PURE MTH 7023 - Pure Mathematics Topic D

North Terrace Campus - Semester 2 - 2014

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/students/honours

  • General Course Information
    Course Details
    Course Code PURE MTH 7023
    Course Pure Mathematics Topic D
    Coordinating Unit Pure Mathematics
    Term Semester 2
    Level Postgraduate Coursework
    Location/s North Terrace Campus
    Units 3
    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/students/honours
    Course Staff

    Course Coordinator: Dr Hang Wang

    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 Functional Analysis.

    Syllabus

    Motivated by the development of the calculus of variations, integral equations, approximation theory and quantum physics in the early years of the 20th century, functional analysis has grown into a broad field of modern analysis. It studies infinite-dimensional linear topological spaces, in particular Hilbert spaces, and properties of maps between such spaces, using analysis, algebra and geometry. The course focuses on important examples and properties of Hilbert spaces and operators on Hilbert spaces, leading to the spectral theorem for Hermitian operators. Time permitting, we will explore some applications in group representation theory, Fourier analysis and other advanced topics. The course provides a foundation for further studies in both mathematics and physics.

    The topics to be covered includes a review of Hilbert spaces; bounded linear operators on Hilbert spaces and fundamental theorems in functional analysis; compact operators and other examples of operators; Gelfand transform and spectral decomposition; application to group representations, application to Fourier analysis.

    The prerequisites for the course are Topology and Analysis III, and Integration and Analysis III. A good background in linear algebra and real and complex analysis is desirable, as well as some familiarity with groups and manifolds.

    Learning Outcomes

    Students completed the course will be able to:

    1. fully comprehend the statement of fundamental theorems of functional analysis and the concepts relating to them;

    2. demonstrate knowledge of the tools of the theory of Hilbert spaces;

    3. be able to solve functional analysis problems using modern analysis;

    4. recognise a problem in other areas of mathematics that can be solved using functional analysis;

    5. demonstrate an understanding of the spectral theorem in functional analysis and its role in other relavent subjects in pure mathematics and physics.

    6. demonstrate basic understanding and skill in group representation theory and fourier analysis.

    7. demonstate 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. all
    The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. all
    An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. all
    Skills of a high order in interpersonal understanding, teamwork and communication. 7
    A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. 4, 5, 7
  • Learning Resources
    Required Resources
    None.
    Recommended Resources
    1. J.B. Conway: A Course in Functional Analysis.
    2. W. Rudin: Functional Analysis.
    3. V.S. Sunder: Functional Analysis: Spectral Theory.
    4. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume 1, Functional Analysis.
    5. G. Folland: A Course in Abstract Harmonic Analysis.
  • Learning & Teaching Activities
    Learning & Teaching Modes
    The course consists of 30 lectures and 6 assignments. The students are expected to participate actively in the lectures and complete the assignments on time (the assignments will be collected every two weeks). Upon students' need there will be extra tutorials to solve problems and to learn more details or topics beyond the lectures.
    Workload

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

    Activity Quantity Workload Hours
    Lectures     30 114
    Assignments     6 42
    Total 156
    Learning Activities Summary
    1. Review of Hilbert space (2 lectures).
    2. Bounded linear operators on Hilbert space (4 lectures).
    3. Fundamental theorems of functional analysis (5 lectures).
    4. Compact operators and other types of operators (4 lectures).
    5. Spectral theorem (6 lectures).
    6. Applications to group representation theory (4 lectures).
    7. Applications to Fourier analysis (5 lectures).
  • 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 Learning Outcomes
    Assignments 30% all
    Test 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 1 Week 2 5%
    Assignment 2 Week 3 Week 4 5%
    Assignment 3 Week 5 Week 6 5%
    Assignment 4 Week 7 Week 8 5%
    Assignment 5 Week 9 Week 10 5%
    Assignment 6 Week 11 Week 12 5%
    Submission
    Assignments will be collected at the beginning of a lecture, every two weeks. 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
  • Fraud Awareness

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