Differential Galois Theory

• The course will be run on moodle. To sign up, you need to request a key from me via email.
• Lectures: Mon 16:15-17:45 (moodle)
• Instructor: Tobias Dyckerhoff
• Office: Geomatikum 338
• Office Hours: Wed 13:00-14:00 (click here to enter BBB office hour)
• Email: tobias.dyckerhoff AT uni-hamburg.de

Tutorial

• Wed 14:15-15:45 (BBB as explained in moodle)

Lectures

Problem Sets

Problem Set 1
Problem Set 2
Hints for Problem Set 2
Problem Set 3
Problem Set 4
Problem Set 5
Problem Set 6
Problem Set 7
Problem Set 8
Problem Set 9
Problem Set 10

Course description

Content

• Classical Galois theory relates algebraic properties of the roots of a given polynomial to properties of its Galois group. For example, the roots can be expressed in terms of radicals if and only if the Galois group is solvable. Differential Galois theory provides an analogous perspective, allowing for a study of the differential-algebraic properties of the solutions of a linear differential equation in terms of its so-called differential Galois group. In contrast to the classical context, where the Galois group is a permutation group acting on the roots, the differential Galois group is a linear algebraic group acting on the vector space of solutions.
• In the course, we will provide an introduction to differential Galois theory from scratch. Needed background material from algebraic geometry, algebraic groups, and algebraic topology will be discussed in the tutorial along with detailed examples and problems adapted to the lectures.
• The course will be mostly self-contained, but towards the end, we will discuss one or more selected advanced topics based on the interest of the participants:
  • Tannakian categories
  • Integrability questions in mathematical physics
  • Riemann Hilbert Correspondence
  • Stokes phenomena

Literature

• Galois theory of linear differential equations (M. Van der Put, M.F. Singer)
• Galois’ Dream: Group Theory and Differential Equations (M. Kuga)
• Lectures on differential Galois theory (A. Magid)
• Differential Galois Theory and Non-Integrability of Hamiltonian Systems (J.J. Morales-Ruiz)

Prerequisites

• Algebra (Bachelor): Rings, Fields, Galois theory
• Analysis (Bachelor): Basics of ordinary differential equations such as the Picard-Lindelöf theorem
• Recommended, but not strictly necessary: Function Theory (Bachelor): Basics of the theory of functions of one complex variable.
• Additional background material such as some rudiments of algebraic geometry and algebraic groups will be introduced and discussed in the lectures/tutorials.

Exam

• Problem Set Policy.
  • There will be one problem set each week.
  • You can submit in groups of two.
  • You will need 50 % of the total points to be admitted to the oral exam.
• Exam.
  • Oral exam.