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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
Have an understanding of:
- Euler characteristic and the topological classification of surfaces;
- how surfaces arise as complex algebraic curves.
- Riemannian metrics and curvature for surfaces;
- the Gauss-Bonnet theorem;
Be able to:
- calculate Euler characteristic and identify surfaces described combinatorially;
- compute lengths, angles, areas for a given Riemannian metric;
- apply the Gauss-Bonnet theorem;
- compute principal curvatures, mean curvature, Gaussian curvature for surfaces in Euclidean space;
- do simple calculations with algebraic plane curves.
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: 8 August 2023