OverviewIn this module we will study plane algebraic curves and the way that they arise in applications to other parts of mathematics and physics. Examples include the use of elliptic functions to solve problems in mechanics (e.g. the pendulum, or Euler's equations for rigid body motion), spectral curves of separable Hamiltonian systems, and algebraic curves over finite fields that are used in cryptography. The geometrical properties of a curve are not altered by coordinate transformations, so it is important to identify quantities that are invariant under such transformations. For curves, the most basic invariant is the genus, which is most easily understood in terms of the topology of the associated Riemann surface: it counts the number of handles or "holes". The case of genus zero (corresponding to the Riemann sphere) is well understood, but curves of genus one (also known as elliptic curves) lead to some of the most interesting and difficult problems in modern number theory.
This module appears in:
Method of assessment
70% examination, 30% coursework
Complex Algebraic Curves, Frances Kirwan, LMS Student Texts 23, Cambridge, 1992, ISBN-100521423538.
Algebraic Curves and Riemann Surfaces, Rick Miranda, Graduate Studies in Math., vol. 5, AMS, 1995, ISBN 0-8218-0268-2.
Lectures on elliptic curves. J.W.S. Cassels, LMS Student Texts 24, Cambridge, 1991, ISBN-100521425301.
Algebraic Aspects of Cryptography, N. Koblitz, Springer, 1998, ISBN 978-3-540-63446-1.
A Course of Modern Analysis, E.T. Whittaker and G.N. Watson, Cambridge, fourth edition, 1927 (reprinted 2005), ISBN 0-521-58807-3.
The Arithmetic of Elliptic Curves, Joseph H. Silverman, Graduate Texts in Mathematics 106, Springer, 1986, ISBN 0-387-96203-4.
The intended subject specific learning outcomes. On successfully completing the module students will be able to:
1 rigorous thinking.
2 calculating with and visualization of geometrical objects.
3 systematic observation, generalization and techniques of proof.
4 the use of geometrical methods in other areas of mathematics and physics.
5 algebraic and analytical techniques for understanding geometry.
6 classification of objects according to their topological and geometrical properties.
7 connecting abstract mathematics to the real world.
8 proficient use of mathematical software such as Maple and MAGMA to masters level.
The intended generic learning outcomes. On successfully completing the module students will be able to:
1 an enhanced ability to correctly formulate geometrical problems and solve them efficiently;
2 enhanced skills in understanding and communicating mathematical results and conclusions;
3 a holistic view of mathematics as a problem solving and intellectually stimulating discipline;
4 an appreciation of algorithms and computational methods in geometry.
5 matured in their problem formulating and solving skills;
6 consolidated a variety of analytical and algebraic tools to model and classify geometrical objects and configurations.