Irrational Toric Varieties

Classical toric varieties come in two flavours:  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.
When A is homogeneous, it is projective.   Applications of 
mathematics have long studied the positive real part of a 
toric variety as the main object, where the points A may
be arbitrary points in R^n.  For example, in 1963 Birch 
showed that 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 with Garcia-Puente, Zhu, Postinghel, and 
Villamizar, and work in progress with Pir.