PURE MTH 7002 - Pure Mathematics Topic B

North Terrace Campus - Semester 1 - 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 7002
    Course Pure Mathematics Topic B
    Coordinating Unit Pure Mathematics
    Term Semester 1
    Level Postgraduate Coursework
    Location/s North Terrace Campus
    Units 3
    Course Staff

    Course Coordinator: Dr Pedram Hekmati

    Course Timetable

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

  • Learning Outcomes
    Course Learning Outcomes
    Course Learning Outcomes

    1. Demonstrate understanding of the fundamental concepts of algebraic topology, including homotopy and homology, and their role in mathematics.

    2. Demonstrate familiarity with a range of examples of these structures.

    3. Compute the fundamental group and homology of topological spaces at a reasonable level of difficulty.

    4. Comprehend the statement and proof of the basic results in algebraic topology, such as the Brouwer fixed-point theorem.

    5. Apply algebraic topology techniques to solve diverse problems in engineering and other mathematical contexts.

    6. Demonstrate proficiency 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. 1,2,3,4,5,6
    An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. 3,5
    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
    A commitment to the highest standards of professional endeavour and the ability to take a leadership role in the community. 6
    An awareness of ethical, social and cultural issues within a global context and their importance in the exercise of professional skills and responsibilities. 5,6
  • Learning Resources
    Required Resources
    Recommended Resources
    There are many excellent books on algebraic topology. Below is a selection suitable for an introductory course:

    - A Hatcher. Algebraic Topology. Cambridge University Press, 2002. Free electronic version available at http://www.math.cornell.edu/~hatcher

    - J P May. A Concise Course in Algebraic Topology. University of Chicago Press, 1999. Free electronic version available at http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf

    - G E Bredon. Topology and Geometry. Springer GTM 139, 1993.

    - P Kirk and J F Davis. Lecture Notes in Algebraic Topology. Graduate Studies in Mathematics, 35, 2001.  Free electronic version available at http://indiana.edu/~lniat/m621notessecondedition.pdf

  • Learning & Teaching Activities
    Learning & Teaching Modes
    The lecturer will present the course material in 30 lectures. Students are expected to attend all lectures and actively engage with the material during the lectures. Interaction and discussion of any difficulties that arise during the lecture is encouraged. Fortnightly homework assignments help students strengthen their understanding of the theory and its applications, and timely feedback through these problems allows students to gauge their progress.

    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 60
    Total 150
    Learning Activities Summary
    Week 1    Review of point-set topology; basic notions of homotopy theory.
    Week 2    Fundamental groups; properties, simple connectivity, homotopy invariance.

    Week 3    Fundamental groups (cont.); example computations, the fundamental theorem of algebra.

    Week 4    Fundamental groups (cont.); amalgamated product of groups, the Seifert-van Kampen theorem

    Week 5    Covering spaces; lifting of paths and homotopies, deck transformations.

    Week 6    Covering spaces (cont.); regular and universal covering spaces. Higher homotopy groups.

    Week 7    Homology; simplices, singular chains, homology groups, chain complexes, homotopy invariance.

    Week 8    Homology (cont.); long and short exact sequences, relative homology, Mayer-Vietoris sequence.

    Week 9    Homology (cont.);  the Brouwer fixed-point theorem, example computations, the Jordan-Brouwer separation theorem.

    Week 10  Cohomology; singular cochains, homotopy invariance, relative cohomology.

    Week 11  Cohomology (cont.);  cup product, universal coefficient theorem, the Eilenberg-Steenrod axioms. 

    Week 12  Additional topics (e.g. orientations, Poincare duality) and review.
  • 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 task Task type Due Weighting Learning outcomes
    Homework assignments Formative and summative Weeks 2,4,6,8,10,12 30% All
    Examination Summative Examination period 70% All
    Assessment Related Requirements
    An aggregate score of 50% is required to pass the course.
    Assessment Detail
    There will be a total of six homework assignments, distributed during each odd week of the semester and due at the end of the following week. Each will cover material from the lectures and sometimes go beyond, so that students may have to undertake some additional research. Assignments will be returned within two weeks.
    Homework assignments must be submitted on time, either directly to the lecturer or in the box outside the lecturer's office. Students may be excused from an assignment for medical or compassionate reasons. Documentation is required and the lecturer must be notified as soon as possible. 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:

    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

    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.

The University of Adelaide is committed to regular reviews of the courses and programs it offers to students. The University of Adelaide therefore reserves the right to discontinue or vary programs and courses without notice. Please read the important information contained in the disclaimer.