# 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.