Noncommutative Minkowski Superspace:
I describe below one of my past projects, the main thrust of my research has diverged slightly from this topic.. but if you are interested I would be happy to dust it off and start something new if we can find a time and a place.
My first research as a math graduate student was on a gauge field theory in noncommutative Minkowski spacetime. This is actually much weirder than the mathematics of SUSY project I described above. However, I do believe I know of a few fairly benign physics projects I could help you calculate if you wanted to explore the world of deformation quantization in physics. Conceptually, deformation quantization is a different can of worms than my paper but the sort of calculations involved are very similar. Noncommutative geometry in physics is very new, it is likely if we chose the calculation carefully we could get a publication out of it. This would not hurt if you are aiming towards getting into a good graduate school. Let me tell you what I did, but again you could do some project which is only tangentially related to my work. I wanted to understand if it was physically sensible to write a supersymmetric (SUSY) field theory for which the underlying spactime is noncommutative (actually in the space I study there is no supersymmetry,but when the noncommutative deformation is set to zero we recover the usual SUSY field theory). There is no experimental verification of SUSY at this time, but many physicsists hope it may be detected about 2010. Noncommutative geometry places constraints on space itself, so that space is no longer continuous below some certain scale. Essentially what this means is that we have done away with the point, hence Noncommutative geometry is pointless. The theory I worked out naturally extended ideas from both SUSY and Noncommutative geometry to encorporate them simultaneously. If your interested I've posted a preprint of my work below.
Noncommutative Minkowski Superspace paper
Another current project of mine is a joint work with an ex-fellow graduate student friend of mine, Dr. J.D. Brown. We are working out how n-symplectic geometry behaves over a super symplectic manifold. I suspect that this area of inquiry could provide many projects in the years to come. More on this later, Lord willing.
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Last Modified: 7-16-08