University of Toronto's Symplectic Geometry Seminar

September 15, 2006, 3:10pm
Bahen 6183

John J. Millson

University of Maryland

The toric geometry of triangulated polygons in Euclidean space

Abstract: There are (many) toric degenerations (one for each triangulation of a fixed convex planar n-gon = topological equivalence class of pair of pants decomposition of the n times punctured two sphere) of the moduli space of projective equivalence classes of n ordered points on the projective line constructed via commutative algebra and combinatorics. In fact these degenerations are closely related to "phylogenetic trees" now being studied in mathematical biology and combinatorics. The combinatorial descriptions give no clue about what the toric fibers actually look like as spaces acted on by a torus. The point of my talk (joint work wih Ben Howard and Chris Manon) is to construct the above toric fibers geometrically in terms of moduli spaces of spatial Euclidean n-gons with fixed side-lengths modulo a coarsening of Euclidean congruence. The compact part of the torus action is given by bending n-gons along the diagonals of the triangulation. Our construction is a translation of a construction of Jacques Hurtubise and Lisa Jeffrey.