MATH4281: Topics in Combinatorics IV

• Algebra II (MATH2581)

• None

• None

• To introduce key concepts in algebraic combinatorics and study these concepts through a body of examples.
• To introduce a topic in modern combinatorics.

• Catalan numbers, Dyck paths, triangulations, noncrossing set partitions.
• Symmetric group, statistics on permutations, length and inversions.
• Partitions, Young diagrams, Young tableaux, Schensteds correspondence.
• Partially ordered sets and lattices.
• One of the following topics:
• 1. Coxeter Combinatorics: introduction to the theory of Coxeter groups and root systems, and exploring their applications and connections to actively developing areas of current research.
• 2. The Dimer Model: introduction to the study of perfect matchings of a graph, with main focus on the combinatorial aspects of the model, as well as on some of the probabilistic insights.
• 3. Theory of Alternating Sign Matrices: introduction to generalisations of permutation matrices and their connections to partitions, plane partitions, symmetric functions, hypergeometric series, and to the six vertex model a simplified model of ice from statistical mechanics, including insights into very recent activity in the field.
• 4. Combinatorics of Symmetric Group Representations: introduction to the combinatorial approach of Vershik and Okounkov to the representation theory of symmetric groups, and related topics in combinatorics such as longest increasing subsequence, symmetric functions, and the Littlewood-Richardson rule.

Subject-specific Knowledge:

• By the end of the module students will:
• Be able to solve novel and/or complex problems in the given subject.
• Have a knowledge of a specialised and complex topic in algebraic combinatorics.
• Be able to reproduce theoretical mathematics related to this module at a level appropriate for Level 4, including key definitions and theorems.

Subject-specific Skills:

• Students will have developed advanced technical and scholastic skills in the areas of Algebraic Combinatorics.

Key Skills:

• Students will have highly specialised skills in the following areas: problem solving, abstract reasoning.

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Subject material assigned for independent study develops the ability to acquire knowledge and understanding without dependence on lectures.
• Problem classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
• Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Eight written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

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