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.
Research Papers
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.
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.
Extended TQFT, Gauge Theory, and 2Linearization
(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.
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.
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.
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.
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Cubical nCategories and Finite Limits Theories (arXiv:1001.2628; unpublished)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.
