With T. Nevins. *W*-Symmetry of the Adelic Grassmannian.

We give a geometric construction of the *W*_{1+∞}-vertex algebra
as the infinitesimal form of a factorization structure on an adelic Grassmannian.
This gives a concise interpretation of the higher symmetries and
Backlund-Darboux transformations of the KP hierarchy and its multicomponent
extensions in terms of a version of "*W*_{1+∞}-geometry":
the geometry of D-bundles on smooth curves, or equivalently torsion-free
sheaves on cuspidal curves.