Galois Groups of Differential Equations

My research is partially funded through the Lise Meitner Grant "Galois Groups of Differential Equations" M 2582-N32 of the Austrian Science Fund FWF (Funding period: 7/2019-6/2023).

The main goal of this project is to further our understanding of the symmetries among the solutions of linear differential equations with rational function coefficients. The Galois group of a such a differential equation is a linear algebraic group, i.e., it can be identified with a subgroup of a general linear group defined by polynomials in the matrix entries. As we vary over all linear differential equations with rational function coefficients, these groups fit together to form a proalgebraic group, called the absolute differential Galois of the rational function field. This rather large group encodes all symmetries among all solutions of linear differential equations with rational function coefficients. It can be seen as a differential analog of the absolute Galois group of the field of rational numbers. 

The Shafarevich conjecture in inverse Galois theory states that the absolute Galois group of the maximal abelian extension of the field of rational numbers is a free profinite group. A differential analog of Shafarevich's conjecture, known as Matzat's conjecture, suggests that the absolute differential Galois group of the field of rational functions is a free proalgebraic group. The main aim of this project is to make progress towards this conjecture. Several special cases of the conjecture have already been established. In particular, the conjecture has been proved in the important classical case, where the field of coefficients for the rational functions is the field of complex numbers.  

Update 9/2022: Matzat's conjecture is proved in full generality: The absolute differential Galois group of a one-variable function field over an algebraically closed field of characteristic zero is a free proalgebraic group. See the preprint ArXiv:2209.01581 with Ruyong Feng.

For more details please see:

• A poster describing the project.

• Free proalgebraic groups, M. Wibmer, Épijournal de Géométrie Algébrique, Volume 4, Article no.1, 1-36, 2020, link, ArXiv:1904.07455

• Free differential Galois groups, A. Bachmayr, D. Harbater,  J. Hartmann and M. Wibmer, Transactions of the American Mathematical Society 374(6), 4293-4308, 2021, link, ArXiv:1904.07806

• The differential Galois group of the rational function field, A. Bachmayr, D. Harbater,  J. Hartmann and M. Wibmer, Advances in Mathematics, Volume 381, 2021, link, ArXiv:2004.05906

• Subgroups of free proalgebraic groups and Matzat's conjecture for function fields, M. Wibmer, to appear in Israel Journal of Mathematics, ArXiv:2102.02553

• Regular singular differential equations and free proalgebraic groups, M. Wibmer, ArXiv:2209.01764

• Differential Galois groups, specializations and Matzat's conjecture, with R. Feng, ArXiv:2209.01581