  An action of a compact Lie group is called equivariantly formal, if the
Leray--Serre spectral sequence of its Borel fibration degenerates at the
E_2-term. This term is as prominent as it is restrictive.
  In this article, also motivated by the lack of junction between the notion of
equivariant formality and the concept of formality of spaces (surging from
rational homotopy theory) we suggest two new variations of equivariant
formality: "MOD-formal actions" and "actions of formal core".
  We investigate and characterize these new terms in many different ways
involving various tools from rational homotopy theory, Hirsch--Brown models,
$A_\infty$-algebras, etc., and, in particular, we provide different
applications ranging from actions on symplectic manifolds and rationally
elliptic spaces to manifolds of non-negative sectional curvature.
  A major motivation for the new definitions was that an almost free action of
a torus $T^n\curvearrowright X$ possessing any of the two new properties
satisfies the toral rank conjecture, i.e. $dim H^*(X;Q)\geq 2^n$. This
generalizes and proves the toral rank conjecture for actions with formal orbit
spaces.

Algebraic Topology;Differential Geometry