Singularity Theory (MAGIC084)
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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.
Autumn 2015 (Monday, October 5 to Friday, December 11)
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.
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.
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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.
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