401-3129-67L  Differential Galois Theory

SemesterAutumn Semester 2017
LecturersP. S. Jossen
Periodicitynon-recurring course
Language of instructionEnglish



Courses

NumberTitleHoursLecturers
401-3129-67 VDifferential Galois Theory2 hrs
Fri10:15-12:00ETZ E 7 »
P. S. Jossen

Catalogue data

AbstractAlgebraic 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.
ObjectiveWe introduce differential Galois theory and mathematical concepts surrounding it. We formulate and prove an important case of the Riemann-Hilbert correspondence.
ContentWe 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.
LiteratureM. 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)
Performance assessment as a semester course
ECTS credits4 credits
ExaminersP. S. Jossen
Typesession examination
Language of examinationEnglish
RepetitionThe performance assessment is offered every session. Repetition possible without re-enrolling for the course unit.
Mode of examinationoral 30 minutes
Additional information on mode of examinationPrüfungssprache: 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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Offered in

ProgrammeSectionType
Doctoral Department of MathematicsGraduate SchoolWInformation
Mathematics BachelorSelection: Algebra, Topology, Discrete Mathematics, LogicWInformation
Mathematics MasterSelection: Algebra, Topology, Discrete Mathematics, LogicWInformation