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Author: William A. Stein
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MECCAH: Computation of Curt McMullen

Some Computations of Curt McMullen on MECCAH

During the last couple of months I have used meccah intensively to study a problem in billiards and moduli space.

Just as billiards in a square can be analyzed by passing to a torus, billiards in more complicated polygons can be analyzed by passing to Riemann surfaces of higher genus. Especially remarkable are "lattice polygons", which give rise to algebraic curves in moduli space, and which admit a very complete dynamical analysis.

In 1989 Veech showed the regular n-gons are lattice polygons. The corresponding Riemann surfaces have genus tending to infinity. In 2001 I found an infinite family of lattice polygons in genus two, and showed each such polygon has an associated discriminant D>0.

Lately I formulated a precise conjecture about how to construct all lattice polygons in genus two. This conjecture can be checked one discriminant at a time. By automating this check, I was able to verify the conjecture for all discriminants D <= 400.

The "discovery" of the proof for this range of D took several weeks of computation, but now that it is found, the proof can be verified in about 2 hours. These matters are discussed in a paper in preparation, titled:

Teichmueller curves in genus two: The regular decagon.

(In the conjecture there is one exceptional case, which comes from the regular 10-gon when D=5.)