## PURE MTH 7066 - Pure Mathematics Topic E

### North Terrace Campus - Semester 2 - 2019

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 7066 Pure Mathematics Topic E Mathematical Sciences Semester 2 Postgraduate Coursework North Terrace Campus 3 Y
##### Course Staff

Course Coordinator: Associate Professor Nicholas Buchdahl

##### Course Timetable

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

• Learning Outcomes
##### Course Learning Outcomes
In 2019, the title of this course is Global Differential Geometry.

Overview
In that undergraduate calculus usually comprises the notions of differentiation with applications, integration with applications, and the relationship between differentiation and integration described by the Fundamental Theorem of Calculus, the first semester course "Manifolds, Lie groups, and Riemannian geometry" can be viewed as an advanced analogue of differentiation with applications, whereas this second semester course can be viewed as an advanced analogue of integration with applications, together with the relationship between these advanced forms of differentiation and integration, usually known as "Stokes theorem".

Prerequisites
Apart from Multivariable & Complex Calculus II (or equivalent), there are no formal prerequisites for the course. However, it will be assumed that students have a firm grasp of linear algebra (as might be obtained from Algebra II) and general topology (as might be obtained from Topology & Analysis III). Experience in formal mathematical reasoning (as might be obtained from Real Analysis II) will be helpful. There will be some overlap with the Semester I course Manifolds, Lie groups, and Riemannian geometry, but this will restricted mainly to review of key notions.

Learning Outcomes
On successful completion of this course, students will:
1. understand the notion of a differential form on a manifold as well as the related concepts of exterior derivative and pull-back;
2. understand how to integrate a differential form over an oriented manifold;
3. understand the statement of Stokes' theorem and how it relates to the Fundamental Theorem of Calculus;
4. understand the notion of de Rham cohomology, and be able to compute this in simple cases;
5. understand the notion of Cech cohomology and how this is related to de Rham cohomology;
6. understand the notion of vector bundles, and the related notion of a connection on a vector bundle;
7. understand the statement and proof of the Gauss-Bonnet theorem.

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 freely available on-line resources covering the material in the course. Four books amongst many that present the ideas succinctly are:

R. Bott & L. W. Tu, Differential forms in algebraic topology.
Y. Choquet-Bruhat, C. DeWitt-Morette: Analysis, manifolds, and physics. (1982)
J. M. Lee: Introduction to smooth manifolds. (2012)
M. Spivak: Calculus on manifolds. (1971)
• 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                                                  87.5
Assignments                                     5                                                    68.5
Total                                                                                                    156

##### Learning Activities Summary
Topics covered:

• Review of:
• Differentiation in euclidean space
• manifolds and tangent spaces
• derivatives of functions between manifolds;
1. Differential forms on manifolds and exterior differential calculus
2. Partitions of unity
3. Integration on manifolds
4. Stokes theorem
5. de Rham cohomology
6. Cech cohomology
7. Vector bundles
8. Connections and curvature
9. Characteristic classes, particularly of line bundles
10. The Gauss-Bonnet theorem.
Time permitting, Donaldson's theorem on (definite) intersection forms of smooth four-dimensional manifolds will be covered, but if so, this material will not be examinable.
• 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 60% all Mid-semester test Summative Week 6 20% 1,2,3 Homework assignment Formative and summative One week after assigned 20% all
##### Assessment Related Requirements
An aggregate score of at least 50% is required to pass the course.
##### Assessment Detail
There will be a total of 5 homework assignments, due at the end of the week following their distribution.
##### Submission
Homework assignments must be given to the lecturer in person or emailed as a pdf file. 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:

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
• Fraud Awareness

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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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