## PURE MTH 7038 - Pure Mathematics Topic A

### North Terrace Campus - Semester 1 - 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 7038 Pure Mathematics Topic A Mathematical Sciences Semester 1 Postgraduate Coursework North Terrace Campus 3 Y Ongoing assessment, exam
##### Course Staff

Course Coordinator: Associate Professor Thomas Leistner

##### 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 Manifolds, Lie groups, and Riemannian geometry

Syllabus

The aim of the course is to introduce the students to some fundamental concepts in Differential Geometry. Differential Geometry is the theory that generalises all aspects of multivariable calculus in Euclidean n-space to more general spaces. The fundamental concepts introduced in the course will be grouped into 3 chapters:

1) Smooth manifolds

Smooth manifolds are generalisations of Euclidean space. They are topological spaces that have the property that they can be covered by neighbourhoods that are homeomorphic to open balls Euclidean space, they are “locally Euclidean”

Topics:
- review of multivariable differentiation and topology
- topologogical, differentiable and smooth manifolds
- tangent spaces and the tangent bundle of a manifold
- smooth maps between manifolds and their differentials
- submanfolds
- vector fields, their Lie brackets and flows

2) Riemannian Geometry

Riemannian geometry  generalises the “geometry” of Euclidean space, i.e., the notions of length, distance and angles, to manifolds. It was observed by Bernhard Riemann that manifolds  can be equipped with these geometric quantities, now called Riemannian manifolds, and that the vanishing of the curvature tensor decides whether a Riemannian manifold is "flat", i.e., locally has the same geometry as Euclidean space.

Topics:
- the notion of a Riemannian metric
- The Levi-Civita connection of a Riemannian metric and its curvature tensor
- geodesics (shortest curves)
- isometries

3) Lie groups

Lie groups are manifolds with a compatible group structure. They are a generalisation of groups of matrices and used in many areas of mathematics and physics to describe symmetries (of a manifold). They were introduced by Sophus Lie in order to describe continuous groups of transformations of Euclidean space.

Topics
:
- Lie groups and their Lie algebras
- Lie group homomorphisms
- Lie subgroups

Learning Outcomes

The topics in these four chapters are reduced to the introduction of the fundamental concepts and a derivation of their properties, with the aim to equip the students with a set of concepts and tools which enables them to pursue their carreer in pure mathematics or theoretical physics.
On successful completion of this course, students will be able to:

1) define and recognise a smooth manifold, a Lie group and its Lie algebra,
2) differentiate smooth maps between manifolds and commute vector fields,
3) work with a Riemannian metric and determine its Levi-Civita connection and its curvature and derive the geodesic equation,
4) familiar with many examples of Riemannian manifolds and Lie groups.

Prerequisites

The course requires an adequate knowledge in linear algebra and multivariable calculus. Knowledge of group theory is

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:

• J. M. Lee, Introduction to Smooth Manifolds, Springer 2006
• J. M. Lee, Riemannian Manifolds. An Introduction to Curvature, Springer 1997
• F. Warner, Foundations of Differentiable manifolds and Lie groups, Springer 1983
• B. O'Neill, Semi-Riemannian Geometry. With Applications to Relativity, Academic Press 1983

##### Online Learning
The 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 Lecture 30 90 Assignments 5 66 Total 156
##### Learning Activities Summary
1) Introduction to of smooth manifolds, smooth maps, and vector fields (10 Lectures)
2) Riemannian geometry (12 lectures)
3) Lie groups and their Lie algebras, subgroups, homomorphisms (8 lectures)

• 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 70% all Homework assignment Formative and summative One week after assigned 30% 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:

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

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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