PURE MTH 4013 - Pure Mathematics Topic D - Honours

North Terrace Campus - Semester 2 - 2017

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 4013
    Course Pure Mathematics Topic D - Honours
    Coordinating Unit School of Mathematical Sciences
    Term Semester 2
    Level Undergraduate
    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 Mathai Varghese

    Course Timetable

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

  • Learning Outcomes
    Course Learning Outcomes
    In 2017, the topic of this course is Differential Topology.

    Syllabus:

    This course is part of the geometry/topology sequence. However, its methods also underlie part of the basic theory of partial differential equations, which appears, roughly speaking, as an infinite-dimensional extension of these ideas.

    Aims:

    1. To prove the Whitney embedding and immersion theorems of manifolds into Euclidean space;
    2. To define regular and singular values of smooth maps and outline Sard's theorem with applications;
    3. To define the degree of a smooth map, the Hopf invariant, linking numbers and applications;
    4. To define the (local) index of vector fields and prove the Poincare-Hopf Theorem;
    5. A brief introduction to Morse theory, Morse inequalities, surgery and cobordism;
    6. Framed cobordism, the Pontryagin-Thom construction and proof of the Hopf degree theorem;
    7. To define and study transversality, intersection theory and Lefschetz fixed-point theorem;
    8. Advanced topics from vector bundles, connections, characteristic classes, K-theory, h-cobordism theorem. 

    Assumed knowledge:
    Topic A (semester 1)
    Topic B (semester 1)
    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
    None.
    Recommended Resources
    1. Differential topology, V Guillemin; Alan Pollack. ISBN/ISSN: 0132126052 Englewood Cliffs, N.J : Prentice-Hall 1974, 222 pp.

    2. Differential topology, Morris W. Hirsch. ISBN/ISSN: 3540901485; 0387901485 Graduate Texts in Mathematics, 33. Springer-Verlag, New York, 1994. x+222 pp.

    3. Topology from the differentiable viewpoint. John W. Milnor, Based on notes by David W. Weaver. The University Press of Virginia, Charlottesville, Va. 1965 ix+65 pp.

    4. Differential forms in algebraic topology. R. Bott, L. Tu. Graduate Texts in Mathematics,82. Springer-Verlag, New York-Berlin, 1982. xiv+331 pp. ISBN: 0-387-90613-4

    5. Vector bundles and K-theory by Allan Hatcher
    Online Learning
    This 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
    1) Review of vector bundles and connections.
    2) Elements of the Chern-Weil theory of characteristic classes.
    3) Introducton to G-equivariant cohomology where G is a compact Lie group.
    4) Introduction to the Thom class and its Chern-Weil representative.
    5) Review the implicit and inverse function theorem.
    6) To prove existence of embeddings of compact smooth manifolds into Euclidean space
    7) To define the degree of a smooth map and give standard topological applications
    8) To define and study transversality results
    9) To study regular and singular values of smooth maps and Sard's theorem
    10) A brief introduction to Morse theory
    11) A brief introduction to surgery theory
    12) A brief introduction to K-theory
    Other topics will be included if time permits
  • 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
    Examination                      70% 
    Homework assignments      30%
    Assessment Related Requirements
    An aggregate score of 50% is required to pass the course.
    Assessment Detail
    There will be a total of 5 homework assignments, distributed during each even week of the semester and due at the end of the following 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

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