupoidification of the Heisenberg algebra, and its
2linear representations.
