Geometry

This subject has the following teaching availabilities in 2016:

Estimated total time commitment of 170 hours

and one of

For the purposes of considering request for Reasonable Adjustments under the Disability Standards for Education (Cwth 2005), and Student Support and Engagement Policy, academic requirements for this subject are articulated in the Subject Overview, Learning Outcomes, Assessment and Generic Skills sections of this entry.

It is University policy to take all reasonable steps to minimise the impact of disability upon academic study, and reasonable adjustments will be made to enhance a student's participation in the University's programs. Students who feel their disability may impact on meeting the requirements of this subject are encouraged to discuss this matter with a Faculty Student Adviser and Student Equity and Disability Support: http://services.unimelb.edu.au/disability

This subject introduces three areas of geometry that play a key role in many branches of mathematics and physics. In differential geometry, calculus and the concept of curvature will be used to study the shape of curves and surfaces. In topology, geometric properties that are unchanged by continuous deformations will be studied to find a topological classification of surfaces. In algebraic geometry, curves defined by polynomial equations will be explored. Remarkable connections between these areas will be discussed.

Topics include: Topological classification of surfaces, Euler characteristic, orientability.Introduction to the differential geometry of surfaces in Euclidean space:smooth surfaces, tangent planes, length of curves, Riemannian metrics, Gaussian curvature, minimal surfaces, Gauss-Bonnet theorem.Complex algebraic curves, including conics and cubics, genus.

On completion of this subject, students should

Have an understanding of:

• Euler characteristic and the topological classification of surfaces;
• Riemannian metrics and curvature for surfaces;
• the Gauss-Bonnet theorem;
• how surfaces arise as complex algebraic curves.

Be able to:

• calculate Euler characteristic and identify surfaces described combinatorially;
• compute lengths, angles, areas for a given Riemannian metric;
• compute principal curvatures, mean curvature, Gaussian curvature for surfaces in Euclidean space;
• apply the Gauss-Bonnet theorem;
• do simple calculations with algebraic plane curves.

Two or three written assignments due at regular intervals during semester amounting to a total of up to 50 pages (20%), and a 3-hour written examination in the examination period (80%).

None

N. Hitchin, Geometry of surfaces, Oxford University lecture notes, available online.

M. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, 1976.

F. Kirwan, Complex algebraic curves, Cambridge University Press, 1992.

This subject potentially can be taken as a breadth subject component for the following courses:

• Bachelor of Commerce
• Bachelor of Environments
• Bachelor of Music

You should visit learn more about breadth subjects and read the breadth requirements for your degree, and should discuss your choice with your student adviser, before deciding on your subjects.

In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:

• problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
• analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
• collaborative skills: the ability to work in a team;
• time-management skills: the ability to meet regular deadlines while balancing competing commitments.

This subject is available for science credit to students enrolled in the BSc (both pre-2008 and new degrees), BASc or a combined BSc course.