Universität Regensburg   IMPRESSUM   DATENSCHUTZ
Fakultät für Mathematik Universität Regensburg

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Differential Galois Theory
Semester
SoSe 2020

Lecturer
Niko Naumann

Type of course (Veranstaltungsart)
Vorlesung

Contents
This course is an introduction to the Galois theory of (linear, homogeneous) differential
equations. Similar to a polynomial equation in one variable, studied in every basic course in
algebra, such an equation admits a Galois-group which captures essential information about the
equation. The groups appearing are (linear) algebraic groups and there will be a parallel course by
Ertl/Schäppi outlining their basic theory (which is not mandatory to follow this one). A first
major application of classical Galois theory is the result that a general equation of degree at
least 5 cannot be solved by radicals. Similarly in spirit, we will prove here that the indefinite
integral \int exp(-x^2) dx does not admit an elementary solution. Time permitting, we will give an
overview of further developments, e.g. to the foundations through Tannakian categories or about
(algebraic) D-modules.

Literature
1) Kolchin, E. R., Differential algebra and algebraic groups. Pure and Applied Mathematics, Vol.
54. Academic Press, New York-London, 1973.\\ 2) Magid, Andy R., Lectures on differential Galois
theory. University Lecture Series, 7. American Mathematical Society, Providence, RI, 1994.\\ 3)
Deligne, P., Catégories tannakiennes. The Grothendieck Festschrift, Vol. II,
111–195, Progr. Math., 87, Birkhäuser Boston, Boston, MA, 1990.\\ 4) Borel, A. et al.
Algebraic D-modules. Perspectives in Mathematics, 2. Academic Press, Inc., Boston, MA, 1987

Recommended previous knowledge
Linear algebra, algebra and commutative algebra.

Time/Date
Tuesday, 10 am.

Location
M101

Course homepage
https://elearning.uni-regensburg.de/course/view.php?id=40884
(Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern)

Registration
  • Preliminary registration for the organisation of exercise classes: at the end of the previous
    semester via EXA or LSF (see announcement by the department)
  • Registration for the exercise classes: in class
  • Registration for course work/examination/ECTS: FlexNow
Course work (Studienleistungen)
  • Successful participation in the exercise classes:
Examination (Prüfungsleistungen)
  • Written exam: Duration: 120 minutes, Date: TBD, re-exam: Date: TBD
Modules
BAlg(2), BV, MV, MArGeo, LA-GyAlg

ECTS
6