Topology Seminar (Introductory Talk): Astonishingly symmetric flat surfaces

Seminar | April 19 | 2:10-3 p.m. | 736 Evans Hall

 Alex Wright, Stanford

 Department of Mathematics

The group SL(2,Z) of two by two integer matrices with determinant one acts on the torus $R^2/Z^2$ by affine maps. Similarly, a finite index subgroups of SL(2,Z) acts on any square-tiled surface. You might reasonably think that square-tiled surfaces are the only flat surfaces with an affine action of such a large group of matrices (called a lattice), but you'd be wrong. For example, Veech discovered in the 1980s that the regular 2n-gon with opposite sides identified has this property. We'll review Veech's discovery, and its motivation and consequences. Then we'll explain discoveries of McMullen and Moller from the 2000's that the affine symmetries are related to symmetries of a larger dimensional torus called the Jacobian, which is studied in algebraic geometry and number theory. No background will be assumed.