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MAST90068
Groups, Categories and Homological Algebra

Semester II 2012

Lecturer: Arun Ram, 174 Richard Berry, email: aram@unimelb.edu.au

Time and Location:
       Lecture: Monday 10:00 - 12:00 Richard Berry 215
       Lecture/Practical Friday 11:00-12:00 Richard Berry 215

Announcements

  • No books, notes, calculators, ipods, ipads, phones, etc at the exam.
  • Consultation hours are 12:00-3:00pm on Mondays.
  • Prof. Ram reads email but generally does not respond.
  • The start of semester pack includes: Housekeeping (pdf file), Plagiarism (pdf file), Plagiarism declaration (pdf file), Academic Misconduct (pdf file), SSLC responsibilities (pdf file).
  • It is University Policy that:           

    “a further component of assessment, oral, written or practical, may be administered by the examiners in any subject at short notice and before the publication of results.  Students must therefore ensure that they are able to be in Melbourne at short notice, at any time before the publication of results” (Source: Student Diary).

    Students who make arrangements that make them unavailable for examination or further assessment, as outlined above, are therefore not entitled to an alternative opportunity to present for the assessment concerned (i.e. a ‘make-up’ examination).
  • Printing arrangements from computer Lab G70: Students must use their Unicards to print documents. Locations for Unicard uploaders can be found at: http://www.studentit.unimelb.edu.au/images/facilities/autoloaders.gif

Subject Outline

The handbook entry for this course is at https://handbook.unimelb.edu.au/view/2011/MAST90068. The objective of the course listed there are:

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

Main Topics

  • (1) Lie groups and Lie algebras
  • (2) Vector fields and the exponential map
  • (3) Examples of Lie groups and Lie algebras
  • (4) Categories, Functors and Natural transformations
  • (5) Examples of Categories, Universal objects
  • (6) Products, Pushouts, Limits and colimits
  • (7) Adjoint functors and examples
  • (8) Homology
  • (9) Tor and Ext
  • (10) Homology and Groups
  • (11) Spectral sequences

Assessment

Assessment will be based on up to 60 pages of assignments (60%: three assignments worth 20% each, due early, mid and late in semester), a two-hour written examination (40%, in the examination period).

The plagiarism declaration is available here. The homework assignments are as follows:

  • Assignment 1: Due 20 August 2012: From Chapter I of T. Bröcker and T. tom Dieck, Representations of Compact Lie groups, Springer, 1985, ISBN 3-540-13678-9, do
    • Section 1 ex 4, 8, 12, 14;
    • Section 2 ex 1,5,6,8;
    • Section 3 ex 1, 4, 6;
    • Section 4 ex 6 (also do the fundamental group part), 7, 11;
  • Assignment 2: Due 1 October 2012: From S. Maclane, Categories for the Working Mathematician, Springer 1971, ISBN 0-387-98403-8, do
    • Ch I, Sect 3 ex 1, 5, Sect 4 ex 3, Sect 5 ex 4,5,6
    • Ch II Sect 3 ex 3,5, Sect 4 ex 1,3,5, Sect 5 ex 4, 8
    • Ch III Sect 1 ex 1,3,7, Sect 2 ex 1, Sect 3 ex 2, 6, Sect 4 ex 1,2
    • Ch IV Sect 1 ex 3, Sect 2, ex 1,2,3,6
  • Assignment 3: Due 29 October 2012: From J. Rotman, An Introduction to Homological Algebra, Second edition, Springer, 2009, ISBN 978-0-387-24527-0, do
    • Ex 6.5, 6.12, 6.17, 6.18, 6.21
    • Ex 7.4, 7.6, 7.7, 7.12, 7.17, 7.24
    • Ex. 9.12, 9.14, 9.15, 9.22, 9.31, 9.33, 9.34
    • Ex. 10.2, 10.4, 10.5, 10.6 , 10.7 , 10.8
  • The final exam will be 2 hours.

Resources part I: recommended Texts

  • see http://researchers.ms.unimelb.edu.au/~aram@unimelb/Teaching/RepThy2010/RepThy2010.html
  • see http://researchers.ms.unimelb.edu.au/~aram@unimelb/notes.html
  • Wikipedia
  • T. Bröcker and T. tom Dieck, Representations of Compact Lie groups, Springer, 1985. ISBN 3-540-13678-9
  • S. Maclane, Categories for the Working Mathematician, Springer 1971. ISBN 0-387-98403-8
  • J. Rotman, An Introduction to Homological Algebra, Second edition, Springer, 2009. ISBN 978-0-387-24527-0

Resources part II: Lectures and lecture notes

  • Lecture 1/2, 23 July 2012: Brocker-Tom Dieck, Chapter 1 Section 1 (lecture covered by Lawrence Reeves) -- for lecture notes see Lie Groups and Lie algebras
    • Math Grammar: Definitions, Theorems and How to do Proofs (pdf file)
    • Examples of proofs written in proof machine (pdf file)
  • Lecture 3, 27 July 2012: More Lie group and Lie algebra examples (lecture covered by Lawrence Reeves)

  • Lecture 4/5, 30 July 2012: Brocker-Tom Dieck, Chapter 1 Section 2
  • Lecture 6, 3 August 2012: Adjoint representation specifics -- handwritten lecture notes (pdf file)

  • Lecture 7/8, 6 August 2012: Brocker-Tom Dieck, Chapter 1 Section 3
  • Lecture 9, 10 August 2012: Fibre bundles and cocycles

  • Lecture 10/11, 13 August 2012: Brocker-Tom Dieck, Chapter 1 Section 4
  • Lecture 12, 17 August 2012: injectivity and surjectivity in a category setting

  • Lecture 13/14, 20 August 2012: Maclane, Chapter 1
  • Lecture 15, 24 August 2012: Prac class

  • Lecture 16/17, 27 August 2012: Maclane, Chapter 2
  • Lecture 18, 31 August 2012: Prac class

  • Lecture 19/20, 3 September 2012: Maclane, Chapter 3
  • Lecture 21, 7 September 2012: Prac class

  • Lecture 22/23, 10 September 2012: Maclane, Chapter 4
  • Lecture 24, 14 September 2012: Prac class

  • 17-29 September: MID SEMESTER BREAK

  • Lecture 25/26, 1 October 2012: Rotman, Chapter 6
  • Lecture 27, 5 October 2012: Prac class

  • Lecture 28/29, 8 October 2012: Rotman, Chapter 7
  • Lecture 30, 12 October 2012: Prac class

  • Lecture 31/32, 15 October 2012: Rotman, Chapter 9
  • Lecture 33, 19 October 2012: Prac class

  • Lecture 34/35, 22 October 2012: Rotman, Chapter 10
  • Lecture 36, 26 October 2012: Prac class