MusicTheorist said:
Still, for some reason I'm just not that interested in doing a project on music. I'm not sure why. It tends to be that when I really want to do something (in this case string theory) it's extremely hard to get myself out of that mindset. Maybe I just have to do a little convincing.
I bet I can convince you to stay away from string theory.
If you want to approach string theory in any meaningful sense, you can't just rely on Kaku and Green's elegant universe- that is, pop sci books and features. Especially if this is a 3-year research contest/event I would hope that you could offer more than a book report on Green. The problem is, these treatments, though readable, offer none of the meat of higher physics; they're totally stripped of math (stripped of the actual physics and *reasoning*), and what you're left with is a very blurry, hand-wavey picture and you certainly can't draw any meaningful conclusions from that alone. In short, you need to look at what string theorists themselves write about string theory amongst themselves, which means YOU NEED THE MATH.
So what math, exactly, would you need to learn in order to make any sort of headway with string theory? It sounds like you've got algebra and trigonometry under your belt, which is a good start. Let's crack those books.
First a solid background in calculus is necessary for basic-level physics. Learning up to multivariable calculus is at least a year's worth of material, normally. Hand-in-hand with calc is usually taught linear algebra, which is a generalization of systems of algebraic equations, and fundamental to understanding quantum mechanics.
After calc you'd have to learn differential equations, which is the language of classical physics. This is not a trivial matter, ususally a mid-level college course or two.
Most of the language of analyzing fields and equations in the standard model and beyond uses the tools of functional analysis: real and complex analysis are high-level undergrad courses if not graduate level mathematics. These are very abstract notions and complex analysis relies on an understanding of imaginary numbers.
After all that basic framework is out of the way, you can start learning differential geometry and group theory, specifically lie groups, lie algebra and topology. Now we're approaching the language that string theory is written in.
To tackle ST itself, you would have to understand the fiber bundle formalism of the standard model, which also requires knowledge of cohomolgy. It would also behoove you to study up the basic ideas in extensions of the standard model: supersymmetry is required for string theory, but you should also have a working understanding of kaluza-klein-type extra dimensions.
Somewhere along the way you'll have to pick up statistics, most likely, for quantum. Oh yeah, and all the physics between here and there- which is to say, start from advanced Newtonian mechanics (lagrangians), extend this to an understanding of quantum mechanics, then quantum field theory, and then the standard model.
Now on the other hand, a physical treatment of music is both fascinating and within your immediate mathematical reach from the sounds of it. But don't take my word for it...
something like that could be well within your reach.
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