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General


Description

The course is an introduction to Singularity Theory, which is also known as Catastrophe Theory. It provides tools to study sharp changes, bifurcations and metamorphoses taking place in various systems depending on parameters under continuous changes of these parameters. You will become familiar with basic notions and theorems used in Singularities. A technical part of the course will be devoted to reduction of functions to local normal forms, which is a far-reaching generalisation of the classification of extrema of functions well-known from school.

Semester

Autumn 2015 (Monday, October 5 to Friday, December 11)

Timetable

  • Wed 13:05 - 13:55

Prerequisites

There are no prerequisites beyond a standard undergraduate curriculum: elements of group theory, linear algebra, real and complex analysis. Some knowledge of differentiable manifolds, Lie groups and Lie algebras would be helpful but is not compulsory.

Syllabus

Inverse and implicit function theorems; Morse lemma; manifolds; tangent bundles; vector fields; germs of functions and mappings; derivative of a mapping between manifolds; critical points and critical values of mappings; Sard's lemma. Equivalence of map-germs; stable map-germs of a plane into a plane; transversality; jet spaces; Thom's transversality theorem. Local algebra of a singularity; local multiplicity of a mapping; Preparation theorem. Finite determinacy, Tougeron’s theorem; versal deformations of functions. Beginning of the classification of function singularities; Newton diagram; quasihomogeneous and semi-quasihomogeneous functions; ruler rotation method; simple functions; Arnold’s spectral sequence; boundary function singularities.

Lecturer


Victor Goryunov
Email goryunov@liverpool.ac.uk
Phone (0151) 7944041
Interests Singularity theory, Vassiliev invariants


Students


Photo of Zhe Chen
Zhe Chen
(Durham)
Photo of Alessio Cipriani
Alessio Cipriani
(Liverpool)
Photo of Maxime Fairon
Maxime Fairon
(Leeds)
Photo of Doaa FILALI
Doaa FILALI
(Cardiff)
Photo of Calum Horrobin
Calum Horrobin
(Loughborough)
Photo of Stephen Nand Lal
Stephen Nand Lal
(Liverpool)
Photo of Konstantinos Palapanidis
Konstantinos Palapanidis
(Southampton)
Photo of Bogna Pawlowska
Bogna Pawlowska
(Exeter)
Photo of Pedro Peres
Pedro Peres
(Exeter)
Photo of Sam Povall
Sam Povall
(Liverpool)
Photo of Courtney Quinn
Courtney Quinn
(Exeter)
Photo of Marzia Romano
Marzia Romano
(Northumbria)
Photo of ANON STUDENT
ANON STUDENT
(*External)


Bibliography


Modern Birkh‰User Classics: Singularities of Differentiable Maps ...Arnold, Gusein-Zade and Varchenko
Singularity Theory. I. Reprint of the original English edition from the series Encyclopaedia of Mathematical SciencesArnold, Goryunov, Layshko
Curves and Singularities: A Geometrical Introduction to Singularity TheoryBruce and Giblin
Stable Mappings and Their Singularities [By] M. Golubitsky [And] V. GuilleminGolubitsky and Author)


Note:

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Assessment



The assessment for this course will be via a single take-home paper in January with 2 weeks to complete and submit online. The paper will have three questions, each worth 25 points. To pass the exam you will need 40 points out of the total 75.

Singularity Theory

Files:Exam paper
Released: Monday 4 January 2016 (628.6 days ago)
Deadline: Friday 15 January 2016 (616.6 days ago)


Recorded Lectures


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