PURE MTH 7038 - Pure Mathematics Topic A
North Terrace Campus - Semester 1 - 2017
General Course Information
Course Code PURE MTH 7038 Course Pure Mathematics Topic A Coordinating Unit School of Mathematical Sciences Term Semester 1 Level Postgraduate Coursework Location/s North Terrace Campus Units 3 Available for Study Abroad and Exchange Y Course Description 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 Coordinator: Dr Thomas Leistner
The full timetable of all activities for this course can be accessed from Course Planner.
Course Learning OutcomesIn 2017, the topic of this course is Manifolds, Lie groups and Lie algebras.
The aim of the course is to introduce the students to the theory of Lie groups and Lie algebras. This requires a brief introduction into the basics of smooth manifolds which will be given at the beginning of the course.
Manifolds, Lie groups and Lie algebras are fundamental concepts in both mathematics and theoretical physics. The theory of Lie groups and Lie algebras was developed in the late nineteenth century by Sophus Lie, Wilhelm Killing and others when Lie groups appeared as symmetries of differential equations. Soon it was realised that they can be treated by purely algebraic means yielding the concept of a Lie algebra. In physics Lie groups and Lie algebras are important in describing symmetries of physical systems and in gauge theories.
Based on the notion of a smooth manifold, which will be given at the beginning of the course, the course continues with an introduction to the theory of Lie groups, the relation between Lie groups and Lie algebras, the exponential map and Lie subgroups. The we will briefly look at the action of a Lie group on a manifolds and the related notion of a homogeneous space.
In the second part of the course we wil focus on Lie algebras and study them using the distinction into nilpotent, solvable, simple and semisimple Lie algebras. In the last part of the course the classification of semisimple complex Lie groups via root systems and Dynkin diagrams will be presented.
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 and understand how the exponential map relates them to each other,
2) differentiate Lie group homomorphisms to the corresponding Lie algebra homomorphism,
3) decide whether subsets in a Lie group are Lie subgroups by using Cartan's theorem,
4) distinguish between nilpotent, solvable, semisimple and simple Lie groups using effectively several criteria such as Lie's theorem, Engel's theorem, Cartan's criterion,
5) use Dynkin diagrams and root spaces in order to classify the simple complex Lie algebras.
The course requires an adequate knowledge in linear algebra and multivariable calculus. Knowledge of basic group theory is
helpful but not mandatory. Its recommended that students in this course also take PURE MTH 4012 - Pure Mathematics Topic B - Differential Geometry by Michael Eastwood, in which many example and further aspects of smooth manifolds will be studied in detail.
University Graduate Attributes
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
Required ResourcesThere are no required resources for this course.
Recommended ResourcesThere 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
• F. Warner, Foundations of Differentiable manifolds and Lie groups, Springer 1983
• B. O'Neill, Semi-Riemannian Geometry. With Applications to Relativity, Academic Press 1983
• W. Rossmann, Lie groups: an introduction through linear groups, Oxford UP 2006
• K. Erdmann, M. J. Wildon, Introduction to Lie Algebras, Springer, 2006
• J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer 1972
• A. W. Knapp, Lie Groups Beyond an Introduction, BirkhäÂuser, 1996
Online LearningThe course will have an active MyUni website.
Learning & Teaching Activities
Learning & Teaching ModesThe 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 6 66 Total 156
Learning Activities Summary1) Introduction to of smooth manifolds and vector fields (6 Lectures)
2) Lie groups and their Lie algebras, subgroups, homomorphisms (9 lectures)
3) Lie group actions and homogeneous spaces (3 lectures)
4) Structure theory of Lie algebras: nilpotent, solvable and semisimple Lie algebras (6 lectures)
5) Classification of complex semisimple Lie algebras (6 lectures)
The University's policy on Assessment for Coursework Programs is based on the following four principles:
- Assessment must encourage and reinforce learning.
- Assessment must enable robust and fair judgements about student performance.
- Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
- Assessment must maintain academic standards.
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 RequirementsAn aggregate score of 50% is required to pass the course.
Assessment DetailThere will be a total of 6 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.
SubmissionHomework 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) Grade Mark Description 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.
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