damned if i know

Getting a good look in Belem, Portugal



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.

Research Papers

  • 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.

  • 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 2-Linearization (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.

  • 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.

  • 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.

  • 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.

Expository Papers

These are some shorter papers explaining some of the same material as in the above, but for different audiences.
  • Cubical n-Categories and Finite Limits Theories (arXiv:1001.2628; unpublished)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.