MAS 441 / MAS 6360

Optics and Symplectic Geometry

2019 - 2020

All the materials for this course are on these web pages. There are no documents on MOLE.

Course material

Symplectic geometry arose from the study of optics and the study of mechanics. (In MAS 441 we won't consider mechanics.) In the past 20 years or so it has become a major part of modern mathematics.

When a ray of light passes through a lens (such as the lens in a pair of spectacles, or in a telescope) its direction is changed -- or refracted -- and the change in direction corresponds to a symplectic matrix. The definition of a symplectic matrix looks similar to that of an orthogonal matrix and just as an orthogonal matrix preserves the dot product, and hence distances and angles, symplectic matrices preserve a `symplectic structure', which we'll define early on in the course. Although symplectic matrices are defined in a similar way to orthogonal matrices, their properties are very different.

Symplectic geometry enters modern mathematics in several different ways. For example, the set of all straight lines in the plane, or in any Rn, has a symplectic structure. Symplectic structures arise in the study of Lie algebras, and in complex geometry. (We will touch on the first two of these examples.)

Symplectic geometry is part of differential geometry, but MAS 441 is very different to MAS 336. MAS 336 focuses on the curvature properties of curves and surfaces in three dimensions.

MAS 441 does not assume any knowledge of the physical properties of light. What is required is actually quite small, and will be covered as we need it.

If you look on the web or in books for information about symplectic geometry you will probably be told that the subject developed from theoretical mechanics. In fact Hamilton (1805-1865), who developed what is called the Hamiltonian formalism in mechanics, first developed his theory for optics, and only afterwards realized that the same mathematical ideas apply to mechanics. This is not at all obvious and is a striking example of how a mathematical theory can have applications which are unrelated to each other. The underlying phenomenon is that the solutions of a variational problem have a symplectic structure.

Prerequisites and other courses

The prerequisite course is MAS 211 Advanced Calculus and Linear Algebra.

If you haven't taken MAS 211, it may still be possible to take MAS 441. In this case, you should get in touch with me before the start of semester, to discuss your background.

Other relevant course:

To summarize: Provided you have taken MAS 211, you are welcome to enrol in MAS 441. It is helpful, but not necessary, to have taken MAS 336.