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 Posted: Fri Mar 09, 2007 9:00 am
Here's a question that's pretty open to interpretation. What do you think are the fields of math in which every major should have an introduction? More than an introduction? What about in physics, or other sciences (graph theory, for instance, is pretty essential for computer science)?
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Posted: Fri Mar 09, 2007 10:39 am
topology up to tychonoff's theorem.
real analysis up to bolzano-weirerstrass, but maybe also talking about uniform continuity and swapping integrals and sums.
measure theory up to fubini's. (edit: maybe martingales as well)
a smattering of finite, discrete and combinatorics.
group theory plus more general algebra.
axiomatic set theory, maybe, but I don't see it as essential (Euler didn't have it, Gauss didn't have it, Archimedes didn't have it etc though they didn't have the other stuff either...)
Edit: that's my list for a pure mathematics major. no idea what an applied type person should go for.
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Posted: Fri Mar 09, 2007 10:47 am
jestingrabbit topology up to tychonoff's theorem. real analysis up to bolzano-weirerstrass, but maybe also talking about uniform continuity and swapping integrals and sums. measure theory up to fubini's. (edit: maybe martingales as well) a smattering of finite, discrete and combinatorics. group theory plus more general algebra. axiomatic set theory, maybe, but I don't see it as essential (Euler didn't have it, Gauss didn't have it, Archimedes didn't have it etc though they didn't have the other stuff either...) Edit: that's my list for a pure mathematics major. no idea what an applied type person should go for. Basic statistics for certain. Perhaps a little bit of complexity theory...
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Posted: Fri Mar 09, 2007 11:02 am
yeah, some sort of treatment of dynamics too, but its not like there's a signpost sort of result... or maybe there is and I don't know it. I want to say the Birkhoff ergodic theorem (or Hurewicz ergodic theorem would be even better) but thats because I'm deeply into ergodic theory (and nonsingular ergodic theory in particular).
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Posted: Sun Mar 11, 2007 9:04 am
The track most of my friends have followed is
All basic calculus (up through generalized Stokes' theorem) plus basic linear algebra
Group theory, ring theory, field theory, Galois theory (though sometimes all of that squashed into a single class that vaguely covers set theory)
Fourier analysis on finite intervals, the real line, Euclidean spaces, and finite fields
Complex analysis (basic stuff)
Differential geometry
Real analysis, focusing on Lebesgue and measure theory
Point-set topology, usually followed by algebraic topology
Of course, those are in no particular order, save that Fourier-complex-real analysis is a 3-semester sequence. Oh, and that basic calc+linear algebra comes first and is required to get into the math program.
Oddly enough, number theory, statistics, and combinatorics are notably absent. Most of the people I know who take those classes are those who happen to like math a lot, but are majoring in computer science. That, and there are a bunch of physics majors who take Fourier analysis every year, and sometimes lower-level complex analysis. But after that, they kind of shun math classes.
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Posted: Sun Mar 11, 2007 9:15 am
Swordmaster Dragon Here's a question that's pretty open to interpretation. What do you think are the fields of math in which every major should have an introduction? More than an introduction? What about in physics, or other sciences (graph theory, for instance, is pretty essential for computer science)? I can only speak for physicist. Real Analysis! Statistics! Complex analysis, group theory, and set theory become useful later on if you delve into electromagnetism and Quantum electrodynamics. An introduction to tensors and fibre bundles and whatnot is useful for cosmology, Quantum information, and the study of structural properties (though that's more for engineers). Fourier analysis is essential for optics.
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Posted: Mon Mar 12, 2007 12:42 pm
Ah, right, physics I had forgotten about that. For physics, I think that
Real analysis, spec. diff eq and Fourier
Complex analysis
Group theory
Differential geometry
are the essential classes. I don't know so much about statistics, though. I'm used to statistics either being given from an entirely computational approach or an entirely combinatoric approach, and I don't think either is *quite* right for physics, given modern interpretations of quantum mech.
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Posted: Tue Mar 13, 2007 8:38 am
Swordmaster Dragon Ah, right, physics I had forgotten about that. For physics, I think that Real analysis, spec. diff eq and Fourier Complex analysis Group theory Differential geometry are the essential classes. I don't know so much about statistics, though. I'm used to statistics either being given from an entirely computational approach or an entirely combinatoric approach, and I don't think either is *quite* right for physics, given modern interpretations of quantum mech. Statistics becomes useful when dealing with data analysis. You need to know how to interpret your data and estimate the error and reliability etc.
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Posted: Thu Mar 15, 2007 7:41 am
I'm not saying it's not important; I'm saying that I don't think we offer a class that teaches statistics in a way that's awesome for sciences. I think there's an economics class in statistical interpretation, and a math class in combinatorics. Neither of those are quite what you'd want.
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