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Arbeitsgruppe Algebra
Algebra

Singularity Theory / Singularitätentheorie (summer term 2020)

Content

This lecture is concerned with the so-called hypersurface singularities, these are special critical points of a differentiable or holomorphic function in several variables. In that sense the lecture is a natural continuation of the analysis courses. The central topic will be the classification of certain such singularities. These are essentially the so-called "elementary catastrophes" of R.Thom. To reach this classification, we will use several tools from areas like algebra (lokal rings), analysis (inverse function theorem) or differential geometry (tangent spaces and transversality). All of these notions are rather elementary and will be recalled if neccessary.

This lecture will be given in either German or English, depending on the audience.

There will be a weekly exercise sheet. Working actively on solving the problems from these sheets is an essential part of the lecture.

Prerequisits for this lecture is a good account of the content of the lectures " Analysis 1,2" and " Lineare Algebra und analytische Geometrie 1 und 2".

Literature

  • V.I.Arnold, S.M. Gusein-Zade, A.N. Varchenko: Singularities of Differentiable Maps, Volume 1: The Classification of Critical Points Caustics, Wave Fronts. Birkhäuser, 1982
  • D.P.L. Castrigiano, S.A. Hayes: Catastrophe theory, 2. Auflage. Westview Press, 2003
  • V.I.Arnold: Catastrophe theory. 2. Auflage, Springer-Verlag, 1986.
  • Th. Bröcker, L. Lander: Differentiable germs and catastrophes. Cambridge University Press 1975
  • Th. de Jong, G. Pfister: Local analytic geometry. Vieweg 2000
  • W. Ebeling: Funktionentheorie, Differentialtopologie und Singularitäten. Vieweg 2001

Lecture notes

Excercise sheets

Lecture

Due to the spread of the corona virus and the consequent lockdown of the university, this lecture will be done via video conference, using the Zoom plattform. The lecture is taking place every Wednesday, from 09:15 - 10:45. Details about how to connect are send out by email. If you want to join, please send a message to Christian Sevenheck.

Exercise classes (by Valeria Bertini)

Exercise classes will be be done via Zoom as well, every Friday, 15.30-17.00. The first class is scheduled for April, 17th. Please send your solution to the exercise sheets to Valeria Bertini.

Whiteboards from exercise classes