At the Perimeter Institute in Waterloo, Canada.
At the 17th Oporto Meeting on Geometry, Topology and Physics at Universidade do Porto, Portugal.
Abstract: Topologically massive 3d gravity is a theory that comes with all of the basic conceptual difficulties of quantizing full-fledged 4d general relativity: it is diffeomorphism invariant and, unlike 3d Einstein gravity, has propagating degrees of freedom. So far, relatively little is known about quantizing this system. But even a linearized analysis reveals a wealth of interesting features, particularly when a cosmological constant is introduced. For example, with a negative cosmological constant, the theory's massive graviton modes have masses branched according to chirality. There are also special values of the topological mass parameter where the massive gravitons exhibit null propagation, and other values where the linearized theory becomes equivalent to topologically massive electrodynamics. I will review the basic features of topologically massive gravity, and discuss some recent results peculiar to the theory in the presence of a cosmological constant.
(Based on joint work with S. Carlip, S. Deser, and A. Waldron. See the papers Cosmological Topologically Massive Gravitons and Photons and Topologically Massive AdS Gravity for details and references.)
A slightly different version of the talk above, this one given at the Quantum Geometry and Quantum Gravity Conference (QG2), at the University of Nottingham, UK.
This was an invited talk at the Workshop on Categorical Groups in Barcelona. The first part was a blackboard talk; the second was a slide presentation.
Abstract: The general theory of 2-group representations is a straightforward categorification of group representation theory: representations are 2-functors, intertwining operators are pseudonatural transformations, and 2-intertwiners between these are modifications. However, while the category of vector spaces is the standard target category for group representations, one of the major challenges of 2-group representation theory has been to find a suitable target 2-category. For example, one would like a 2-category in which Lie 2-groups have an interesting variety of representations. One promising proposal is the 2-category Meas of "measurable categories", which first appeared in the work of Crane, Sheppeard, and Yetter. I will explain this 2-category, whose objects can be thought of as infinite-dimensional "higher Hilbert spaces". I will then describe representations of 2-groups on these higher Hilbert spaces, together with the intertwiners and 2-intertwiners between them. Examples will clarify their geometric meaning. I will also briefly discuss how 2-group representations in Meas have already shown up in physics, particularly in the state sum model of Baratin and Freidel.
At the workshop TRACES'07 (Applications of traces to algebra, analysis and categorical logic) at the University of Ottawa.
Abstract: The notorious "infinities" in quantum field theory are intimately related to the general notion of trace. For example, the problematic Feynman diagrams in field theory are the ones involving "feedback loops" like those in trace diagrams. Other examples include 2d gauge theories with noncompact gauge group, which have divergences whenever we try taking the "trace" of a cobordism. Indeed, the existence of certain kinds of traces is a major distinction between well-behaved topological field theories and their real real-world counterparts in physics. In this talk I will explore the relationships between traces and topology in quantum field theories. In particular, I will consider examples from 2d Yang-Mills theory, and from p-form electromagnetism in p+1 dimensions, which has divergences related to certain higher dimensional analogues of "trace".
At the 23rd Pacific Coast Gravity Meeting, held at Caltech.
A talk in the quantum gravity seminar at UC Riverside. This talk was partly an introduction to the basics of Kleinian geometry, but focussing on various conceptions of "spacetime", including the absolute space and time of Newtonian gravity, Galilean spacetime, Minkowski spacetime, and de Sitter spacetime. Someday, I might get some notes from this talk scanned in.
A seminar talk at the University of Ottawa. I spoke about higher dimensional analogs of braids and their relation to `quandles', which can be thought of as the sort of algebraic gadget one gets by starting with a group and forgetting how to multiply, remembering only how to conjugate. I also sketched how all this relates to "strings" in a topological field theory called BF theory. I explained the pictures on the handout, which I drew for this paper.
A talk at the Category Theory Octoberfest 2006, University of Ottawa, October 21-22, 2006.
