Favourite Field of Math?

For the discussion of math. Duh.

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What's your favourite field of math?

Algebra
8
9%
Analysis
16
19%
Applied Mathematics/Physics
15
17%
Discrete Mathematics
11
13%
Geometry
6
7%
Logic
6
7%
Number Theory
11
13%
Set Theory
2
2%
Statistics
2
2%
Topology
5
6%
Other
4
5%
 
Total votes: 86

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Favourite Field of Math?

Postby Qoppa » Tue Nov 27, 2007 5:53 pm UTC

What's your favourite field of math? Where abouts do your interests lie in the mathematical realm? I tried to include all the major fields in the poll (taken mostly from Wikipedia of course). I know there is some overlap, and I probably forgot to include your interest area in the poll, but do your best.

Try to avoid "other" unless your interests ABSOLUTELY cannot be generalized under one of the poll options; it's just so... uninformative.

I did not vote, because I have not had enough exposure yet to really any of them to really choose anything.

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Re: Favourite Field of Math?

Postby SimonM » Tue Nov 27, 2007 6:14 pm UTC

Currently, I am saying number theory. This it because it is a vastly interesting topic which is accessible to people who haven't had that much exposure to advanced mathematics.
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Re: Favourite Field of Math?

Postby Indon » Tue Nov 27, 2007 6:35 pm UTC

Number theory, because other fields within mathematics seem to me like too much work. Though, this may change in the future because the more I look into one branch of mathematics, the more I find myself needing to dabble in all the others to understand.
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Re: Favourite Field of Math?

Postby SimonM » Tue Nov 27, 2007 6:38 pm UTC

Indon wrote:Number theory, because other fields within mathematics seem to me like too much work. Though, this may change in the future because the more I look into one branch of mathematics, the more I find myself needing to dabble in all the others to understand.


Too much work? Number theory can have just as much or more work in it if you delve deep enough
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Re: Favourite Field of Math?

Postby crazyjimbo » Tue Nov 27, 2007 6:41 pm UTC

I like the finite field F47 myself. [/obligatory bad pun]

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Re: Favourite Field of Math?

Postby Indon » Tue Nov 27, 2007 6:56 pm UTC

SimonM wrote:Too much work? Number theory can have just as much or more work in it if you delve deep enough


I think it's bias from high school. Yes, number theory is complex. But it doesn't seem like work; I enjoy playing around with it.
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Re: Favourite Field of Math?

Postby SimonM » Tue Nov 27, 2007 7:22 pm UTC

Indon wrote:
SimonM wrote:Too much work? Number theory can have just as much or more work in it if you delve deep enough


I think it's bias from high school. Yes, number theory is complex. But it doesn't seem like work; I enjoy playing around with it.


Fortunately, I have reached that level with all maths (except Geometry dammit). I can pretty much find something to play with in all the bits I study
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Re: Favourite Field of Math?

Postby Govalant » Tue Nov 27, 2007 7:38 pm UTC

Analysis, which can be applicated to pretty much all physics.
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Re: Favourite Field of Math?

Postby JayDee » Tue Nov 27, 2007 11:23 pm UTC

Geometry, ever since high school. Especially since late high school, where I seemed to be the only one any good at Geometry.

I've just enrolled in first year maths for Uni though, so give me a couple of years and my answer might change.

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Re: Favourite Field of Math?

Postby Buttons » Tue Nov 27, 2007 11:54 pm UTC

Govalant wrote:Analysis, which can be applicated to pretty much all physics.

Really? I don't really know a lot of physics, but analysis always struck me as very non-applied. Can you give an example?

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Re: Favourite Field of Math?

Postby Govalant » Wed Nov 28, 2007 12:33 am UTC

Buttons wrote:
Govalant wrote:Analysis, which can be applicated to pretty much all physics.

Really? I don't really know a lot of physics, but analysis always struck me as very non-applied. Can you give an example?



Maybe I exagerated with *all* of physics. But just a while ago I calculated the speed necessary to shoot a rocket so that it nevers comes back to earth. Since gravity changes with displacement is sort of tricky, but once you see it you realize it is easy (Without air drag. I'm not sure how complicated it would be with friction.)

