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Groups, Categories & Homological Algebra (MAST90068)
Graduate courseworkPoints: 12.5Not available in 2017
Overview
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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;
- Free groups, presentations, free products (with amalgamation);
- Basic category theory: categories, functors, natural transformations, adjoints. (Co)products, universal objects, (co)limits, especially pushouts and pullbacks;
- Homological algebra: (pro/in)jective objects, resolutions, chain complexes, homotopy, the snake lemma. Applications: Ext, Tor, group homology;
- Noncommutative algebra: semisimple rings, modules, Wedderburn theorem.
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: 3 November 2022
Eligibility and requirements
Prerequisites
The following, or equivalent.
Code | Name | Teaching period | Credit Points |
---|---|---|---|
MAST30005 | Algebra | Semester 1 (On Campus - Parkville) |
12.5 |
Corequisites
None
Non-allowed subjects
None
Inherent requirements (core participation requirements)
The University of Melbourne is committed to providing students with reasonable adjustments to assessment and participation under the Disability Standards for Education (2005), and the Assessment and Results Policy (MPF1326). Students are expected to meet the core participation requirements for their course. These can be viewed under Entry and Participation Requirements for the course outlines in the Handbook.
Further details on how to seek academic adjustments can be found on the Student Equity and Disability Support website: http://services.unimelb.edu.au/student-equity/home
Last updated: 3 November 2022
Assessment
Additional details
Up to 40 pages of assignments (30%: three assignments worth 10% each, due early, mid and late in semester), a 3-hour written examination (70%, in the examination period).
Last updated: 3 November 2022
Dates & times
Not available in 2017
Time commitment details
170 hours
Last updated: 3 November 2022
Further information
- Texts
Prescribed texts
None
Recommended texts and other resources
Representations of compact Lie groups,by Theodor Bröcker, Tammo tom Dieck, Springer Graduate Texts in Mathematics, 1985.
Representations and Cohomology, I and II, by David J. Benson, Cambridge University Press, 1998.
An introduction to homological algebra, by Charles Weibel, Cambridge University Press, 1995. - Related Handbook entries
This subject contributes to the following:
Type Name Course Master of Science (Mathematics and Statistics) Course Doctor of Philosophy - Engineering Course Master of Philosophy - Engineering Course Ph.D.- Engineering Informal specialisation Mathematics and Statistics - Available through the Community Access Program
About the Community Access Program (CAP)
This subject is available through the Community Access Program (also called Single Subject Studies) which allows you to enrol in single subjects offered by the University of Melbourne, without the commitment required to complete a whole degree.
Entry requirements including prerequisites may apply. Please refer to the CAP applications page for further information.
Last updated: 3 November 2022