## PURE MTH 7038 - Pure Mathematics Topic A

### North Terrace Campus - Semester 1 - 2015

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 7038 Pure Mathematics Topic A Pure Mathematics Semester 1 Postgraduate Coursework North Terrace Campus 3 Y
##### Course Staff

Course Coordinator: Professor Michael Murray

##### Course Timetable

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

• Learning Outcomes
##### Course Learning Outcomes
In 2014, the topic of this course is DIFFERENTIAL GEOMETRY.

Syllabus

This course is concerned with the generalisation of multivariable calculus to settings more general than Euclidean spaces. It provides the foundation for advanced studies in analytical mathematics and physics, amongst other fields. The topics covered include a review of multivariable calculus in Euclidean spaces; manifolds; differentiation of functions between manifolds; differential forms; the general form of Stokes' theorem; cohomology of manifolds; vector bundles; connections, curvature and characteristic classes. The course concludes with a proof of the celebrated Gauss-Bonnet theorem, bringing together most of the new ideas presented during the semester.

Assumed knowledge for the course is some form of multivariable calculus and a working knowledge of linear algebra.

Learning Outcomes

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

1. define and recognise a differentiable manifold, and see how the properties of a differentiable function between two manifolds is reflected in the properties of its derivative;
2. differentiate, integrate and pull back differential forms on manifolds;
3. state and apply the general form of Stokes' theorem;
4. define and use de Rham and Cech cohomology groups of a manifold, and calculate these in simple cases;
5. recognise real and complex vector bundles on manifolds, and construct connections on these;
6. calculate the curvature of a connection, and explain the relationship between curvature and characteristic classes of the underlying vector bundle;
7. fully comprehend the statement and proof of the Gauss-Bonnet theorem for abstract surfaces.

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. all
The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. all
An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. 1,3,6
A proficiency in the appropriate use of contemporary technologies. 7
A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. 1,7
• Learning Resources
##### Required Resources
There are no required resources for this course.
##### Recommended Resources
There are many excellent books on differential geometry in the Barr Smith library. The following is a short selection of some that are very compatible with the objectives and the level of this course:

1.  Bott & Tu:  Differential forms in algebraic topology.

2.  Chern: Differentiable manifolds.

3.  Choquet-Bruhat, DeWitt-Morette, Dillard-Bleick:  Analysis, manifolds, and physics (rev. ed.).

4.  Guillemin & Pollack:  Differential topology.

5.  Lee:   Introduction to smooth manifolds.
• 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.  An optional mini-project, which may contribute to the final grade, is available to students who wish to extend their learning of a particular topic.

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

The following table is a guide to the workload for each component of the course. As a very rough estimate, the optional miniproject can be expected to add around 15 hours.

 Activiity Quantity Workload hours Lectures 30 90 Assignments 6 54 Test 1 10 Total 154
##### Learning Activities Summary
1.  Review of multivariable calculus; inverse & implicit function theorems;
integration (lectures 1-4);
2.  Differential forms in Rn (lectures 5-7);
3.  Differentiable manifolds (lectures 8-11);
4.  Differential forms on manifolds (lectures 12-16);
5.  Stokes' theorem (lectures 17-18);
6.  de Rham and Cech cohomology (lectures 19-22);
7.  Vector bundles (lectures 23-25);
8.  Connections (lectures 26-28);
9.  The Gauss-Bonnet theorem (lectures 29-30).

If there is sufficient interest, several more lectures may be given on gauge theory and its application to the topology of smooth 4-manifolds. These lectures would be optional, and the material in them would not be assessed in any way.
• 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 Assignments Summative and formative Evenweeks 10-20% All Test Summative Midsemester 10-20% 1,2 Examination Summative Examinationperiod 50-70% All Mini-project Sumative andformative End of teaching 0-15% 1
##### 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. The best 5 of the 6 assignments will be used towards the final grade, totalling between 10 and 20% of it.

Weightings for each of the assessment items specified in the table presented in the Assessment Summary will be applied individually to maximise each student's final mark.
##### Submission
Homework assignments must either be given to the lecturer in person or left in the box outside the lecturer's office by the given due time. Failure to meet the deadline without reasonable and verifiable excuse  may result in a significant penalty for that assignment.  The last day on which a miniproject may be submitted is the last teaching day of the semester.

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

M10 (Coursework Mark Scheme)
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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