Solution algebras of differential equations and quasi-homogeneous   varieties: a new differential Galois correspondence
  We develop a new connection between Differential Algebra and Geometric Invariant Theory, based on an anti-equivalence of categories between solution algebras associated to a linear differential equation (i.e. differential algebras generated by finitely many polynomials in a fundamental set of solutions), and affine quasi-homogeneous varieties (over the constant field) for the differential Galois group of the equation.   Solution algebras can be associated to any connection over a smooth affine variety. It turns out that he spectrum of a solution algebra is an algebraic fiber space over the base variety, with quasi-homogeneous fiber. We discuss the relevance of this result to Transcendental Number Theory. 
Yves Andre
math.AG