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MAST 30024
Geometry

Lecturer: Craig Westerland, 168 Richard Berry, phone: 8344 9712, email: C.Westerland@ms.unimelb.edu.au

Time and Location: Lecture Tuesday 10-11, Wednesday 4:15-5:15, Friday 12-1; Practice class Tuesday 2:15-3:15, all in 213 Richard Berry.
Consultation hours: Tuesday 11-12 and 3:15-5:00.

Subject Outline

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.

Main Topics

The material covered will be drawn from the following:
  • Topological classification of surfaces
  • Euler characteristic
  • Orientability
  • Smooth surfaces
  • Tangent planes
  • Length of curves
  • Riemannian metrics
  • Gaussian curvature
  • Minimal surfaces
  • Gauss-Bonnet theorem
  • Complex algebraic curves

References

  • N. Hitchin, Geometry of surfaces, Oxford University lecture notes, available online here, under the heading "Geometry of Surfaces."
  • M. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, 1976.
  • F. Kirwan, Complex algebraic curves, Cambridge University Press, 1992.
  • Jason Leung's notes from last year's class. Part 1: topology, Part 2: differential topology, Part 3: differential geometry.

Assessment

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%).

Homework assignments:
  • Homework 1, due Friday 26 August. Solutions.
  • Homework 2, due Friday 14 October. Solutions.
  • Homework 3, due Friday 28 October. Solutions.
Practice problems:
  • Prac 1, Tuesday, 2 August. Solutions.
  • Prac 2, Tuesday, 9 August. Solutions.
  • Prac 3, Tuesday, 16 August. Solutions.
  • Prac 4, Tuesday, 23 August. Solutions.
  • Prac 6, Tuesday, 13 September. Solutions.
  • Prac 7, Tuesday, 4 October. Solutions.
  • Prac 8, Tuesday, 11 October. Solutions.
  • Prac 9, Tuesday, 18 October. Solutions.
The plagiarism declaration is available here.