Differential isomorphism and equivalence of algebraic varieties

Let us call two (complex irreducible) varieties X and Y differentially isomorphic if the rings of (global) differential operators on X and Y are isomorphic. A natural question whether differentially isomorphic varieties are in fact isomorphic is answered in the negative, and the answer depends drastically on geometry of the underlying varieties. In the case of curves, there is a remarkable connection to the theory of integrable systems which allows one to settle the problem completely. In higher dimensions, we know only a few non-trivial examples. The talk will attempt to survey the present state of our knowledge about this and related questions. If time permits, some conjectures and applications to other areas (including analysis of PDEs and representation theory) will be discussed.

[This is joint work with George Wilson (Imperial College, London); for a readable survey see math.AG/0304320.]