Anderson defined MV polytopes as the moment polytopes of
Mirkovic-Vilonen cycles, which show up in the geometric Satake
approach to geometric representation theory. Kamnitzer gave a closed-form
description of MV polytopes (which I'll recall), and Baumann-Kamnitzer
gave a way of associating them to generic representations of
preprojective algebras of quivers, which show up (as I'll recall)
in the Lusztig approach to geometric representation theory.
(Since then, these two approaches have been related by BLPW symplectic duality.)
So where are the cycles, not just the polytopes?

I'll present a conjecture due to me and Kamnitzer, and time permitting,
its partial resolution through recent work of Braverman-Finkelberg-Nakajima
on symplectic duality.