## PURE MTH 4013 - Pure Mathematics Topic D - Honours

### North Terrace Campus - Semester 2 - 2018

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 Pure Mathematics Topic D - Honours School of Mathematical Sciences Semester 2 Undergraduate North Terrace Campus 3 Y 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 Michael Eastwood

##### Course Timetable

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

• Learning Outcomes
##### Course Learning Outcomes
In 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 with 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.

Learning Outcomes:

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

1. Define and give key examples of homogeneous smooth manifolds equipped with various geometric structures;
2. Classify the invariant differential operators on the sphere under its various symmetries;
3. Explain what are the differential geometries based on these various homogeneous models;
4. In particular, define what are projective, conformal, and CR geometries;
5. State and prove Liouville's Theorem for these geometries;
6. Define and construct various vector bundles for these geometries equipped with their natural connections;
7. Construct the basic invariants of projective and conformal differential geometry.

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
None.
##### Recommended Resources
There are many excellent books on differential geometry in the Barr Smith Library. Browse the shelves, especially around 514.764, or consult from the following short selection.

* T. Aubin, A Course in Differential Geometry, GSM 27, American Mathematical Society 2001.
* A. Cap and J. Slovak, Parabolic Geometries I, Surveys and Monographs 154, American Mathematical Society 2009.
* S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-time, Cambridge University Press 1973.
* S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, GSM 34, American Mathematical Society 2001.
* J.M. Lee, Manifolds and Differential Geometry, GSM 107, American Mathematical Society 2009.
* P.W. Michor, Topics in Differential Geometry, GSM 93, American Mathematical Society 2008.

##### 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.

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

Lectures          30              90
Assignments      6              66
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
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 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.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M11 (Honours Mark Scheme)
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

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 studentâ€™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.

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