What is symplectic geometry? One answer is that it is the mathematical theory which has developed from studying the properties of light rays and light waves. I plan to start by considering light rays passing through systems of lenses (such as the lenses in spectacles, or in telescopes). Lenses refract light rays - that is, they change their direction - and the change in direction caused by each lens corresponds to a symplectic matrix. The definition of a symplectic matrix is in some respects like that of an orthogonal matrix. Remember that orthogonal matrices preserve distances and describe rotations. 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 and we will spend several lectures on their theory.
In the second part of the course I expect to consider caustics and focusing. When a beam of light is reflected from a curved surface it often focuses to a point, or along a curve. For example, a parabolic mirror focuses incoming light to a single point and on a bright day a cup of tea will often show a bright pattern on the side of the cup. The mathematical theory behind these phenomena is the starting point of much current work in mathematics.
The course will 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 Hamiltonian formalism is now a fundamental part of symplectic geometry. We will not cover any mechanics in this course.
MAS 336, Differential Geometry, is a prerequisite primarily because the last part of MAS 441 uses basic notions for curves in the plane --- unit tangent and normal vectors, and curvature. Also, symplectic geometry --- the great subject to which this course is a small introduction --- is part of differential geometry, one of the richest and most influential subjects in mathematics.
However the amount of MAS 336 which is needed is actually quite small. Much of MAS 336 is concerned with understanding curvature (for curves in the plane and surfaces in 3-space): there are different ways in which curvature arises, and it can play different roles. For MAS 441, it is possible just to master the basic formulas for curves and surfaces, without studying the consequences and interpretations given in MAS 336. If I agree to you taking MAS 441 without MAS 336, I will provide material to read up in preparation.
If you are a Level 2 student and considering taking MAS 441 in Level 4, then I urge you to take MAS 336 first.
If you are already in Level 4 and want to take MAS 441, but did not take MAS 336, then you will need to have a good average in Level 2 and Level 3. If your average is at the lower end of the MMath scale, I will probably not agree to you taking MAS 441.