This module is not currently running in 2024 to 2025.
In this module we will study certain configurations with symmetries as they arise in real world applications. Examples include knots described by "admissible diagrams" or chemical structures described by "colouring patterns". Different diagrams and patterns can describe essentially the same structure, so the problem of classification up to equivalence arises. This will be solved by attaching "invariants" which are then put in "normal form" to distinguish them. The syllabus will be as follows: (a) Review of basic methods from linear algebra, group theory and discrete mathematics; (b) Permutation groups, transitivity, primitivity, Burnside formula; (c) Finitely generated Abelian groups; (d) Applications to knot theory, Reidemeister moves, the Abelian knot group; (e) Examples, observations, generalizations and proofs; (f) General Polya-enumeration (as an extension of the Burnside formula).
42-48 hours
80% Examination, 20% Coursework
G Burde & H Zieschang, Knots. (De Gruyter Studies in Mathematics, 1985, Walter de Gruyter, ISBN
3-11-008675-1)
LH Kauffman, On Knots. (Princeton, 1987, ISBN 0-691-08435-1)
A Kerber, Applied finite group actions. (Springer, 1999, ISBN/ISSN 3540659412)
WBR Lickorish, An introduction to knot theory. (Springer, 1997, ISBN/ISSN 038798254X)
V Manturov, Knot Theory. (Chapman & Hall, 2004, ISBN 1-415-31001-6)
K Murasugi, Knot theory and its applications. (Birkhäuser, 1996, ISBN/ISSN 0817638172)
See the library reading list for this module (Canterbury)
The Intended Subject Specific Learning Outcomes. On successful completion of this module students will have increased their knowledge, understanding, intuition and computational expertise in:
(a) rigorous thinking
(b) detecting symmetries and common patterns
(c) systematic observation, generalization and techniques of proof
(d) using group theory to calculate with symmetries
(e) distinction and classification of objects up to equivalences and symmetries
(f) the use of \normal forms" and \invariants" to distinguish symmetry classes
(g) combinatorial analysis and enumeration of symmetry classes and group orbits
(h) proficient use of mathematical software such as Maple and MAGMA to masters level
The Intended Generic Learning Outcomes. We expect students successfully completing the module to have
(i) an enhanced ability to correctly formulate classification problems and solve them efficiently;
(ii) enhanced skills in understanding and communicating mathematical results and conclusions;
(iii) a holistic view of mathematics as a problem solving and intellectually stimulating discipline;
(iv) an appreciation of algorithms and computational methods in algebra and group theory.
On completion of the module students will have:
_ matured in their problem formulating and solving skills;
_ consolidated a variety of tools from abstract algebra to model and classify concrete objects and configurations.
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