PURE MTH 4038 - Pure Mathematics Topic A - Honours

North Terrace Campus - Semester 1 - 2021

• General Course Information
Course Details
Course Code PURE MTH 4038 Pure Mathematics Topic A - Honours School of Mathematical Sciences Semester 1 Undergraduate North Terrace Campus 3 Y Honours students only Please contact the School of Mathematical Sciences for further details.
Course Staff

Course Coordinator: Professor Michael Eastwood

Course Timetable

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

• Learning Outcomes
Course Learning Outcomes
In 2021, the topic of this course is DIFFERENTIAL GEOMETRY

Introduction

Differential geometry is a classical subject. It is the mathematical study of geometry using calculus and differential equations. The motivation for the subject comes from navigation and map making. In 1569, the cartographer Mercator experimentally discovered his celebrated map projection in which courses of constant bearing are represented by straight lines. This was a hundred years before Newton and Leibniz introduced the differential calculus needed to understand Mercator's projection fully. Nowadays, there are many map projections with various subtle and useful properties but none is totally accurate. The fundamental reason for this is Gauss' Theorema Egregium proved in 1821, which says that a quantity now called the Gaussian curvature is intrinsically defined by measurements of distance alone. As the Gaussian curvature of the unit sphere is everywhere 1 and that of the plane is everywhere 0, any map of the earth necessarily distorts distance. In 1854, Riemann extended Gaussian curvature to higher dimensions. The resulting Riemann curvature tensor was developed and extended over the next 50 years and the differential geometry of smooth metrics was born. This theory is central in both mathematics and physics; for example, Einstein's equations of general relativity are just restrictions on the curvature of space-time. The terminology of modern differential geometry reflects its historical roots: one speaks of a manifold defined by an atlas of smooth charts et cetera and even the word geometry comes from Greek, roughly meaning measurement of earth.

Topics

* Motivation (and for smooth manifolds#)
~ Celestial mechanics
~ Mathematical physics

* Surfaces in three-space
~ Inverse and Implicit Function Theorems in three dimensions
~ Euclidean normalisation: Gaussian and mean curvature
~ Affine normalisation: the Pick invariant
~ Statement of Gauss' Theorema Egregium

* Smooth manifolds#
~ Flora and fauna: vector fields, tangent and co-tangent bundles
~ Calculus on manifolds: exterior derivative, de Rham complex
~ Vector bundles: connections and curvature

* Riemannian geometry
~ Torsion and curvature
~ Proof of Theorema Egregium
~ The Levi-Civita connection and Riemannian curvature
~ Examples
- Spaces of constant curvature: hyperbolic space
- Kaehler geometry: complex projective space

* Non-Riemannian geometry
~ Lorentzian geometry: general relativity, Einstein's equations
~ Projective differential geometry: invariants and curvature
~ Conformal differential geometry: invariants and curvature
~ Other differential geometries

# There will be some overlap with Dr David Baraglia's Topics B course on Lie Algebras.

Learning Outcomes

On successful completion of this course, students will be able to
1. Define Riemannian manifolds, and understand curvature both conceptually and computationally;
2. Define other differential geometric structures and understand key examples;
3. State and prove Gauss' Theorema Egregium;
4. Understand the construction of the Levi-Civita connection and its role in Riemannian differential geometry;
5. Construct the basic invariants of projective and conformal differential geometry.

Prerequisites

The course requires an adequate knowledge of linear algebra and multivariable calculus. It is recommended, but not required, that students in this course also take PURE MTH 4012 - Pure Mathematics Topic B - Honours on Lie Algebras given by Dr David Baraglia, in which many further examples of smooth manifolds will occur in the context and mathematical study of symmetry.

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
Required Resources
There are no required resoures for this course but lecture notes will be provided.
Recommended Resources
There are many excellent books on differential geometry in the Barr Smith Library. Browse the shelves, especially around 514.764, or consult from the following short selection.

* T. Aubin, A Course in Differential Geometry, American Mathematical Society 2001.
* S. Gallot, D. Hulin, and J. Lafontaine, Riemannian Geometry, Springer 1987, 1990, 2004.
* S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-time, Cambridge University Press 1973.
* S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, American Mathematical Society 2001.
* J.M. Lee, Manifolds and Differential Geometry, American Mathematical Society 2009.
* J.M. Lee, Introduction to Smooth Manifolds, Springer 2003, 2013.
* P.W. Michor, Topics in Differential Geometry, American Mathematical Society 2008.

Online Learning
This course will have an active MyUni website.
• 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.

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 Summary
1) Introduction to smooth manifolds and associated gadgets (10 Lectures)
2) Riemannian geometry (10 lectures)
3) Projective, conformal, and other differential geometries (10 lectures)

Specific Course Requirements
None.
• 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 Examination Summative Examination period 70% all Homework assignment Formative and summative One week after assigned 30% all

Assessment Related Requirements
An aggregate score of 50% is required to pass the course.
Assessment Detail
There will be a total of 6 homework assignments, given out at intervals of about two weeks.
Submission
Homework 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:

M11 (Honours Mark Scheme)
Fail A mark between 1-49 F
Third Class A mark between 50-59 3
Second Class Div B A mark between 60-69 2B
Second Class Div A A mark between 70-79 2A
First Class A mark between 80-100 1
Result Pending An interim result RP
Continuing Continuing CN

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
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