This is an announcement for the paper "Almost-invariant and essentially-invariant halfspaces" by Gleb Sirotkin and Ben Wallis. Abstract: In this paper we study sufficient conditions for an operator to have an almost-invariant half-space. As a consequence, we show that if $X$ is an infinite-dimensional complex Banach space then every operator $T\in\mathcal{L}(X)$ admits an essentially-invariant half-space. We also show that whenever a closed algebra of operators possesses a common AIHS, then it has a common invariant half-space as well. Archive classification: math.FA Mathematics Subject Classification: 15A03, 15A18, 15A60, 47L10, 47A10, 47A11, 47A15 Remarks: 11 pages. Keywords: functional analysis, Banach spaces, surjectivity The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1509.07428 or http://arXiv.org/abs/1509.07428