Differential Galois theory is an algebraic theory for linear differential equations, in analogy to classical Galois theory. It was proposed by Picard and Vessiot, developed by Kolchin, and is closely tied to the theory of linear algebraic groups. Patching techniques have been used in inverse Galois theory and more recently in other areas of algebra and arithmetic geometry. The talk gives an introduction to differential Galois theory and to patching. Using patching methods, we will deduce new properties of differential Galois extensions over function fields of Riemann surfaces. As time permits, we will also point out connections to local-global principles for homogeneous spaces. Speaker(s): Julia Hartmann (University of Pennsylvania)