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:

- Experiment 7:
*A conformal metric, starting off with lots of torsion, but with one of the three (scalar) forms of curvature zero* - Experiment 7a:
*Again, conformal, but this time with no torsion, and with scalar curvature.* - Experiment 9:
*non-conformal, lots of high frequency modes, large amplitude, lots of torsion.*

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.

Back to the fun research description, or the technical one.

SPB