What will I learn in a course on Groups and Symmetries?

[Howers]

Hello. As some of you know I'm a chemistry student, but I plan to take some math for the hell of it next summer. I've come across a course called "Groups and Symmetries" and intend to take it, mainly because it is one of the few upper maths avaialbe in the summer. I've never heard of this type of math, and I am assuming it is some form of upper algebra. If anyone can tell me what I would learn in this kind of course I would appretiate it. And would this course have math useful in physics and chemistry? Here is the descprition:

Congruences and fields. Permutations and permutation groups. Linear groups. Abstract groups, homomorphisms, subgroups. Symmetry groups of regular polygons and Platonic solids, wallpaper groups. Group actions, class formula. Cosets, Lagrange’s theorem. Normal subgroups, quotient groups. Classification of finitely generated abelian groups. Emphasis on examples and calculations.

 

[Chris Hillman]

At least three reasons why should take this course

Hi, Howers,

Three gifts from the mathematical gods to students of chemistry:

1. Wallpaper groups and space groups, which are required to classify the symmetries of crystalline substances. Looks like the course in question will cover wallpaper groups, which should give you the background to read about space groups on your own.

2. Polya enumeration, for which the original application was enumerating alkanes. (See http://algo.inria.fr/libraries/autocomb/Polya-html/Polya.html for this topic dressed up in the language, more or less, of "structors", aka "combinatorial species".) Polya enumeration could well be included as a special topic in the course you are considering, so you should try to see the professor in advance and request this topic.

3. Elementary group theory is a prerequisite for a course on representation theory, which is needed to analyze the vibrations of molecules.

This is only a partial list: I could add many more topics. In particular, group theory is also needed for quantum mechanics, which as you no doubt know is needed for models of atoms. Also, group theory is necessary background for many important mathematical methods, such as Lie's symmetry analysis of (systems of) PDEs and ODEs, which is pretty much the only general method for attacking nonlinear systems of differential equations (no magic bullet, but often useful). So if you ever found yourself studying say the Belousov-Zhabotinsky spatially oscillating reaction (which can be modeled by an interesting system of nonlinear PDEs), you'd want to have this tool handy. If you got interested in quasicrystals you'd want to know about the group theoretical take on Fourier transforms, and so on...

 

[Howers]

Wow, thanks!

I had no idea there so much chemistry can be modeled by math. I thought quanta was the end. Good thing this will have a practical use.