Universität Wien
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250025 VO Differential Algebra and Fuchsian Differential Equations (2024W)

5.00 ECTS (3.00 SWS), SPL 25 - Mathematik

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Details

Language: English

Lecturers

Classes (iCal) - next class is marked with N

  • Tuesday 01.10. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 02.10. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 08.10. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 09.10. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 15.10. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 16.10. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 22.10. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 23.10. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 29.10. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 30.10. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 05.11. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 06.11. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 12.11. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 13.11. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 19.11. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 20.11. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 26.11. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 27.11. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 03.12. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 04.12. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 10.12. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 11.12. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 17.12. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 07.01. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 08.01. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 14.01. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 15.01. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 21.01. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 22.01. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 28.01. 11:30 - 13:00 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 29.01. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock

Information

Aims, contents and method of the course

Consider the set R of polynomials or (germs of) holomorphic functions or formal power series in n complex variables x_1,..., x_n. With pointwise addition and multiplication, R becomes a commutative ring which is noetherian (every ideal is finitely generated) and an integral domain (no zero-divisors) and unique factorization domain. It is the central object of study in the field of Commutative Algebra.

There is an additional algebraic structure on R, given by taking partial derivatives of functions. No limit procedure is needed to compute them, due to the known formulas of how to differentiate a polynomial or power series. Each of them defines a derivation on R, i.e., a linear map from R to R satisfying the Leibniz or product rule. This is what one calls a "differential ring", i.e., a ring together with a derivation. The study of such rings is called "Differential Algebra", an extension of classical Commutative Algebra. In this setting, one can now study differential equations from a purely algebraic perspective. Already in one single variable this is a fascinating subject, starting with Lazarus Fuchs' celebrated theory of regular singularities and the associated differential equations.

In the course, we will develop gently the basic concepts, illustrate them with many examples, and pass on on both a very concrete and a very abstract level to describe the solutions of the differential equations. A realm of classical ordinary differential equations (Gauss' hypergeometric equation, Euler equations, Legendre equation, Heun equation, Apéry equation, ...) fall in this class and produce a beautiful theory where analytically flavored results are obtained by purely algebraic means. By the interpretation of ordinary linear differential equations as linear recurrences for sequences of numbers there also appears a strong link to combinatorics and generating functions, as well as to transcendence questions in number theory.

The course is intended for Master's and PhD students interested and specialized in algebra, combinatorics, number theory, or analysis.

Assessment and permitted materials

Minimum requirements and assessment criteria

Examination topics

Reading list

Association in the course directory

MALV; MANV

Last modified: We 31.07.2024 11:06