Algebraic theory of linear differential equations, Picard-Vessiot theory, Differential Galois groups, local theory of differential equations, the Frobenius method, Newton polygons, Connections and local systems, Riemann-Hilbert correspondence on ℙ¹.
Objective
We introduce differential Galois theory and mathematical concepts surrounding it. We formulate and prove an important case of the Riemann-Hilbert correspondence.
Content
We study linear differential equations from an algebraic perspective, introducing differential rings, fields and differential modules (so-called Picard-Vessiot theory), and very soon the Galois group of a differential equation. We relate then the algebraic theory with the analytic theory, which leads us to the classical Riemann-Hilbert correspondence. In particular we will prove that differential equations on the complex projective line ℙ¹ with regular singularities in a finite set S correspond to representations of the fundamental group of ℙ¹∖S. If time permits, we have a look at differential equations in positive characteristic.
Literature
M. van der Put and M. F. Singer, Galois theory of linear differential equations, Grundlehren der Math. Wiss. Vol 328, Springer 2003
Performance assessment
Performance assessment information (valid until the course unit is held again)
The performance assessment is offered every session. Repetition possible without re-enrolling for the course unit.
Mode of examination
oral 30 minutes
Additional information on mode of examination
Prüfungssprache: Deutsch oder Englisch / Language of examination: English or German. 30 Minuten Vorbereitungszeit und 30 Minuten Prüfung (ein Kandidat bereitet vor, während der andere geprüft wird) / 30 minutes preparation and 30 minutes exam (one candidate prepares during the 30 minutes oral exam of the previous candidate).
This information can be updated until the beginning of the semester; information on the examination timetable is binding.
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