PURE MTH 4012 - Pure Mathematics Topic B - Honours

North Terrace Campus - Semester 1 - 2019

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 4012
    Course Pure Mathematics Topic B - Honours
    Coordinating Unit School of Mathematical Sciences
    Term Semester 1
    Level Undergraduate
    Location/s North Terrace Campus
    Units 3
    Available for Study Abroad and Exchange Y
    Restrictions Honours students only
    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: Dr Guo Chuan Thiang

    Course Timetable

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

  • Learning Outcomes
    Course Learning Outcomes
    In 2019, the topic of this course is Functional Analysis

    Introduction

    Motivated by the development of calculus of variations, integral equations, approximation theory and quantum physics in the early twentieth century, functional analysis has grown into a broad field of modern analysis. Broadly speaking, it is the study of infinite dimensional linear topological spaces as well as properties of linear maps among these spaces. Of particular importance are linear operators on Hilbert spaces, as they play a fundamental role in quantum mechanics, partial differential equations, signal processing, ergodic theory, dynamics, and many other branches of mathematics, physics and engineering. Besides these applications, the subject of functional analysis also has important connections to geometry, topology and number theory.


    The course will cover the following topics.

    1. Fundamentals of Hilbert spaces (4 lectures).
    2. Bounded linear operators on Hilbert spaces and some important theorems in functional analysis (7 lectures).
    3. Projections, unitaries, self-adjoint and normal operators  (5 lectures).
    4. Compact operators and its spectral decompositions (4 lectures).
    5. Spectral theory of self-adjoint operators (6 lectures).
    6. Advanced topics which may be added to the core topics (sample: quantum mechanics, unbounded operators, operator algebras, representation theory, Fourier analysis, Fredholm index and Toeplitz operators) (4 lectures).

    Learning Outcomes

    On successful completion of this course, students should
    1. Understand properties of Hilbert spaces and their bounded linear operators; know how to apply these properties;
    2. Be able to identify and work on key examples involving Hilbert space analysis;
    3. Understand the concept of the spectrum of an operator, and compute the spectrum of specific examples;
    4. Be able to state and prove the spectral theorem for compact and for self-adjoint operators;
    5. Be aware of concrete applications of functional analysis in specialised topics and connections to other areas of mathematics.

    Prerequisites

    The course requires knowledge of point-set topology, basic measure theory, and a firm background in linear algebra. Good knowledge of real analysis and some exposure to complex analysis would be desirable. It is recommended that students have taken Topology and Analysis III previously and are familiar with basic group theory.
    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)
    Deep discipline knowledge
    • informed and infused by cutting edge research, scaffolded throughout their program of studies
    • acquired from personal interaction with research active educators, from year 1
    • accredited or validated against national or international standards (for relevant programs)
    all
    Critical thinking and problem solving
    • steeped in research methods and rigor
    • based on empirical evidence and the scientific approach to knowledge development
    • demonstrated through appropriate and relevant assessment
    all
  • Learning Resources
    Required Resources
    Course lecture notes will be provided, partly adapted from previous versions of this course (credit: Dr. Hang Wang)
    Recommended Resources
    J. Conway, A course in functional analysis (Good introduction at advanced undergraduate/beginning graduate level)
    M. Reed and B. Simon, Methods of modern mathematical physics. I, Functional analysis (Good foundation for mathematical quantum theory)
    W. Rudin, Functional analysis (Quite abstract, deals with general topological vector spaces and Banach spaces)

    Other references:
    K. Yosida, Functional analysis
    G. Folland: Real analysis: Modern techniques and their applications

    Online Learning
    The course will have an active MyUni website.
  • Learning & Teaching Activities
    Learning & Teaching Modes
    The lecturer guides the students through the course material in 30 lectures. Students are expected to engage with the material in the lectures. Interaction with the lecturer and discussion of any difficulties that arise during the lecture is encouraged. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them 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

    No information currently available.

  • 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
    Assessment taskTask typeDueWeightingLearning outcomes
    Examination Summative Examination period 70% All
    Homework assignments Formative and summative One week after assigned 30% All
    Assessment Related Requirements
    An aggregate score of 50% is required to pass the course.
    Assessment Detail
    There will be a total of 6 homework assignments, distributed every four lectures and due within one week. Each will cover material from the lectures and, in addition, will sometimes go beyond that so that students may have to undertake some additional research.
    Submission
    Homework assignments must be given to the lecturer in person or emailed as a pdf. Failure to meet the deadline without reasonable and verifiable excuse may result in a significant penalty for that assignment.
    Course Grading

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

    M11 (Honours Mark Scheme)
    GradeGrade reflects following criteria for allocation of gradeReported on Official Transcript
    Fail A mark between 1-49 F
    Third Class A mark between 50-59 3
    Second Class Div B A mark between 60-69 2B
    Second Class Div A A mark between 70-79 2A
    First Class A mark between 80-100 1
    Result Pending An interim result RP
    Continuing Continuing CN

    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

    Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's student’s disciplinary procedures.

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