Adèles and the geometry of schemes

By a result of André Weil vector bundles on a curve can be described as a
double quotient of the set of invertible matrices over the ring of adèles.
In this talk we will discuss an extension of this description to arbitrary
Noetherian schemes, and perfect complexes. As a corollary we obtain that
every Noetherian scheme can be reconstructed from Beilinson’s cosimplicial
ring of adèles.