Irrational Toric Varieties

Classical toric varieties are among the simplest objects in algebraic geometry. They arise in an elementary fashion as varieties parametrized by monomials whose exponents are a finite subset $\mathcal{A}$ of $\mathbb{Z}^n$. They may also be constructed from a rational fan $\Sigma$ in $\mathbb{R}^n$. The combinatorics of the set $\mathcal{A}$ or fan $\Sigma$ control the geometry of the associated toric variety. These toric varieties have an action of an algebraic torus with a dense orbit. Applications of algebraic geometry in geometric modeling and algebraic statistics have long studied the nonnegative real part of a toric variety as the main object, where the set $\mathcal{A}$ may be an arbitrary set in $\mathbb{R}^n$. These are called irrational affine toric varieties. This theory has been limited by the lack of a construction of an irrational toric variety from an arbitrary fan in $\mathbb{R}^n$. We construct a theory of irrational toric varieties associated to arbitrary fans. These are $(\mathbb{R}_>)^n$-equivariant cell complexes dual to the fan. Such an irrational toric variety is projective (may be embedded in a simplex) if and only if its fan is the normal fan of a polytope, and in that case, the toric variety is homeomorphic to that polytope. We use irrational toric varieties to show that the space of Hausdorff limits of translates an irrational toric variety associated to a finite subset $\mathcal{A}$ of $\mathbb{R}^n$ is homeomorphic to the secondary polytope of $\mathcal{A}$.Comment: This thesis is a longer version of arXiv:1807.0591

Recent Progress in Irrational Conformal Field Theory

In this talk, I will review the foundations of irrational conformal field theory (ICFT), which includes rational conformal field theory as a small subspace. Highlights of the review include the Virasoro master equation, the Ward identities for the correlators of ICFT and solutions of the Ward identities. In particular, I will discuss the solutions for the correlators of the $g/h$ coset constructions and the correlators of the affine-Sugawara nests on $g\supset h_1 \supset \ldots \supset h_n$. Finally, I will discuss the recent global solution for the correlators of all the ICFT's in the master equation.Comment: 16 pages, Latex, UCB-PTH-93/25, LBL-34610, talk presented at the conference "Strings 1993", Berkeley, May 23-2

New Rotation Sets in a Family of Torus Homeomorphisms

We construct a family $\{\Phi_t\}_{t\in[0,1]}$ of homeomorphisms of the two-torus isotopic to the identity, for which all of the rotation sets $\rho(\Phi_t)$ can be described explicitly. We analyze the bifurcations and typical behavior of rotation sets in the family, providing insight into the general questions of toral rotation set bifurcations and prevalence. We show that there is a full measure subset of $[0,1]$, consisting of infinitely many mutually disjoint non-trivial closed intervals, on each of which the rotation set mode locks to a constant polygon with rational vertices; that the generic rotation set in the Hausdorff topology has infinitely many extreme points, accumulating on a single totally irrational extreme point at which there is a unique supporting line; and that, although $\rho(t)$ varies continuously with $t$, the set of extreme points of $\rho(t)$ does not. The family also provides examples of rotation sets for which an extreme point is not represented by any minimal invariant set, or by any directional ergodic measure.Comment: Author's accepted version. The final publication is available at Springer via http://dx.doi.org/10.1007/s00222-015-0628-