Abstract: In two dimensions, the quantum gauge theory pioneered by Yang and Mills is known to be exactly solvable, and in fact "almost" a topological quantum field theory - aside from the topology of a spacetime cobordism, it requires only the total area as input. In this talk I show that in the abelian case of Yang-Mills theory, also known as electromagnetism, this fact remains true under categorification. More precisely, "n-form electromagnetism" is a categorification of electromagnetism in which the group U(1) is promoted to a strict abelian n-group. I describe n-form electromagnetism in (n+1)-dimensional spacetime as a "volumetric field theory" - a symmetric monoidal functor from the category of smooth (n+1)-dimensional cobordisms with volume to the category of Hilbert spaces.
This was related to my talk at CT earlier this year; both talks relate to my research on p-form electromagnetism.
A longer version of the talk at Octoberfest, given in the Topology Seminar at UCR, and focusing more on the topology than the higher category theory.
Abstract: In two dimensions, the quantum gauge theory pioneered by Yang and Mills is known to be exactly solvable, and in fact "almost" a topological quantum field theory - aside from the topology of a spacetime cobordism, it requires only the total area as input. In this talk I show that in the abelian case of Yang-Mills theory, also known as electromagnetism, this fact remains true under categorification. More precisely, "n-form electromagnetism" is a categorification of electromagnetism in which the group U(1) is promoted to a strict abelian n-group. I describe n-form electromagnetism in (n+1)-dimensional spacetime as a "volumetric field theory" - a symmetric monoidal functor from the category of smooth (n+1)-dimensional cobordisms with volume to the category of Hilbert spaces.
A talk in the quantum gravity seminar at the Perimeter Institute.
Abstract: Gravity in 2+1 dimensions has the remarkable property that momenta live most naturally not in Minkowski vector space but in the 3d Lorentz group SO(2,1) itself. Having group-valued momentum has interesting consequences for particles, including exotic statistics and a modified classification of elementary particle types. These results generalize immediately to 3d BF theory with arbitrary gauge group. Better yet, they generalize to 4d BF theory, where matter shows up as string-like defects. These "strings" exhibit exotic statistics governed not by the usual braid group, but by its higher dimensional cousin: the "loop braid group". I discuss these statistics as well as the classification of elementary "string types" in 4d BF theory.
A blackboard lecture given to Dan Christensen's research group at the University of Western Ontario. I talked about how momentum in 3d general relativity naturally takes values not in a vector space, but in the 3d Lorentz group SO(2,1). I discussed consequences of this fact, including exotic statistics and a modification of the classification of elementary particles. I also sketched how all this generalizes to momenta of particles in 3d BF theory, and to strings in 4d BF theory. Much of what I discussed is in the paper Exotic Statistics for Strings in 4d BF Theory.
A talk at the International Category Theory Conference, CT 2006
Abstract: One active area of research in the application of category theory to physics is 'higher gauge theory'---a generalization of the gauge theories of physics in which groups are replaced by 'n-groups'. So far higher gauge theory has mainly been studied for n = 2, but we can go further in the strict abelian case, which is called n-form electromagnetism, since it involves replacing the connection 1-form of ordinary electromagnetism by an n-form. In this talk, I describe a discrete version of n-form electromagnetism as a `chain field theory'---a theory analogous to topological quantum field theory, but with chain complexes replacing manifolds. More precisely, I define a symmetric monoidal category analogous to the category of cobordisms, but with chain complexes as objects and cospans as morphisms. A chain field theory is defined to be a symmetric monoidal functor from this category into the category of Hilbert spaces. The main result is that the physical theory of n-form electromagnetism on discrete spacetime gives such a functor.
A talk in the Topology Seminar at U.C. Riverside. This covered much of the same material as the paper Exotic Statistics for Strings in 4d BF Theory.
A talk in the Quantum Gravity Seminar at UCR.
A talk in the Quantum Gravity Seminar at UCR.
A talk at Loops '05, at the Albert Einstein Institute (AEI) in Golm, Germany.
A talk given to undergraduate physics majors at my alma mater, Abilene Christian University, Abilene, Texas. At the end of this talk, I had some extra time and wound up talking about how the second pair of Maxwell equations is related to Poincare duality.
A joint math/physics talk sponsored by the Math Department at the University of Colorado at Colorado Springs.
My oral qualifying exam at UCR.
This two part series of blackboard lectures was given in the Mathematical Physics Seminar at UCR, organized by Michel Lapidus.