PURE MTH 7038  Pure Mathematics Topic A
North Terrace Campus  Semester 1  2021

General Course Information
Course Details
Course Code PURE MTH 7038 Course Pure Mathematics Topic A Coordinating Unit Mathematical Sciences Term Semester 1 Level Postgraduate Coursework Location/s North Terrace Campus Units 3 Available for Study Abroad and Exchange Y Assessment Ongoing assessment, exam 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 spacetime. 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#)
~ Navigation and cartography
~ Celestial mechanics
~ Mathematical physics
* Surfaces in threespace
~ 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 cotangent bundles
~ Calculus on manifolds: exterior derivative, de Rham complex
~ Vector bundles: connections and curvature
* Riemannian geometry
~ Torsion and curvature
~ Proof of Theorema Egregium
~The LeviCivita connection and Riemannian curvature
~ Examples
 Spaces of constant curvature: hyperbolic space
 Kaehler geometry: complex projective space
* NonRiemannian 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 LeviCivita 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 7002  Pure Mathematics Topic B 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.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
Required Resources
There are no required resources 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 Spacetime, 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
The 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.Workload
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)

Assessment
The University's policy on Assessment for Coursework Programs is based on the following four principles:
 Assessment must encourage and reinforce learning.
 Assessment must enable robust and fair judgements about student performance.
 Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
 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.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 149 Fail P 5064 Pass C 6574 Credit D 7584 Distinction HD 85100 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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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.

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