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Groups, Categories & Homological Algebra (MAST90068)

Graduate courseworkPoints: 12.5On Campus (Parkville)

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Overview

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Semester 2
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As well as being beautiful in its own right, algebra is used in many areas of mathematics, computer science and physics. This subject provides a grounding in several fundamental areas of modern advanced algebra including Lie groups, combinatorial group theory, category theory and homological algebra.
The material complements that covered in the subject Commutative and Mutlilinear Algebra without assuming it as prerequisite.

Intended learning outcomes

On completion of this subject, students should have an understanding of:

  • The geometry of Lie groups, and important examples coming from linear groups;
  • Lie algebras, the exponential map, and the relation with Lie groups;
  • Basic category theory: categories, functors, natural transformations, adjoints. (Co)products, universal objects, (co)limits, especially pushouts and pullbacks;
  • Noncommutative algebra: semisimple rings, modules, Wedderburn theorem.
  • Homological algebra: (pro/in)jective objects, resolutions, chain complexes, homotopy, the snake lemma. Applications: Ext, Tor, group homology;
  • Free groups, presentations, free products (with amalgamation);

Be able to:

  • prove results about Lie groups and algebras;
  • give presentations of groups and algebras;
  • construct and compute derived functors.

Generic skills

In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:

  • problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
  • analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
  • collaborative skills: the ability to work in a team;
  • time-management skills: the ability to meet regular deadlines while balancing competing commitments

Last updated: 31 January 2024