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 Posted: Sun Nov 05, 2006 2:01 pm
Um... I guess you should look into algebraic geometry and differential topology if you're going to do string theory. Number theory can be interesting, and a lot of the results are really fascinating, especially since, unlike a lot of other branches of math, we often feel that we have an intuitive grasp of numbers and so the bizarre results really stand out.
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Posted: Tue Nov 07, 2006 8:03 pm
Unfortunately, I only have a choice between the uber-basic pre-freshman-level number theory (tho' it's taught by my current Galois theory prof) and the senior-level number theory. Almost all of the sophomore- and junior-level courses focus on analysis, algebra, and set theory. I mean, we're required to take a 4-course analysis sequence junior year, but only 2 courses in algebra and none in number theory...wtf?
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Posted: Tue Nov 07, 2006 8:49 pm
Wait... I'm not quite clear what it is you're asking, especially if you're going into "special topics" for algebra and need to know the prereq material out of complex analysis, topology, and differential geometry. I'm not sure if that question even can be answered--"special topics" can simply cover any portion of very many things. That's true to some extent for all classes, since different professors often emphasize different areas, but it is especially true in this case. Some clear and more specific goal would make your question easier to answer.
For example, if you're trying to understand quantization (you did say you're a physics major, right?), then my "minimal level" would probably include everything necessary to understand the following theorems and their derivations: from analysis, Stone-Weierstraß theorem; from topology and algebra, Gel'fand-Naĭmark theorem; from differential geometry, sufficient familiarity to define the Lagrangian and Hamiltonian in terms of tangent and cotangent bundles, "canonical coordinates" in terms of fibres, etc. The neccessity of having familiarity with Hilbert spaces is, of course, obvious.
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Posted: Sat Nov 11, 2006 10:51 am
That's the thing. I'm not completely sure whether I want to major in math or physics just yet. Luckily, the physics sequence is pretty set-in-stone until senior year, when we start specializing, but the math track is really pretty open-ended. You have to take classes that "cover certain topics", and every semester there are usually 2 or 3 different classes all satisfying a certain requirement. I want to keep studying math to understand the proof and reasoning behind physics, at the very least; whether or not I want something more is basically the question, "Do I want to major in math?"
Aye, special topics in Algebra can cover anything, but in terms of departmentals, it's just another Algebra course. But in terms of the whole "understanding physical argument" thing, I'm not sure if I should take those courses sooner or later. I want to double major in math and physics, but since I'm already on course for a minor in education I don't think I have the time. Regardless of my major, I'll still take courses in both, though.
Here's the breakdown: I'm going to have two humanities, one physics, and two math classes next semester (almost the same as this semester), and I'm trying to figure out which two math courses to take. The math and physics majors here don't collaborate as much as I thought, and the professors in each department seem reluctant to give me a straight answer. The math classes I can take next semester are:
* Uber-basic (freshman) number theory
* Topics in algebra
* Topology
* Analysis I: Partial Diff. Eq. and Fourier transforms
* Analysis III: Integration theory and Hilbert space
I *technically* meet all the requirements for these, though it wouldn't hurt to assume that I should take analysis 1 and 2 before 3.
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Posted: Sun Nov 12, 2006 5:59 pm
If you're physics, then diff. eq is definitely necessary. I recommend the topology; diff. topology and algebraic topology play a big role in modern theoretical physics.
The algebra and the analysis III really depend on what you're doing in physics and your personal interests, although both I recommend taking eventually.
The number theory will help you the least for physics, as far as I know. While I do know a guy doing stuff linking number theory to physics, I have no idea what he's actually doing.
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Posted: Tue Nov 14, 2006 7:46 pm
Urg...diff. eq. is s'posed to be the most *boring* math class we have. It seems like it's a class in using Mathematica to make convincing graphs. I didn't know topology by itself played an important role in physics; I knew differential geometry did, but that's kewl. I think I'm gonna end up taking at least 3 of the 4 semesters of analysis eventually, and it probably wouldn't be smart to start with analysis 3. I can take the algebra course next year or somesuch; the Galois theory was the important part.
Thanks so much, you too. I'm gonna see if I can get into analysis I and topology next semester (tho' analysis I is a 9am. class...urg). Again, you guys have been a huge help.
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Posted: Wed Nov 15, 2006 9:14 pm
Topology doesn't play quite as obvious a role as differential geometry, but it's there. Diff. topology and diff geometry are basically two sides of the same field, and algebraic topology plays into things like holonomy groups and such, which appear in string theory.
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Posted: Fri Nov 17, 2006 1:13 pm
I talked with the physics department head briefly (who is studying string theory) and she told me that to do it from a mathematical perspective, I'll need diff. eq., complex analysis, topology, differential geometry, and algebraic geometry/topology. The math department head said that I should take the topics in algebra course so long as Katz is teaching it (which he is) and try to get into complex analysis so long as Stein is teaching it (which he is). So it looks like I'll end up shooting for those two, quantum I, experimental physics seminar, and either environmental law, religion and evolution, or North Korean history.
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Posted: Wed Nov 29, 2006 7:02 pm
Alright, I chose my classes, finally. Quantum mech, experimental physics, topics in algebra, fourier analysis and partial diff eq, and religion seminar. Even with all these classes, it feels like I'm behind everyone in math. Layra, you're a sophomore like me; what math did you take in high school, and what math have you had in college?
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