PURE MTH 7066 - Pure Mathematics Topic E
North Terrace Campus - Semester 2 - 2023
General Course Information
Course Code PURE MTH 7066 Course Pure Mathematics Topic E Coordinating Unit School of Mathematical Sciences Term Semester 2 Level Postgraduate Coursework Location/s North Terrace Campus Units 3 Available for Study Abroad and Exchange Y Course Description This course is available for students taking a Masters degree in Mathematical Sciences. The course will cover an advanced topic in pure mathematics. For details of the topic offered this year please refer to the Course Outline.
Course Coordinator: Dr Daniel Stevenson
The full timetable of all activities for this course can be accessed from Course Planner.
Course Learning OutcomesIn 2023, the title of this course is Category Theory.
This course is an introduction to category theory. Category theory is a kind of algebra that studies the fundamental structures that occur everywhere in mathematics: objects, relationships between them, relationships between relationships, and so on. Knowledge of basic category theory is useful to all mathematicians and essential to many. For example, modern algebraic geometry and algebraic topology could not exist without category theory. The categorical way of thinking enables us to see common patterns in diverse areas of mathematics and guides us in our search for appropriate definitions and fruitful conjectures. We will pay particular attention to categorical structures in the areas of mathematics that the students in the course have studied previously.
No strict prerequisites, but the more third-year pure mathematics you have done, the better.
1. Demonstrate understanding of and ability to apply the basic concepts and theorems of category theory.
2. Demonstrate awareness and understanding of categorical structures in diverse areas of mathematics.
3. Demonstrate skills in formulating, solving, and communicating mathematical problems.
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)
Attribute 1: Deep discipline knowledge and intellectual breadth
Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.
Attribute 2: Creative and critical thinking, and problem solving
Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.
1, 2, 3
Attribute 3: Teamwork and communication skills
Graduates convey ideas and information effectively to a range of audiences for a variety of purposes and contribute in a positive and collaborative manner to achieving common goals.
Attribute 8: Self-awareness and emotional intelligence
Graduates are self-aware and reflective; they are flexible and resilient and have the capacity to accept and give constructive feedback; they act with integrity and take responsibility for their actions.
Required ResourcesThis is a reading course. The course will make use of the following textbook, which is available online through the university library.
K. Erdmann and M. J. Wildon, Introduction to Lie algebras.
Supplementary notes on Lie groups and their relation to Lie algebras will also be provided.
Recommended ResourcesThere are many excellent references for Lie groups and Lie algebras. The most suitable ones for this course are:
K. Erdmann and M. J. Wildon, Introduction to Lie algebras (this will be our main reference for Lie algebras)
J. E. Humphreys, Introduction to Lie Algebras and Representation Theory (a slightly more advanced reference for Lie algebras)
H. Samelson, Notes on Lie Algebras (covers much the same material as Humphreys)
A. K. Knapp, Lie Groups, Beyond an Introduction (contains much more material than we will cover in this course, but is very well- written)
W. Fulton, J. Harris, Representation Theory, A First Course (covers somewhat different material, but is heavily example based so may be useful as a learning resource for Lie algebras and Lie groups)
Online LearningCourse information and resources will be posted on MyUni.
Learning & Teaching Activities
Learning & Teaching ModesStudents are expected to read and engage with the assigned reading material. There will be a weekly workshop with a mix of lecturing, students working on problems, together and with guidance from the lecturer, and consulting. 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.
Activity Quantity Workload Hours Workshops 12 24 Assignments 5 50 Self-study 82 Total 156
Learning Activities Summary1. Basic concepts of Lie algebras and Lie groups.
2. The relation between Lie algebras and Lie groups.
3. Solvable and nilpotent Lie algebras, Engel's theorem, Lie's theorem.
4. The Killing form, semisimple Lie algebras, Cartan's criterion.
5. Cartan subalgebras, root space decompositions.
6. Root systems and their classification.
7. The classification of complex semisimple Lie algebras.
The University's policy on Assessment for Coursework Programs is based on the following four principles:
- Assessment must encourage and reinforce learning.
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- Assessment must maintain academic standards.
Assessment task Task type Due Weighting Learning outcomes Examination Summative Examination period 60% all Homework assignments Formative and summative Weeks 3,5,7,9,11 40% all
Assessment Related RequirementsA mark of 50 is required to pass the course.
Assessment DetailThere will be five homework assignments, due in Weeks 3, 5, 7, 9 and 11.
SubmissionHomework assignments should be submitted via MyUni.
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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