My Background

Education

• Ph.D.: June 2007 from the Mathematics Department of the University of California, Riverside. My dissertation was entitled "Extended TQFT's And Quantum Gravity" and was completed under the advice and supervision of Dr. John Baez.
• M.Sc.: February 2001 from McGill University, in the Department of Mathematics and Statistics. My Masters Thesis was entitled "Existence and Nonexistence Theorems for Solutions of the Einstein-Dirac-Maxwell Equations", and was written under the supervision of Dr. Niky Kamran.
• B.Math: May 1998 from the Math Faculty at the University of Waterloo; double major in departments of Pure Mathematics and Combinatorics and Optimization.

Teaching

• McGill University (Montreal); T.A. 1998-2000; Lecturer, 2001
• Dawson College (Montreal); Lecturer, 2001-2002
• The University of California (Riverside); T.A. 2002-2007
• The University of Western Ontario (London, Canada); Postdoc and Assistant Professor, 2007-2010

Curriculum Vitae

• My CV proper
• My research statement
• My teaching statement
• A list of my publications)

Research

Papers

• Extended TQFT, Gauge Theory, and 2-Linearization (arXiv:1003.5603; submitted to Journal of Knot Theory and its Ramifications)

  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.

• Cubical n-Categories and Finite Limits Theories (arXiv:1001.2628; unpublished)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.)

• 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 Theory and Applications of Categories (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.

Selected Talks

Relating to spans, groupoids, and 2-vector spaces

• 2-Linearization in Physics and Topology: A talk given at the XIX Oporto Meeting in Faro, Portugal.
• Groupoidification and 2-Linearization, Discrete and Smooth: A talk given at CAMGSD in July 2010.
• Groupoidification and 2-Linearization: A talk given at the mathematics colloquium while I was visiting at Dalhousie in September of 2009.
• Groupoids, Spans and Physics: a talk given at U of Ottawa, January 2009. Presents 1-vector space and 2-vector space views of groupoidified physics.
• 2-Vector Spaces and Groupoids: a talk given for GroupoidFest '08 at UCR, November 2008.

From the Seminar on Groupoids/Stacks/Algebras

• Topological and Lie Groupoids, C*-Algebras and Stacks: An intorductory talk introducing groupoids as the link between two ways of generalizing "space": stacks, and noncommutative geometry.
• Groupoid Representation Theory: The basics of groupoid representations, and leading up to a main theorem about Morita equivalence in different categories.

Relating to ETQFT's and Gravity

• Slides for a talk given at the conference "Connections in Geometry and Physics" organized by the Perimeter Institute for Theoretical Physics, the Fields Institute, and the University of Waterloo, held at the Perimeter Institute in Waterloo, Canada, May 8-10, 2009
• Slides for a talk given at the quantum gravity seminar at the Perimeter Institute, Feb 28, 2008.
• Slides for my dissertation defense talk at UCR in June of 2007. This is more technical, focusing on defining terms and stating my main theorem.

Relating to the Quantum Harmonic Oscillator

• A talk about the categorified harmonic oscillator which I gave at the International Category Theory Conference at White Point, Nova Scotia in June 2006.
• Slides for a talk I gave for for my oral qualifying exam in May 2005