PURE MTH 7066 - Pure Mathematics Topic E

North Terrace Campus - Semester 2 - 2015

Please contact the School of Mathematical Sciences for further details, or view course information on the School of Mathematical Sciences web site at http://www.maths.adelaide.edu.au

  • General Course Information
    Course Details
    Course Code PURE MTH 7066
    Course Pure Mathematics Topic E
    Coordinating Unit Pure Mathematics
    Term Semester 2
    Level Postgraduate Coursework
    Location/s North Terrace Campus
    Units 3
    Available for Study Abroad and Exchange Y
    Course Description Please contact the School of Mathematical Sciences for further details, or view course information on the School of Mathematical Sciences web site at http://www.maths.adelaide.edu.au
    Course Staff

    Course Coordinator: Professor Michael Eastwood

    Course Timetable

    The full timetable of all activities for this course can be accessed from Course Planner.

  • Learning Outcomes
    Course Learning Outcomes
    In 2014, the topic of this course will be Lie Groups and Lie Algebras.

    Syllabus


    The theory of Lie groups and Lie algebras lies at the intersection of several mathematical disciplines: algebra, analysis. geometry and topology.  It is therefore a very rich and beautiful subject, with many applications.  For instance the notion of Lie group is of fundamental importance in differential geometry and mathematical physics.  This course is an introduction to the basic theory underlying the subject; the aim is to give students an appreciation of the subject, and to prepare them for further study in this area.  

    Topics to be covered include: 

    Basic notions of differential geometry; Lie groups - definitions and examples; Lie algebras and the exponential map; homogenous spaces; Lie's Theorem; linear algebra and Lie theory; maximal tori; 
    further topics selected according to interests of students.  

    Prerequisites: as a preparation for the course you should have taken the third year courses PURE MTH 3007 Groups and Rings, PURE MTH 3002 Topology and Analysis and PURE MTH 3022 Geometry of Surfaces. This course also follows on naturally from the honours course PURE MTH 7002 Differential Geometry.

    Learning Outcomes


    1. Demonstrate understanding of the basic notions underlying the theory of Lie groups and Lie algebras.  

    2. Demonstrate familarity with a range of examples of these notions.  

    3. Prove basic results about Lie groups and Lie algebras.  

    4. Demonstrate an understanding of the existence of maximal tori in compact, connected Lie groups, and the role of these in representation theory and classification problems.  

    5. Apply the theory of the course to solve a variety of problems at an appropriate level of difficulty.  

    6. Demonstrate skill in communicating mathematics orally and in writing.   

    University Graduate Attributes

    This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

    University Graduate Attribute Course Learning Outcome(s)
    Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised. 1,2,3,4,5
    The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. 2,4,5
    An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. 5
    Skills of a high order in interpersonal understanding, teamwork and communication. 6
    A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. 1,2,3,4,5,6
    An awareness of ethical, social and cultural issues within a global context and their importance in the exercise of professional skills and responsibilities. 6
  • Learning Resources
    Required Resources
    None.
    Recommended Resources

    The following texts are relevant for the course.  

    J. F. Adams, Lectures on Lie groups, W. A. Benjamin, Inc., New York, 1969 

    R. Carter, G. Segal and I. MacDonald, Lectures on Lie Groups and Lie Algebras, London Mathematical Society Student Texts, 32, Cambridge University Press, Cambridge 1995. 

    T. Bröcker and T. tom Dieck, Representations of compact groups, Graduate Texts in Mathematics, Vol. 98, Springer-Verlag, New York, 1985.
  • Learning & Teaching Activities
    Learning & Teaching Modes
    This course relies on lectures at the primary delivery mechanism for the material. A sequence of written assignments reinforce the students understanding of the lecture material and provides assessment opportunities for students to gauge their progress.
    Workload

    The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

    Activity                                       Quantity                                  Workload Hours

    Lectures                                           30                                                  90 
    Assignments                                     6                                                    66
    Total                                                                                                    156



    Learning Activities Summary

    Week 1: Basic concepts of differential geometry

    Week 2: Introduction to Lie groups; examples 

    Week 3: Lie algebras and the exponential map

    Week 4: The exponential map, continued; homogenous spaces

    Week 5: Lie's Theorem 

    Week 6: Linear algebra and Lie theory

    Week 7: Maximal tori

    Week 8: Maximal tori, continued 

    Weeks 9-12: Further topics selected from (according to student interest): the Peter-Weyl theorem; elementary representation theory; the geometry of the Steifel diagram; unitary groups and symmetric groups
  • Assessment

    The University's policy on Assessment for Coursework Programs is based on the following four principles:

    1. Assessment must encourage and reinforce learning.
    2. Assessment must enable robust and fair judgements about student performance.
    3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
    4. Assessment must maintain academic standards.

    Assessment Summary
    Component Weighting  Outcomes Assessed
    Assignments 30% All
    Exam 70% All
    Assessment Related Requirements
    An aggregate score of at least 50% is required to pass the course.
    Assessment Detail
                                               Distributed               Due Date                Weighting

    Assignment 1                       Week 2                      Week 3                         5% 

    Assignment 2                       Week 4                      Week 5                         5% 

    Assignment 3                       Week 6                      Week 7                         5%

    Assignment 4                       Week 8                      Week 9                         5% 

    Assignment 5                      Week 10                    Week 11                        5% 

    Assignment 6                      Week 11                    Week 12                        5% 

    Submission
    Assignments will have a maximum two week turn-around time for students.  

    Late assignments will not be accepted.
    Course Grading

    Grades for your performance in this course will be awarded in accordance with the following scheme:

    M10 (Coursework Mark Scheme)
    Grade Mark Description
    FNS   Fail No Submission
    F 1-49 Fail
    P 50-64 Pass
    C 65-74 Credit
    D 75-84 Distinction
    HD 85-100 High Distinction
    CN   Continuing
    NFE   No Formal Examination
    RP   Result Pending

    Further details of the grades/results can be obtained from Examinations.

    Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

    Final results for this course will be made available through Access Adelaide.

  • Student Feedback

    The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

    SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

  • Student Support
  • Policies & Guidelines
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