Toric topology

Toric topology studies actions of a compact torus on topological spaces locally equivalent to actions of the standard $n$-torus on open subsets in a complex vector space. These actions arise naturally in many fields of mathematics. There are remarkable functorial constructions of these torus actions leading to several striking applications. These include constructions of new combinatorial invariants of convex polytopes and triangulations of spaces, detecting toric representatives in the cobordism class of any smooth manifold admitting an embedding into a Euclidean space with a complex structure in the normal bundle, and discovery of a new wide family of complex non-Kaehler manifolds.