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Geometry (MAST30024)

Undergraduate level 3Points: 12.5On Campus (Parkville)

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Overview

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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.

Intended learning outcomes

On completion of this subject, students should be able to

  • identify surfaces using topological methods, combinatorial descriptions, Euler characteristic and orientability.
  • compute the Jacobian matrix and regular values of a smooth map between Euclidean spaces.
  • compute differential geometric quantities such as lengths, angles, and areas of Riemannian metrics on smooth surfaces
  • determine principal, mean, and Gaussian curvatures of surfaces in Euclidean space and interpret their geometric significance
  • use the Gauss-Bonnet theorem to relate curvature and topology in the analysis of surfaces.
  • perform basic computations with complex algebraic curves, including conics and cubics, and identify their genus and geometric properties
  • explain how surfaces can arise as complex algebraic curves and describe the connections between differential, topological, and algebraic geometry.

Generic skills

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.

Last updated: 4 February 2026