Differential Galois theory and linear algebraic groups over algebraic function fields

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Differential Galois theory assigns a matrix group, called differential Galois group, to each linear differential equation. The differential Galois group contains information on the differential field generated by the solutions of the equation. It also measures the algebraic relations among the solutions. The differential Galois group of an ordinary linear differential equation is a linear algebraic group. The inverse differential Galois problem asks which linear algebraic groups occur as differential Galois groups over a given differential field. Over rational function fields k(x) over algebraically closed fields k it has been known for almost 20 years that every linear algebraic group occurs. The first part of the project is to show that in addition, the absolute differential Galois group of k(x) is a free proalgebraic group. The approach is based on field patching methods and on the study of differential embedding problems.For a differential equation depending on an additional discrete parameter, one can define its sigma-differential Galois group, which is a subgroup of the differential Galois group. It can be regarded as a refined version of the latter, as it measures difference-algebraic relations among the solutions. Sigma-differential Galois groups are linear differential algebraic groups. In the second part of this project, we will study the corresponding inverse problem over C(x).The third part of this project is devoted to the study of torsors under linear algebraic groups: Using the field patching method, we aim to prove a local-global principle for reduced norms.