PURE MTH 4038 - Pure Mathematics Topic A - Honours

North Terrace Campus - Semester 1 - 2022

This course is available for students taking an honours degree in Mathematical Sciences. The course will cover an advanced topic in pure mathematics. For details of the topic offered this year please refer to the Course Outline.

  • General Course Information
    Course Details
    Course Code PURE MTH 4038
    Course Pure Mathematics Topic A - 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 This course is available for students taking an honours degree in Mathematical Sciences. The course will cover an advanced topic in pure mathematics. For details of the topic offered this year please refer to the Course Outline.
    Course Staff

    Course Coordinator: Professor Finnur Larusson

    Course Timetable

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

  • Learning Outcomes
    Course Learning Outcomes
    In 2022, the topic of this course is DIFFERENTIAL GEOMETRY

    Introduction

    For many purposes in mathematics and physics, for example in the general theory of relativity, we need to extend differentiation and integration from Euclidean spaces to much more general spaces called manifolds. The basic questions to be answered in this course are: What are smooth manifolds? How (and what!) can we differentiate and integrate on them? What is this generalisation of calculus good for?

    The main goal of the course is to set the stage for and prove Stokes' theorem, a vast generalisation of the fundamental theorem of calculus. Stokes' theorem is an important tool for relating local and global properties of manifolds. This is a major theme in modern mathematics, known in technical terms as cohomology. We will develop some basic ideas of algebraic topology from the differentiable viewpoint, using Stokes' theorem and homological algebra, and derive some important applications. (This will complement, but not duplicate, what you may be learning in Pure Topic B.) In the final weeks of the semester, further topics concerning smooth manifolds will be explored, chosen in consultation with the students.

    Topics 

    Differentiation
    Smooth manifolds
    Tangent spaces
    Differential forms and integration on manifolds
    Stokes' theorem
    Cohomology
    Further topics

    Learning Outcomes


    On successful completion of this course, students will be able to:
    1. demonstrate an understanding of the basic theory of smooth manifolds;
    2. demonstrate advanced skills in constructing rigorous mathematical arguments;
    3. apply the theory in the course to solve a variety of problems at an appropriate level of difficulty;
    4. demonstrate advanced skills in writing mathematics;
    5. demonstrate a commitment to academic integrity.

    Assumed knowledge

    The more third-year pure mathematics, the better. At a minimum, Topology & Analysis III.
    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)

    Attribute 1: Deep discipline knowledge and intellectual breadth

    Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.

    1, 2, 3

    Attribute 2: Creative and critical thinking, and problem solving

    Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.

    2, 3

    Attribute 3: Teamwork and communication skills

    Graduates convey ideas and information effectively to a range of audiences for a variety of purposes and contribute in a positive and collaborative manner to achieving common goals.

    2, 4

    Attribute 4: Professionalism and leadership readiness

    Graduates engage in professional behaviour and have the potential to be entrepreneurial and take leadership roles in their chosen occupations or careers and communities.

    2, 4, 5

    Attribute 5: Intercultural and ethical competency

    Graduates are responsible and effective global citizens whose personal values and practices are consistent with their roles as responsible members of society.

    5

    Attribute 7: Digital capabilities

    Graduates are well prepared for living, learning and working in a digital society.

    3, 4

    Attribute 8: Self-awareness and emotional intelligence

    Graduates are self-aware and reflective; they are flexible and resilient and have the capacity to accept and give constructive feedback; they act with integrity and take responsibility for their actions.

    2, 3
  • Learning Resources
    Required Resources
    The course will mostly follow the lecturer's notes, which will be posted on MyUni.  An additional recommended resource is Lee's Introduction to Smooth Manifolds, available on the library website.
    Recommended Resources
    There is a wide range of good textbooks that students might like to explore.  Some of them are listed at the end of the notes.
    Online Learning
    Course information and resources will be posted on MyUni.
  • Learning & Teaching Activities
    Learning & Teaching Modes
    There will be a weekly two-hour workshop with a mix of lecturing, students working on problems, together and with guidance from the lecturer, and consulting. Four homework assignments through the semester will deepen students' understanding of the theory and its applications. A project in the second half of the semester, with a written report, is an opportunity for an individual research experience and development of communication skills.  Students may choose their own project topic in consultation with the lecturer. The examination invites an intensive period of self-study and consolidation of the entire course.
    Workload

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

    Activity Quantity Workload hours
    Workshops 12 24
    Assignments 4 40
    Project 1 20
    Self-study 72
    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 task Task type Due Weighting Learning outcomes
    Examination Summative Examination period 40% all
    Assignments Formative and summative Mondays of Weeks 4, 7, 10, 13 40% all
    Project report Formative and summative Friday of Week 13 20% all

    Assessment Detail

    No information currently available.

    Submission
    Assignments and project report will be produced using LaTeX and submitted as pdf-files via MyUni.
    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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