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.
42-48 lectures and example classes
80% Examination, 20% Coursework
Group theory:
M. Aschbacher: Finite Group Theory (Cambridge Studies in Advanced Mathematics), Cambridge University Press, 2000,
B. Baumslag and B. Chandler: Schaum's Outline of Group Theory, McGraw Hill Professional, 1968,
Knot theory:
C. Livingston, Knot theory, Mathematical Association of America, 1993,
See the library reading list for this module (Canterbury)
On successful completion of this module, H–level students will have increased their knowledge, understanding, intuition and computational expertise in:
(a) detecting symmetries and common patterns;
(b) using group theory to calculate with symmetries;
(c) the distinction and classification of objects up to equivalences and symmetries;
(d) the use of "normal forms" and "invariants" to distinguish symmetry classes.
They will also have:
(e) an enhanced ability to correctly formulate classification problems and solve them efficiently;
(f) an appreciation of algorithms and computational methods in algebra and group theory;
(g) consolidated a variety of tools from abstract algebra to model and classify concrete objects and configurations.
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