Also it can be aplicated to the most basic kinematics. The displacement of a falling object is 1/2 * g* t^2 because that is the second antiderivative of g, which would be the acceleration.

I'm not sure if series would come in handy, but calculus is certain to be a useful tool.
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Re: Favourite Field of Math?

Postby Buttons » Wed Nov 28, 2007 12:37 am UTC

Oh, okay. I guess I don't really think of calculus when I hear "analysis", although obviously the latter is the foundation of the former.

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Re: Favourite Field of Math?

Postby mudge » Wed Nov 28, 2007 3:44 am UTC

I can't decide between geometry or topology! I can't decide! BRAIN ANEURYSM!
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Re: Favourite Field of Math?

Postby btilly » Wed Nov 28, 2007 3:59 am UTC

Buttons wrote:
Govalant wrote:Analysis, which can be applicated to pretty much all physics.

Really? I don't really know a lot of physics, but analysis always struck me as very non-applied. Can you give an example?

I suspect Govalant was confused about what analysis is, but there are still a lot of connections between analysis and physics. For instance the entire theory of Hilbert spaces is the theory behind the Fourier transform, which is very important for physics.

The truth is that analysis shows up again in various areas of theoretical physics. Entire books can and have been written discussing some of the connections. For instance near the bottom of http://math.ucr.edu/home/baez/books.html you'll find a recommendation that Methods of Modern Mathematical Physics is an excellent 4-volume reference on connections between analysis and physics.

(This doesn't, of course, mean that most physicists need to learn analysis. But there are connections between the fields.)
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Re: Favourite Field of Math?

Postby Nimz » Wed Nov 28, 2007 7:08 am UTC

I chose Discrete Mathematics. It's a close call with Number Theory and Topology being runners up. The field of my current research is graph theory, which could probably be categorised under combinatorics and topology. However, since most of the articles I've found concerning the subject come from journals with titles like Discrete Mathematics and Discrete Applied Mathematics, I figured Discrete Mathematics (which would cover combinatorics) fits better than Topology. Number Theory is a runner up because I really like seeing the relationships between numbers. I've seen a few glimpses of the crazy shit Euler did to work out a lot of the Number Theory results we have. Pretty cool nonetheless.
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Re: Favourite Field of Math?

Postby btilly » Wed Nov 28, 2007 7:38 am UTC

Nimz wrote:I chose Discrete Mathematics. It's a close call with Number Theory and Topology being runners up. The field of my current research is graph theory, which could probably be categorised under combinatorics and topology. However, since most of the articles I've found concerning the subject come from journals with titles like Discrete Mathematics and Discrete Applied Mathematics, I figured Discrete Mathematics (which would cover combinatorics) fits better than Topology. Number Theory is a runner up because I really like seeing the relationships between numbers. I've seen a few glimpses of the crazy shit Euler did to work out a lot of the Number Theory results we have. Pretty cool nonetheless.


Um, I would not categorize graph theory as being related to topology in any significant way.

It does have a lot of connections with combinatorics. In fact http://www.math.niu.edu/~rusin/known-math/ groups the two of them together.
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Re: Favourite Field of Math?

Postby dp » Wed Nov 28, 2007 11:40 am UTC

I went for other. I'm interested in numerical analysis, but I'm not all that interested applying it. What I do falls somewhere in the analysis / applied / statistics / probability area. I also want to say that I have always found number theory dull.

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Re: Favourite Field of Math?

Postby Kalak_z » Wed Nov 28, 2007 11:43 am UTC

btilly wrote:Um, I would not categorize graph theory as being related to topology in any significant way.


Then you haven't done enough topology and graph theory. Euler's formula (vertices -edges + faces = 2) tells us something about the euclidean plane as well as about the graph that we have drawn on it. If I take the same graph and draw it on the torus so that it wraps around, then Euler's formula changes to vertices - edges + faces = 1. The number on the right hand side of the equation is referred to as the Euler Characteristic of the surface, and is a topological invariant.

That having been said, my understanding is that the branch of graph theory dealing with embedding graphs on surfaces is not heavily studied anymore. So, there is a good chance that it will not be covered in a graph theory class.

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Re: Favourite Field of Math?

