Experimental Differential Geometry

-Uniformization and Quantum field theory

One of the most interesting projects I'm working on right now is an idea by Jack Gegenberg and Eric Woolgar. The subject of study is uniformization theorems in differential geometry. In particular, we're using a proposition by Jack and Eric (to be detailed later), that uses BF field theory as a method of `flowing' the geometry of a manifold to a homogeneous one. In two dimensions, this leads to the well-known uniformization theorem, ie that all compact orientable Riemann surfaces are just distortions of homogeneous surfaces with n handles.

The method of flowing uses ideas borrow from Ashtekar's new variable (connection representation) formulation of general relativity, and Yang-Mills quantum field theory. We start by representing the metric on a manifold in terms of a connection representation, which is simple a connection that contains a metric, represented by a dyad in two dimensions, and a spin-connection. The spin-connection does not have to be compatible with the metric. The connection is then evolved using a flow that has fixed points at the homogeneous metrics. The conjecture is that the flow always converges to those fixed points for arbitrary initial conditions. If true, then this gives a proof of the uniformization theorem much akin to the Ricci flow proof of Hamilton and Chow (see J. Diff Geom. 35, 723 (1992) or Mathematics and General Relativity, ed J. Isenberg, Contemp. Math 71, AMS (1988)). if this program can be extended to three dimensions, then it may be possible to prove the elusive Thurston Geometrization Conjecture, and produce a classification of all 3-manifolds. This is important both in math, and in physics, where we are interested in summing over all three-geometries in modern formulations of quantum gravity.

I decided to model the flow on a computer. It's a tough problem, as the equations are highly nonlinear, and force one to use conditionally stable methods with small step sizes. The equations are complicated enough that they were implemented by using Maple to write the fortran. At present, the simulation code is written for the two-dimensional toroidal topology case, in Fortran 77. The manifold is split into a rectangular mesh, with a connection on it, and the connection is evolved. The result is saved as a record of cell volumes, connection curvature (with respect to a Yang-Mills/BF energy-momentum), and torsion. The torsion is needed because the connection is often evolved into one in which the spin-connection is not compatible with the metric, until the fixed point is reached. Results are then passed through C code that converts the data into Object-Oriented Geometry Library language, for display by Geomview. This allows us to produce 3D animations of the simulation.

The preliminary results are that the flow works to a certain extent. Presently, we can take a initial condition like this:

where the colours represent torsion (yellow for a compatible spin-connection), and deviation from a flat connection is represented by deviation away from a torus (ie, the shape is not actually the shape of the manifold), and evolve it to this:

in which the metric now has zero curvature, the spin-connection is compatible with the metric, and we thus have a flat torus. We have crude Mpeg movies of some of the runs:

Quality of the movies will improve once we have better image capture and conversion, which is needed because the X version of Geomview does not have an image capture capability yet). I have produced high-quality images at the Centre for Experimental and Constructive Mathematics at SFU, though. The first product of that work is an extremely experimental dynamic research paper that I'm producing, combining the new web language, Java, with the famous LaTeX2HTML converter. This is in the early stages of development, and needs a Java-capable browser, but please feel free to look around.

The flow seems to work for a large range of metrics, but we are finding that the flow has late time instabilities. we believe that we have now pinned the problem down to the numerics, and a buildup of numerical errors. However, in tracing down the problem, we discovered a lot about the flow, including certain conditions on the initial condtions. Thus we may have experimentally discovered important information that may help us understand the flow. This is the use of experimental mathematics. Other interesting properties of the flow have been discovered experimentally, and we're hoping to use these discoveries to point us towards new theorems. If correction terms (ie, viscosity terms) can be found to re-estabilish numerical stability of the flow, work will soon move on to different topologies (multiple handles and the 2-sphere), before we believe we've learned enough to start on the Thurston Conjecture case. I'm also hoping to investigate more advanced methods of numerically evolving the equations, to avoid numerical stability problems.

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