I work on mathematical physics — which I interpret broadly as ‘math that could be of interest in physics, and physics that could be of interest in math’. I’m especially interested in -categories and their applications. I’ve spent a lot of time explaining these subjects on the web. You can see a sampling below. For a lot more stuff, see my website.
Higher-dimensional algebra and topological quantum field theory, with James Dolan, Jour. Math. Phys. 36 (1995), 6073-6105.
Higher-dimensional algebra I: braided monoidal 2-categories, with Martin Neuchl, Adv. Math. 121 (1996), 196-244.
An introduction to -categories, in 7th Conference on Category Theory and Computer Science, eds. E. Moggi and G. Rosolini, Springer Lecture Notes in Computer Science vol. 1290, Springer, Berlin, 1997.
Higher-dimensional algebra II: 2-Hilbert spaces, Adv. Math. 127 (1997), 125-189.
Higher-dimensional algebra III: -categories and the algebra of opetopes, with James Dolan, Adv. Math. 135 (1998), 145-206.
Spin foam models, Class. Quantum Grav. 15 (1998), 1827-1858.
Categorification, with James Dolan, in Higher Category Theory, eds. Ezra Getzler and Mikhail Kapranov, Contemp. Math. 230, American Mathematical Society, Providence, Rhode Island, 1998, pp. 1-36.
An introduction to spin foam models of BF theory and quantum gravity, in Geometry and Quantum Physics, eds. Helmut Gausterer and Harald Grosse, Springer, Berlin, 2000, pp. 25-93.
From finite sets to Feynman diagrams, with James Dolan, in Mathematics Unlimited - 2001 and Beyond, vol. 1, eds. Björn Engquist and Wilfried Schmid, Springer, Berlin, 2001, pp. 29-50.
Higher-dimensional algebra and Planck-scale physics, in Physics Meets Philosophy at the Planck Length, eds. Craig Callender and Nick Huggett, Cambridge U. Press, Cambridge, 2001, pp. 177-195. Also available as a webpage.
Higher-dimensional algebra IV: 2-tangles, with Laurel Langford, Adv. Math. 180 (2003), 705-764.
Quantum quandaries: a category-theoretic perspective, in Structural Foundations of Quantum Gravity, eds. Steven French, Dean Rickles and Juha Saatsi, Oxford U. Press,Oxford, 2006, pp. 240-265. Also available as a webpage
Higher gauge theory, with Urs Schreiber, in Categories in Algebra, Geometry and Mathematical Physics, eds. Alexei Davydov, Michael Batanin, Michael Johnson, Stephen Lack and Amnon Neeman, Contemp. Math. 431, American Mathematical Society, Providence, Rhode Island, 2007, pp. 7-30.
Lectures on -categories and cohomology, with Michael Shulman, to appear in n-Categories: Foundations and Applications, eds. John Baez and Peter May.
The classifying space of a topological 2-group, with Danny Stevenson, to appear in the proceedings of the 2007 Abel Symposium, eds. Nils Baas et al.
Convenient categories of smooth spaces, with Alex Hoffnung.
Categorified symplectic geometry and the classical string, with Alex Hoffnung and Chris Rogers.
Categorified symplectic geometry and the string Lie 2-algebra, with Christopher L. Rogers.
Representations of 2-groups on higher Hilbert spaces, with Aristide Baratin, Laurent Freidel and Derek Wise.
Physics, topology, logic and computation: a Rosetta Stone, with Mike Stay, to appear in New Structures in Physics, ed. Bob Coecke.
Groupoidification made easy, with Alexander E. Hoffnung and Christopher D. Walker.
A history of n-categorical physics, with Aaron Lauda, to appear in proceedings of Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World, ed. Hans Halvorson. Draft version.
Higher gauge theory II: 2-connections, with Urs Schreiber. Draft version.