Postby Ended » Wed Nov 28, 2007 2:14 pm UTC

btilly wrote:I suspect Govalant was confused about what analysis is, but there are still a lot of connections between analysis and physics. For instance the entire theory of Hilbert spaces is the theory behind the Fourier transform...

And indeed behind quantum mechanics.

I chose analysis, although I have quite broad interests and enjoy most areas in one way or another (except for statistics. Evil, evil statistics.).
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Re: Favourite Field of Math?

Postby Robin S » Wed Nov 28, 2007 3:17 pm UTC

I have a broad range of interests in Maths. The main areas which appeal to me are, in no particular order, Chaos Theory, Algebra, Theoretical Physics and Set Theory, but I am also moderately interested in most of the fields listed in the poll.
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Re: Favourite Field of Math?

Postby Govalant » Wed Nov 28, 2007 6:08 pm UTC

btilly wrote:I suspect Govalant was confused about what analysis is, but there are still a lot of connections between analysis and physics. For instance the entire theory of Hilbert spaces is the theory behind the Fourier transform, which is very important for physics.



Analysis includes calculus.
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Re: Favourite Field of Math?

Postby mike-l » Wed Nov 28, 2007 7:45 pm UTC

Blasphemy!

Surely if Set Theory is an option, Category Theory must also be! Besides, arrows are sexier than braces. I voted other.

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Re: Favourite Field of Math?

Postby 22/7 » Wed Nov 28, 2007 8:15 pm UTC

Applied. I'm an engineer, it's what I do? It's also the way my brain works.
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Re: Favourite Field of Math?

Postby Cosmologicon » Wed Nov 28, 2007 9:05 pm UTC

Govalant wrote:Analysis includes calculus.

While that's a very reasonable point of view, I'd like to know if that's what the OP had in mind, because I love calculus and hate analysis.

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Re: Favourite Field of Math?

Postby Robin S » Wed Nov 28, 2007 9:08 pm UTC

Calculus can be derived from analysis, but does that make it analysis if it's studied alone? Practically all of maths can be derived from set theory, but I can't think of anyone who'd call it all set theory.
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Re: Favourite Field of Math?

Postby Unakau » Wed Nov 28, 2007 9:29 pm UTC

Due to my incredible laziness, I did not get the B- that I needed to pass geometry the first time around.

The second time around I passed with flying colors, but there was more to it. I played around with geometry when I was bored in the class, and I ended up playing with dimensions greater than three a lot. I made theories that I later looked up to be correct, and I have been interested in geometry ever since. Ah, geometry. The science of beauty.

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Re: Favourite Field of Math?

Postby Robin S » Wed Nov 28, 2007 9:33 pm UTC

*visual beauty.

Also, it's not really a science, but I was just trying to point out that many people find Maths in itself beautiful - not to mention forms of art which are not strictly, or not at all, beautiful.

Ok, getting us back on topic: what category, out of those listed, would chaos theory fall under (if any)?
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Re: Favourite Field of Math?

Postby btilly » Wed Nov 28, 2007 9:57 pm UTC

Kalak_z wrote:
btilly wrote:Um, I would not categorize graph theory as being related to topology in any significant way.

Then you haven't done enough topology and graph theory. Euler's formula (vertices -edges + faces = 2) tells us something about the euclidean plane as well as about the graph that we have drawn on it. If I take the same graph and draw it on the torus so that it wraps around, then Euler's formula changes to vertices - edges + faces = 1. The number on the right hand side of the equation is referred to as the Euler Characteristic of the surface, and is a topological invariant.

I don't think the question is whether I've studied either subject enough. In fact I've done enough of both to do well on my quals in both topology and combinatorics. (Admittedly that involved a lot more topology than graph theory.) And I'm sure that I could have specialized in either without learning any more than I know now about the connections between them.

That said, I'm familiar with Euler's formula as both a theorem in geometry and topology. But I've never seen it presented as a theorem in graph theory. Indeed I would have never have thought of it that way since it involves the concept of faces, which is not a part of the definition of a graph.

Kalak_z wrote:That having been said, my understanding is that the branch of graph theory dealing with embedding graphs on surfaces is not heavily studied anymore. So, there is a good chance that it will not be covered in a graph theory class.

