## PURE MTH 7066 - Pure Mathematics Topic E

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

Lectures for this course will start in week 2.
• Learning Outcomes
##### Course Learning Outcomes
In 2016, the topic of this course is Lie groups and Lie algebras.

Syllabus

The aim of the course is to introduce the students to the theory of Lie algebras and  Lie groups. These 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, the course will start off with an introduction to the theory of Lie groups, the relation between Lie groups and Lie algebras, the exponential map and Lie subgroups. Then the structure  of Lie algebras is studied further 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.

Learning Outcomes

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

1) define and recognise 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.

Prerequisites

- Topic A (Michael Murray's course in Semester 1)
- Apart from this, the course requires an adequate knowledge in linear algebra and multivariable calculus. Knowledge of basic group theory is helpful but not mandatory.
##### 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
all
• Learning Resources
None.
##### Recommended Resources
• J. M. Lee, Introduction to Smooth Manifolds, Springer 2006
• F. Warner, Foundations of Differentiable manifolds and Lie groups, Springer 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, Birkhauser, 1996
• 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.

##### Workload

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

Activity                                       Quantity                                  Workload Hours

Lectures                                           30                                                  90
Assignments                                     6                                                    66
Total                                                                                                    156

##### Learning Activities Summary
1) Review of smooth manifolds and vector fields (3 Lectures)
2) Lie groups and their Lie algebras, subgroups, homomorphisms (10 lectures)
3) Structure theory of Lie algebras: nilpotent, solvable and semisimple Lie algebras (7 lectures)
4) Classification of complex semisimple Lie algebras (10 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
 Component Weighting Outcomes Assessed Assignments 30% All Exam 70% 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 6 homework assignments, due one week after they are assigned. 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 file. Failure to meet the deadline without reasonable and verifiable excuse may result in a significant penalty for that assignment.

##### Course Grading

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.

• 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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