## Project Details

### Description

This project aims to further the understanding of algebraic properties of solutions of linear
differential equations. The main focus is on linear differential equations whose coefficients are
rational functions.
Li near differential equations are ubiquitous in science and engineering. To perform symbolic
computations with the solutions of linear differential equations it is essential to understand the
algebraic relations among the solutions. The algebraic relations a mong the solutions of a given
linear differential equation are governed by a linear algebraic group, called the differential Galois of
the differential equation. All differential Galois groups of all linear differential equations with
rational function coe fficients fit together to form the absolute differential Galois group of the field
of rational functions. The main goal of this project is to establish an explicit description of this
group. Indeed, following B.H. Matzat we conjecture that this group is a free proalgebraic group on a
set whose cardinality agrees with the cardinality of the field of coefficients of the rational
functions.
The absolute differential Galois group of a differential field can be seen a differential analog
of the absolute Galois group of a field. According to a Theorem of A. Douady, F. Pop and D.
Harbater, the absolute Galois group of the field of rational functions is a free profinite group.
Matzat's conjecture would generalize this theorem.
Important methods for this project include patching techniques and embedding problems.
Out plan to prove Matzat's conjecture has two steps. Firstly we want to characterize free
proalgebraic groups in terms of embedding problems and secondly we would like to show that the
absolute differenti al Galois group satisfies this characterization. For the second step patching
techniques will be used.

Status | Active |
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Effective start/end date | 1/06/19 → 31/05/23 |

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