MATH3968: Semester 2, 2021

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Units MATH3968 Semester 2 2021 [Normal day]

Unit of study_

MATH3968: Differential Geometry (Advanced)

Semester 2, 2021 [Normal day] - Remote

Overview

This unit is an introduction to Differential Geometry, one of the core pillars of modern mathematics. Using ideas from calculus of several variables, we develop the mathematical theory of geometrical objects such as curves, surfaces and their higher-dimensional analogues. Differential geometry also plays an important part in both classical and modern theoretical physics. The unit aims to develop geometrical ideas such as curvature in the context of curves and surfaces in space, leading to the famous Gauss-Bonnet formula relating the curvature and topology of a surface.

Unit details and rules

Unit code	MATH3968
Academic unit	Mathematics and Statistics Academic Operations
Credit points	6
Prohibitions ?	MATH4068
Prerequisites ?	A mark of 65 or greater in 12 credit points of MATH2XXX units of study
Corequisites ?	None
Assumed knowledge ?	(MATH2921 and MATH2922) or MATH2961
Available to study abroad and exchange students	Yes

Teaching staff

Coordinator	Emma Carberry, emma.carberry@sydney.edu.au

Type	Description	Weight	Due	Length
Final exam (Take-home short release)	Final exam Written exam, including computational and proof-based questions.	65%	Formal exam period	2 hours
Outcomes assessed:
Presentation	Tutorial presentation Group presentation of a tutorial problem (with individual assessment)	3%	Multiple weeks	10 minutes
Outcomes assessed:
Assignment	Assignment 1, group component Written assignment	5%	Week 06 Due date: 19 Sep 2021 at 23:59	2 weeks
Outcomes assessed:
Assignment	Assignment 1, individual component Written assignment	10%	Week 06 Due date: 19 Sep 2021 at 23:59	2 weeks
Outcomes assessed:
Assignment	Assignment 2 Written assignment	15%	Week 12 Due date: 07 Nov 2021 at 23:59	2 weeks
Outcomes assessed:
Small continuous assessment	Tutorial problems Write up of a tutorial problem, best 10 submissions count.	2%	Weekly	Variable
Outcomes assessed:
= group assignment ? = Type D final exam ?

Assessment summary

The better mark principle will be used to calculate your final grade.

Detailed information for each assessment can be found on Canvas.

Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name	Mark range	Description
High distinction	85 - 100	Representing complete or close to complete mastery of the material.
Distinction	75 - 84	Representing excellence, but substantially less than complete mastery.
Credit	65 - 74	Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence.
Pass	50 - 64	Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.
Fail	0 - 49	When you don’t meet the learning outcomes of the unit to a satisfactory standard.

For more information see sydney.edu.au/students/guide-to-grades.

For more information see guide to grades.

Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

Deduction of 5% of the maximum mark for each calendar day after the due date.
After ten calendar days late, a mark of zero will be awarded.

Academic integrity

The Current Student website provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

Learning support

Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

Weekly schedule

WK	Topic	Learning activity
Week 01	Parameterised curves, Frenet equations and the Fundamental Theorem	Lecture and tutorial (4 hr)
Week 02	The Theorem of Turning Tangents, basic topology and the Implicit Function Theorem	Lecture and tutorial (4 hr)
Week 03	Regular surfaces, more on the Implicit Function Theorem. Tangent planes and the First Fundamental Form	Lecture and tutorial (4 hr)
Week 04	Area, orientation and the Gauss map	Lecture and tutorial (4 hr)
Week 05	The Second Fundamental Form, principal curvatures, Gauss and mean curvatures	Lecture and tutorial (4 hr)
Week 06	Gauss curvature continued, minimal surfaces	Lecture and tutorial (4 hr)
Week 07	Conformal maps, Gauss’ Theorema Egregium and the Gauss-Mainardi-Codazzi equations	Lecture and tutorial (4 hr)
Week 08	Covariant derivative, parallel transport, geodesics and geodesic curvature	Lecture and tutorial (4 hr)
Week 09	The exponential map, geodesics as length-minimising curves	Lecture and tutorial (4 hr)
Week 10	Theorem of Turning Tangents, Gauss-Bonnet Theorem	Lecture and tutorial (4 hr)
Week 11	Poincaré-Hopf Index Theorem, Morse’s Theorem	Lecture and tutorial (4 hr)
Week 12	Abstract surfaces/manifolds	Lecture and tutorial (4 hr)

Attendance and class requirements

Attendance in tutorials is expected.

Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

Required readings

All readings for this unit can be accessed on the Library eReserve link available on Canvas.

Required text:

Differential Geometry of Curves and Surfaces, 2nd edition, by Manfredo do Carmo, available as an e-book through the library website or on order at the student co-op.

For a more introductory text you may wish to refer to:

Elementary Differential Geometry, by Andrew Crossley.

For the latter part of the course, also useful is:

Differential Forms and Applications, by Manfredo do Carmo.

Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University's graduate qualities and are assessed as part of the curriculum.

Outcomes
Graduate qualities

At the completion of this unit, you should be able to:

LO1. demonstrate knowledge and understanding of fundamental concepts and theorems in differential geometry.
LO2. apply fundamental theorems and concepts of differential geometry in order to solve geometric problems
LO3. understand and apply geometric concepts to analyse examples to draw conclusions
LO4. evaluate geometric quantities such as torsion and curvature
LO5. synthesise knowledge across a range of topics and write valid mathematical proofs.

Graduate qualities

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

GQ1	Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline.
GQ2	Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem.
GQ3	Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context.
GQ4	Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies.
GQ5	Inventiveness Generating novel ideas and solutions.
GQ6	Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues.
GQ7	Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries.
GQ8	Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context.
GQ9	Influence Engaging others in a process, idea or vision.

Outcome map

Learning outcomes	Graduate qualities
Learning outcomes

Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

Quiz removed in order to reduce volume of assessment. More recorded lecture content in response to student feedback.

Additional information

Work, health and safety

We are governed by the Work Health and Safety Act 2011, Work Health and Safety Regulation 2011 and Codes of Practice. Penalties for non-compliance have increased. Everyone has a responsibility for health and safety at work. The Universitys Work Health and Safety policy explains the responsibilities and expectations of workers and others, and the procedures for managing WHS risks associated with University activities.

Disclaimer

The University reserves the right to amend units of study or no longer offer certain units, including where there are low enrolment numbers.

To help you understand common terms that we use at the University, we offer an online glossary.

Current students

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Weekly schedule

Weekly schedule

Attendance and class requirements

Study commitment

Required readings

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Additional information

Additional information

Work, health and safety

Disclaimer

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