PURE MTH 7023 - Pure Mathematics Topic D

North Terrace Campus - Semester 2 - 2020

Please contact the School of Mathematical Sciences for further details.

  • General Course Information
    Course Details
    Course Code PURE MTH 7023
    Course Pure Mathematics Topic D
    Coordinating Unit School of Mathematical Sciences
    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.
    Course Staff

    Course Coordinator: Associate Professor Nicholas Buchdahl

    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 Algebraic topology

    Description: The aim of Algebraic Topology is to use algebraic structures and techniques to classify topological spaces up to homeomorphism. Algebraic objects are associated to topological spaces in such a way that "natural" operations on the latter correspond to "natural" operations on the former - continuous maps correspond to group homomorphisms, homeomorphisms to isomorphisms, etc. In this way, it is often possible to distinguish between different topological spaces by demonstrating that certain associated algebraic objects are not isomorphic. It is rarely the case that the converse can be shown; i.e., that two topological spaces with the same associated algebraic objects are actually homeomorphic, but when this can be done, it is often regarded as a major triumph of the theory.

    Within the realms of algebraic topology, there are several basic concepts that underly the theory and serve as the building blocks and models for subsequent generalisation, the algebraic topology of today being a very broad and highly generalised area that has pervaded much of contemporary mathematics. Such concepts include homotopy, cohomology, and homological algebra and the course will be aimed at providing students with an introduction to these key ideas.

    Key Phrases: Fundamental group, covering spaces, cohomology, homological algebra

    Assumed Knowledge: It will be assumed that you have some familiarity with basic point-set topology (or at least metric spaces) and familiarity with basic notions of abstract algebra (groups, rings, fields etc.) However I will give a review of point-set topology in the first few lectures.

    Learning Outcomes:

    On successful completion of this course, students will be able to

    1) understand the basic notions of homotopy theory such as homotopy of maps, homotopy equivalences, contractible spaces, fibrations
    2) define the fundamental group of a (path connected) topological space and be able to compute fundamental groups of some simple examples using for example the Seifert-van Kampen Theorem,
    3) define the singular cohomology groups of a topological space and their relative versions,
    4) understand and work with basic concepts in homological algebra, including chain complexes and long exact sequences,
    5) compute the cohomology of some topological spaces,
    6) apply the topological invariants constructed in this course to the solution of various problems in topology, for instance, to prove that
    two spaces are not homeomorphic.
    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
    There is no textbook for the course, but the following books are good (legally) free references:

    1. Algebraic Topology by Allan Hatcher
    https://www.math.cornell.edu/~hatcher/AT/ATpage.html
    2. Topology and Groupoids by Ronnie Brown
    https://groupoids.org.uk/pdffiles/topgrpds-e.pdf
    3. A Concise Course in Algebraic Topology by Peter May
    http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf

    Some books in the library that may be useful include

    4. Algebraic Topology: A First Course by Marvin J. Greenberg and John R. Harper, Mathematics Lecture Note Series (515.14 G798a)
    5. A basic course in algebraic topology by William Massey, Graduate Texts in Mathematics 127 (515.14 M416b)

    and more
    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
    Component Weighting Learning Outcomes
    Assignments 30% all
    Exam 70% all
    Assessment Related Requirements
    An aggregate score of at least 50% is required to pass the course.
    Assessment Detail
    There will be a total of 6 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 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:

    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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