Research

My main research interests are mathematical physics, topology, and category theory, and in particular, how "higher-algebraic" ideas from category theory can illuminate questions in quantum gravity and the foundations of quantum mechanics. This has connections to Witten-type path-integral invariants for manifolds, combinatorics, representation theory, and more.

Published Research Papers

• 2-Group Actions and Moduli Spaces of Higher Gauge Theory (with Roger Picken, arXiv:1904.10865)

  Describes a double category of 2-group 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 2-group on a category. (Accepted by Journal of Geometry and Physics, in press)

• Transformation Double Categories Associated to 2-Group 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. 1429-1468, 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. 703-745, 2018, DOI:10.1016/j.jpaa.2017.05.004)

• Cohomological Twisting of 2-Linearization and Extended TQFT (arXiv:1003.5603)

  Uses 2-linearization (see "2-Vector Spaces and Groupoids") to build an Extended TQFT associated to a discrete gauge group. A special case, for closed manifolds, yields the untwisted Dijkgraaf-Witten model. (Journal of Homotopy and Related Structures, vol. 10, no. 3, pp. 127-187,2013, DOI:10.1007/s40062-013-0047-2.)

• 2-Vector Spaces and Groupoids (arXiv:0810.2361; Applied Categorical Structures; DOI: 10.1007/s10485-010-9225-0)

  Constructs a 2-functor from the bicategory of spans of groupoids into the bicategory of 2-Vector spaces. I interpret this as a kind of categorified version of quantization of a physical system. (Applied Categorical Structures, vol. 19, no. 4, pp. 659-707, 2011, https://doi.org/10.1007/s10485-010-9225-0.)

• Double Bicategories and Double Cospans (arXiv:math.CT/0611930; Journal of Homotopy and Related Structures, Vol. 4(2009), No. 1, pp. 389-428)

  Describes a kind of weak cubical 2-category, how a construction using cospans (or spans) gives examples, and how they relate to more familiar 2-categories. (Journal of Homotopy and Related Structures, vol. 4, no. 1, pp. 389-428, 2009)

• Categorified Algebra and Quantum Mechanics (arXiv:math.QA/0601458, Theory and Applications of Categories, Vol. 16, 2006, No. 29, pp 785-854.)

  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. 785-854, 2006, doi:10.1.1.69.5789.)

Expository Papers

• Cubical n-Categories 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 n-categories 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 2-Group 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 2-functors assigning holonomies to edges, and one based on the transformation structure for a global symmetry action of a 2-group 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 2-categories, and shows how this gives rise both to the groupoidification of the Heisenberg algebra, and its 2-linear representations.