Module Handbook

  • Dynamischer Default-Fachbereich geändert auf MAT

Module MAT-41-14-M-7

Singularity Theory (M, 9.0 LP)

Module Identification

Module Number Module Name CP (Effort)
MAT-41-14-M-7 Singularity Theory 9.0 CP (270 h)

Basedata

CP, Effort 9.0 CP = 270 h
Position of the semester 1 Sem. irreg.
Level [7] Master (Advanced)
Language [EN] English
Module Manager
Lecturers
Area of study [MAT-AGCA] Algebra, Geometry and Computer Algebra
Reference course of study [MAT-88.105-SG] M.Sc. Mathematics
Livecycle-State [NORM] Active

Courses

Type/SWS Course Number Title Choice in
Module-Part
Presence-Time /
Self-Study
SL SL is
required for exa.
PL CP Sem.
4V MAT-41-14-K-7
Singularity Theory
P 56 h 214 h - - PL1 9.0 irreg.
  • About [MAT-41-14-K-7]: Title: "Singularity Theory"; Presence-Time: 56 h; Self-Study: 214 h

Examination achievement PL1

  • Form of examination: oral examination (20-30 Min.)
  • Examination Frequency: each semester
  • Examination number: 86400 ("Singularity Theory")

Evaluation of grades

The grade of the module examination is also the module grade.


Contents

  • power series, Theorems of Weierstrass,
  • analytic algebras,
  • elementary theory of coherent sheaves,
  • germ of a complex variety,
  • local compactness theorem for morphisms,
  • invariants of hyper surface singularities,
  • finite determinacy,
  • deformation theory of complete intersections,
  • classification of simple hypersurface singularities.

Competencies / intended learning achievements

Upon successful completion of this module, the students have gained basic knowledge in singularity theory (local analytical and algebraic geometry). Using the example of hypersurface singularities, they know and understand how the theory has been developed further by applying modern methods of algebraic geometry. In particular, the students know and understand the classification of simple or ADE singularities which appear in many different areas of mathematics and theoretical physics.They are able to name the main propositions of the lecture, classify and explain the illustrated relationships.They understand the proofs and are able to reproduce and explain them. In particular, they can critically assess, what conditions are necessary for the validity of the statements.

Literature

  • T. De Jong, G. Pfister: Local Analytic Geometry,
  • G.-M. Greuel, C. Lossen, E. Shustin: Introduction to Singularites and Deformations.

References to Module / Module Number [MAT-41-14-M-7]

Module-Pool Name
[MAT-41-MPOOL-7] Specialisation Algebraic Geometry and Computer Algebra (M.Sc.)
[MAT-RM-MPOOL-7] Pure Mathematics (Advanced Modules M.Sc.)