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When is a gluing of affine schemes affine? I will state a complete but
conjectural answer when the pieces are tori, and the gluing maps are
cluster mutations in the sense of Fomin-Zelevinsky. The combinatorics
of Demazure weaves allows to prove the conjecture in interesting
cases, notably those in which there are finitely many tori, as well as
the simplest ones with infinitely many. This has applications to
open-string Schubert calculus, i.e. computations of the Fukaya
category of G/P. Joint work with M. Gorsky, J. Simental, D. Speyer.