## PURE MTH 4038 - Pure Mathematics Topic A - Honours

### North Terrace Campus - Semester 1 - 2016

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 4038 Pure Mathematics Topic A - Honours School of Mathematical Sciences Semester 1 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 Murray

##### Course Timetable

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

• Learning Outcomes
##### Course Learning Outcomes
In 2016, 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.

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.

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
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.
##### 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.

The following table is a guide to the workload for each component of the course.

 Activity Quantity Workload hours Lectures 30 90 Assignments 5 55 Test 1 11 Total 156
##### 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 26-30);

• 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 Weeks: 3, 5, 7, 9, 11 15% All Test Summative Midsemester 15% 1,2 Examination Summative Examinationperiod 70% All
##### Assessment Related Requirements
An aggregate score of 50% is required to pass the course.
##### Assessment Detail
There will be a total of 5 homework assignments, distributed during each even 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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