Derek Wise

I am a VIGRE postdoc and Visiting Research Assistant Professor at U.C. Davis. My research is in mathematical physics, especially classical and quantum gravity, and applications of geometry, topology, and category theory to physics. Latest news: This quarter at UC Davis, I'm organizing a seminar on "Quantum Gravity and Strings". Here are links to abstracts of upcoming talks (also, past talks). Seminar participants, and others, may also be interested in the new page of quantum gravity resources I'm working on.

Curriculum Vitae Research Papers Talks QG Seminar Teaching

Education

Thesis: Topological Gauge Theory, Cartan Geometry, and Gravity
Advisor: John C. Baez

My main research focus these days is on certain aspects of quantum gravity. More specifically, I've been studying aspects of spin foam quantization and MacDowell-Mansouri gravity.

The following paper explains the geometry behind the MacDowell-Mansouri approach to gravity:

MacDowell-Mansouri Gravity and Cartan Geometry
Derek K Wise gr-qc/0611154

Abstract: The geometric content of the MacDowell-Mansouri formulation of general relativity is best understood in terms of Cartan geometry. In particular, Cartan geometry gives clear geometric meaning to the MacDowell-Mansouri trick of combining the Levi-Civita connection and coframe field, or soldering form, into a single physical field. The Cartan perspective allows us to view physical spacetime as tangentially approximated by an arbitrary homogeneous "model spacetime", including not only the flat Minkowski model, as is implicitly used in standard general relativity, but also de Sitter, anti de Sitter, or other models. A "Cartan connection" gives a prescription for parallel transport from one "tangent model spacetime" to another, along any path, giving a natural interpretation of the MacDowell-Mansouri connection as "rolling" the model spacetime along physical spacetime. I explain Cartan geometry, and "Cartan gauge theory", in which the gauge field is replaced by a Cartan connection. In particular, I discuss MacDowell-Mansouri gravity, as well as its recent reformulation in terms of BF theory, in the context of Cartan geometry.

The past two summers I've been a visitor at the Perimeter Institute in Waterloo, Ontario. I spent much of my time there working with Laruent Freidel and Aristide Baratin on categorical 2-group representation theory, particularly as it relates to their spin foam model. We're now finishing up a paper on this subject, with John Baez.

I've spoken in the quantum gravity seminar at PI during both visits:

• In summer of 2007, I spoke on Cartan Geometry and MacDowell-Mansouri gravity, the subject of the paper above.
• In summer of 2006, I spoke on Exotic Statistics and Particle Types in 3- and 4d BF Theory.

Here are the abstract and slides for my talk at Loops '05.

Here are my notes from John Baez's Quantum Gravity Seminar (most of which are actually not directly about quantum gravity). You can also read some of my notes based on John's lectures on Classical Mechanics.

Abstract: We investigate p-form electromagnetism---with the Maxwell and Kalb-Ramond fields as lowest-order cases---on discrete spacetimes, including the regular lattices commonly used in lattice gauge theory, but also more general examples. After constructing a maximally general model of discrete spacetime suitable for our purpose---a chain complex equipped with an inner product on (p+1)-cochains---we study both the classical and quantum versions of the theory, with either R or U(1) as gauge group. We find results---such as a `p-form Bohm--Aharonov effect'---that depend in interesting ways on the cohomology of spacetime. We quantize the theory via the Euclidean path integral formalism, where the natural kernels in the U(1) theory are not Gaussians but theta functions. As a special case of the general theory, we show p-form electromagnetism in p+1 dimensions has an exact solution which reduces when p = 1 to the abelian case of 2d Yang-Mills theory as studied by Migdal and Witten. Our main result describes p-form electromagnetism as a `chain field theory'---a theory analogous to a topological quantum field theory, but with chain complexes replacing manifolds. This makes precise a notion of time evolution in the context of discrete spacetimes of arbitrary topology.

I am very interested in category theory and its applications to physics, topology, and geometry.

In April 2007, I gave a talk on Loops, Bubbles, and Trace and Divergences in Quantum Field Theory at the workshop TRACES'07 (Applications of traces to algebra, analysis and categorical logic) at the University of Ottawa.

October 20-25, 2006, I was at the University of Ottawa, attending the Category Theory Octoberfest 2006, and talking to Rick Blute.

In June 2006, I spoke at the International Category Theory Conference, CT 2006, at White Point Beach, Nova Scotia. My talk was entitled "Chain Field Theory" and was be based on my work on discrete p-form electromagnetism. However, it focused more on category-theoretic aspects of the work than my previous talks and papers on this subject.

Teaching

In the Fall 2007 quarter at UC Davis, I taught Partial Differential Equations (MAT-118A). This Winter quarter, I'm teaching the second part of the same class (MAT-118B). Here is the schedule of classes for the quarter.

To see a complete list of math and physics courses I've taught or assisted, click here.

External Links