PURE MTH 7023 - Pure Mathematics Topic D
North Terrace Campus - Semester 2 - 2018
General Course Information
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, or view course information on the School of Mathematical Sciences web site at http://www.maths.adelaide.edu.au
Course Coordinator: Professor Michael Eastwood
The full timetable of all activities for this course can be accessed from Course Planner.
Course Learning OutcomesIn 2018, the topic of this course is Symmetry in Differential Geometry.
Description: Many familiar manifolds are "homogeneous," they look the same at each point even when some extra structure is taken into account. A good example is the sphere which its usual "round" metric. To make this precise, the notion of a "Lie group" is useful: its definition combines the concepts of a group and a smooth manifold. Lie groups themselves are homogeneous and are well captured by an infinitesimal and purely algebraic notion known as a "Lie algebra." So this course is about Lie algebras, Lie groups, and their actions on smooth manifolds. The round sphere is homogeneous under the action of its isometries, which is a Lie group that can be described in terms of matrices, as can its Lie algebra (and this will be true for all the Lie groups in this course).
But there are many more homogeneous structures, even on the sphere. For example, the round sphere is also homogeneous under conformal, i.e. angle-preserving, symmetries or projective, i.e. geodesic-preserving, symmetries. Each of these variations comes with its own type of differential geometry. This course will catalogue the various possibilities and explore the associated differential geometries. Of particular interest, especially in physics, are the differential operators that respect these symmetries. Using methods from the theory of Lie algebras (but always expressed in terms of matrices), some classifications of these operators will be obtained.
Key Phrases: Homogeneous space, Homogeneous bundle, Lie group, Lie algebra, Conformal differential geometry, Contact geometry, Parabolic geometry, Invariant differential operator.
Assumed Knowledge: This course naturally follows on from the Differential Geometry honours course in Semester 1. Basic linear algebra and group theory will be useful.
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
Recommended Resources1. Differential Manifolds (Dover Books on Mathematics) by Antoni A. Kosinski
2. Algebraic Topology: A First Course (Mathematics Lecture Note Series) by Marvin J. Greenberg and John R. Harper
3. Algebraic Topology by Allan Hatcher
4. Vector bundles and K-theory by Allan Hatcher
Learning & Teaching Activities
Learning & Teaching ModesThe course consists of 30 lectures and 6 assignments. The students are expected to participate actively in the lectures and complete the assignments on time (the assignments will be collected every two weeks). Upon students' need there will be extra tutorials to solve problems and to learn more details or topics beyond the lectures.
The information below is provided as a guide to assist students in engaging appropriately with the course requirements.
Activity Quantity Workload Hours Lectures 30 114 Assignments 6 42 Total 156
Learning Activities SummaryReview (2 lectures)
Lectures per topic:
1. 2 lectures
2. 2 lectures
3. 4 lectures
4. 4 lectures
5. 6 lectures
6. 4 lectures
7. 4 lectures
8. 2 lectures
Total = 30 lectures
The University's policy on Assessment for Coursework Programs is based on the following four principles:
- Assessment must encourage and reinforce learning.
- Assessment must enable robust and fair judgements about student performance.
- Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
- Assessment must maintain academic standards.
Component Weighting Learning Outcomes Assignments 30% all Exam 70% all
Assessment Related RequirementsAn aggregate score of at least 50% is required to pass the course.
Distributed Due Date Weighting Assignment 1 Week 1 Week 2 5% Assignment 2 Week 3 Week 4 5% Assignment 3 Week 5 Week 6 5% Assignment 4 Week 7 Week 8 5% Assignment 5 Week 9 Week 10 5% Assignment 6 Week 11 Week 12 5%
SubmissionAssignments will be collected at the beginning of a lecture, every two weeks. Late assignments will not be accepted.
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.
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.
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