# MATHS 725 : Lie Groups and Lie Algebras

## Science

### Course Prescription

Symmetries and invariants play a fundamental role in mathematics. Especially important in their study are the Lie groups and the related structures called Lie algebras. These structures have played a pivotal role in many areas, from the theory of differential equations to the classification of elementary particles. Strongly recommended for students advancing in theoretical physics and pure mathematics. Recommended preparation: MATHS 333.

### Course Overview

This is a core graduate course that is recommended for a pure mathematics graduate student and also for many students taking physics, applied mathematics or engineering that may be interested in applying the ideas of continuous symmetries to problems in science.

It explores and develops the interaction between algebra, analysis, geometry, and topology and is extremely useful for those wishing to obtain a conceptual and practical understanding of many of the tools used in other areas of mathematics such as ordinary partial differential equations, partial differential equations as well as the applications of these things to, for example quantum mechanics.

### Course Requirements

Prerequisite: MATHS 320 and 332

### Capabilities Developed in this Course

 Capability 1: Disciplinary Knowledge and Practice Capability 2: Critical Thinking Capability 3: Solution Seeking

### Learning Outcomes

By the end of this course, students will be able to:
1. Understand that the subject combines ideas from analysis, geometry, topology and algebra. (Capability 1 and 2)
2. Know how to design and execute proofs of theorems in this area. (Capability 1, 2 and 3)
3. Have a working knowledge of the basic tools of the exponential map and how it connects analytic properties with algebraic properties. (Capability 1, 2 and 3)
4. Have knowledge of how to combine tools and techniques from many different areas of mathematics in order to prove new results and obtain a heightened level of understanding. (Capability 1, 2 and 3)
5. Display a high level of knowledge of mathematical concepts. (Capability 1, 2 and 3)

### Assessments

Assessment Type Percentage Classification
Assignments 30% Individual Coursework
Final Exam 70% Individual Examination
1 2 3 4 5
Assignments
Final Exam
The final grade for each student will be calculated using the plussage system, taking the maximum of two scores: EITHER 30% from assignments and 70% from the final exam, OR 100% from the final exam.
A minimum of 35% in the exam is required to pass the course.
The use of calculators in the test and exam is not permitted.

### Tuākana

Tuākana Science is a multi-faceted programme for Māori and Pacific students providing topic specific tutorials, one-on-one sessions, test and exam preparation and more. Explore your options at
https://www.auckland.ac.nz/en/science/study-with-us/pacific-in-our-faculty.html
https://www.auckland.ac.nz/en/science/study-with-us/maori-in-our-faculty.html

For further information on the Tuākana Maths programme, please visit
https://www.auckland.ac.nz/en/science/study-with-us/pacific-in-our-faculty.html
https://www.auckland.ac.nz/en/science/study-with-us/maori-in-our-faculty.html

This course is a standard 15-point course and students are expected to spend 10 hours per week involved in each 15-point course that they are enrolled in. For this course, you can expect a total of 36 hours of lectures, 30 hours of reading and thinking about the content, and 84 hours of work on assignments and/or exam preparation.

### Delivery Mode

#### Campus Experience

• Attendance is expected at scheduled activities.
• Lectures will be available as recordings if room has appropriate recording facilities.
• The activities for the course are scheduled as a standard weekly timetable delivery.

### Learning Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.

Course books and notes will be provided.

### Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

The student feedback was positive. The only proposed improvement was to include an indexing or table of content in the lecture notes for the second half of the course, which will be provided.

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.

### Class Representatives

The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.

You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

### Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website

### Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page .

This should be done as soon as possible and no later than seven days after the affected test or exam date.

### Learning Continuity

In the event of an unexpected disruption, we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, and if disruption occurs you should refer to the university website for information about how to proceed.

The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.

### Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter .

### Disclaimer

Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course students may be asked to submit coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.

Published on 28/10/2022 11:28 a.m.