Nonrational Toric Geometry I - Symplectic Toric Quasifolds Minicourse

Explore the fascinating world of nonrational toric geometry in this 53-minute minicourse on symplectic toric quasifolds. Delve into the historical development and core concepts of toric quasifolds, which generalize toric varieties to simple convex polytopes that are not rational. Examine key notions such as quasilattices, quasirationality, and quasitori through various examples, including quasispheres, Penrose tilings, quasicrystals, regular convex polyhedra, and irrational Hirzebruch surfaces. Learn about the generalized Delzant construction, symplectic reduction, and nonrational cuts. Gain insights from recent research findings presented by Dr. Elisa Prato from Universita Degli Studi Firenze, in collaboration with other experts in the field.

Intro
starting point
symplectic toric manifolds
smooth polytopes
integrality
how does the Delzant construction work?
examples
simple rational convex polytopes
general simple convex polytopes
generalized Delzant construction
quasilattice
quasirationality
quasitorus
symplectic toric quasifolds
important remark
the quasisphere
the chart around the point S
transition
the Penrose kite (Battaglia-P. 2010)
the complex case (Battaglia-P. 2001)
symplectic reduction (Battaglia-P. 2019)
remarks on nonrational cuts
cutting a Penrose kite in half
from a kite and dart tiling to a rhombus tiling
cutting in a completely arbitrary direction
generalized Hirzebruch surfaces as symplectic cuts
bibliography
other related works