## PURE MTH 7038 - Pure Mathematics Topic A

### North Terrace Campus - Semester 1 - 2023

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.

• 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: Professor Finnur Larusson

##### Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
##### Course Learning Outcomes
In 2023, the topic of this course is DIFFERENTIAL GEOMETRY

Introduction

For many purposes in mathematics and physics, for example in the general theory of relativity, we need to extend differentiation and integration from Euclidean spaces to much more general spaces called manifolds. The basic questions to be answered in this course are: What are smooth manifolds? How (and what!) can we differentiate and integrate on them? What is this generalisation of calculus good for?

The main goal of the course is to set the stage for and prove Stokes' theorem, a vast generalisation of the fundamental theorem of calculus. Stokes' theorem is an important tool for relating local and global properties of manifolds. This is a major theme in modern mathematics, known in technical terms as cohomology. We will develop some basic ideas of algebraic topology from the differentiable viewpoint, using Stokes' theorem and homological algebra, and derive some important applications. (This will complement, but not duplicate, what you may be learning in Pure Topic B.) Students will pursue further topics in the form of individual projects.

Topics

Differentiation
Smooth manifolds
Tangent spaces
Differential forms and integration on manifolds
Stokes' theorem
Cohomology

Learning Outcomes

On successful completion of this course, students will be able to:
1. demonstrate an understanding of the basic theory of smooth manifolds;
2. demonstrate advanced skills in constructing rigorous mathematical arguments;
3. apply the theory in the course to solve a variety of problems at an appropriate level of difficulty;
4. demonstrate advanced skills in writing mathematics;
5. demonstrate a commitment to academic integrity.

Assumed knowledge

The more third-year pure mathematics, the better. At a minimum, Topology & Analysis III.

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.

1, 2, 3

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.

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.

2, 4

Graduates engage in professional behaviour and have the potential to be entrepreneurial and take leadership roles in their chosen occupations or careers and communities.

2, 4, 5

Attribute 5: Intercultural and ethical competency

Graduates are responsible and effective global citizens whose personal values and practices are consistent with their roles as responsible members of society.

5

Attribute 7: Digital capabilities

Graduates are well prepared for living, learning and working in a digital society.

3, 4

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.

2, 3
• Learning Resources
##### Required Resources
Course lecture notes will be provided.
##### Recommended Resources
Some standard functional analysis textbooks are:

M. Reed and B. Simon, Methods of modern mathematical physics. I, Functional analysis (Good foundation for mathematical quantum theory)
W. Rudin, Functional analysis (Quite abstract, graduate level, deals with general topological vector spaces and Banach spaces)

Other references:
K. Yosida, Functional analysis
G. Folland: Real analysis: Modern techniques and their applications
##### Online Learning
Course information and resources will be posted on MyUni.
• Learning & Teaching Activities
##### Learning & Teaching Modes
There will be a weekly two-hour workshop with a mix of lecturing and students solving and presenting problems. Three homework assignments through the semester will deepen students' understanding of the theory and its applications. A project in the second half of the semester, with a written report, is an opportunity for an individual research experience and development of written communication skills. Students will choose their own project topic in consultation with the lecturer. Presentations in the workshops will develop oral communication skills.

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

 Activity Quantity Workload hours Workshop attendance 12 24 Workshop preparation 12 24 Assignments 3 30 Project 1 20 Self-study 52 Total 150
##### Learning Activities Summary

• 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 Assignments Formative and summative Mondays of Weeks 5, 9, 13 30% all Project report Summative Friday of Week 13 40% all Workshop presentations Formative Weeks 1-12 20% all Active participation in workshops Formative Weeks 1-12 10% all

##### Assessment Detail
There will be a total of 6 homework assignments, given out at intervals of about two weeks.
##### Submission
Assignments and project report will be produced using LaTeX and submitted as pdf-files via MyUni.

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.

```
```