Geometry Seminar

Date: March 31, 2017

Time: 4:00PM - 5:00PM

Location: BLOC 628

Speaker: Ata Firat Pir, TAMU

Title: Irrational Toric Varieties

Abstract: Classical toric varieties come in two flavors: Normal toric varieties are given by rational fans in R^n. A (not necessarily normal) affine toric variety is given by finite subset A of Z^n. Toric varieties are well understood and they can be approached in a combinatorial way, making it possible to compute examples of abstract concepts. Applications of mathematics have long studied the positive real part of a toric variety as the main object, where the points in A may be arbitrary points in R^n. In 1963, Birch showed the such an irrational toric variety is homeomorphic to the convex hull of the set A. Recent work showing that all Hausdorff limits of translates of irrational toric varieties are toric degenerations suggested the need for a theory of irrational toric varieties associated to arbitrary fans in R^n. These are R^n_>-equivariant cell complexes dual to the fan. Among the pleasing parallels with the classical theory is that the space of Hausdorff limits of the irrational projective toric variety of a finite set A in R^n is homeomorphic to the secondary polytope of A. This talk will sketch this story of irrational toric varieties. It represents work in progress with Sottile.