Beyond some discussion of planar graphs and the four colour theorem, it didn't come up. (On my own I did go through the proof that maps on a torus can be coloured with 7 colours.)

I never encountered the subject in topology.

However I'll note that for any pair of basic areas of mathematics it is generally possible to find a specialty that combines them in some interesting way. This does not, however, mean that the two areas are related in any fundamental way. Just that someone has explored a combination of the two.

I don't know the history of the two. If you inform me that historically there were a lot of connections, I'll believe you. But there are lots of cases where historically two fields were connected however today they are entirely independent. Even if there is a shared history, I'd argue that today they are different fields. (Another example of two fields that did that would be the study of symmetric polynomials in combinatorics and algebraic integers in number theory. Both fields have forgotten that they came from the interesting properties of polynomials that are symmetric in the roots of some other polynomials, and now have nothing to do with each other.)
Govalant wrote:
btilly wrote:I suspect Govalant was confused about what analysis is, but there are still a lot of connections between analysis and physics. For instance the entire theory of Hilbert spaces is the theory behind the Fourier transform, which is very important for physics.


Analysis includes calculus.

Analysis includes calculus in the same way that algebra includes arithmetic.
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Re: Favourite Field of Math?

Postby Robin S » Wed Nov 28, 2007 9:59 pm UTC

Or, perhaps, number theory, since there isn't really much to prove about arithmetic itself, as such.
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Re: Favourite Field of Math?

Postby Cosmologicon » Wed Nov 28, 2007 10:06 pm UTC

See, now, I think of number theory as the same as arithmetic, so this just heightens my confusion. (I do, however, understand what you'd mean by a more limited definition of arithmetic, I just wish for my sake you'd explain yourself when you use it.) I certainly don't think of the Fundamental Theorem of Arithmetic as something that's part of algebra. So, btilly, what are you saying? In what way does algebra comprise arithmetic?

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Re: Favourite Field of Math?

Postby Robin S » Wed Nov 28, 2007 10:08 pm UTC

Presumably in the sense that "arithmetic" (meaning number theory, in this case) is a subset of algebra, as it can be derived using the axioms of algebra combined with some more specific ones.
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Re: Favourite Field of Math?

Postby Cosmologicon » Wed Nov 28, 2007 10:11 pm UTC

So algebra presumably includes calculus too?

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Re: Favourite Field of Math?

Postby Robin S » Wed Nov 28, 2007 10:12 pm UTC

Algebra contains a huge proportion of Maths, if you use btilly's definition of "includes".
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Re: Favourite Field of Math?

Postby btilly » Wed Nov 28, 2007 10:49 pm UTC

Cosmologicon wrote:See, now, I think of number theory as the same as arithmetic, so this just heightens my confusion. (I do, however, understand what you'd mean by a more limited definition of arithmetic, I just wish for my sake you'd explain yourself when you use it.) I certainly don't think of the Fundamental Theorem of Arithmetic as something that's part of algebra. So, btilly, what are you saying? In what way does algebra comprise arithmetic?

By arithmetic I meant "elementary arithmetic". Meaning being able to add, subtract, multiply and divide common representations of ordinary numbers, and knowing the basic properties of said operations.

The properties are, of course, the laws of algebra. The operations do, of course, rely heavily on the laws of algebra. And, of course, without some sort of arithmetic we'd lack the basic substance of algebra. The two subjects are intimately connected.

However I don't normally think of arithmetic as a part of algebra. It is more of a practical application that you assume is understood to be a foundation. I suspect other people think of it similarly.
Robin S wrote:Or, perhaps, number theory, since there isn't really much to prove about arithmetic itself, as such.

From the point of view of analysis, there isn't really much to prove about calculus itself, as such. Which is explicitly part of the reason I drew that parallel.

That said, arithmetic is a far more complicated subject than most people realize. If you don't believe me, try to get from the Peano axioms to the rational numbers and you'll suddenly appreciate how much you take for granted. But the interesting research on it these days tends to happen in the CS department. For instance google for Karatsuba multiplication to learn something about arithmetic that you never saw in elementary school.
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Re: Favourite Field of Math?

Postby Bassoon » Wed Nov 28, 2007 11:12 pm UTC

While I've selected Applied Math/Physics, I like quite a few topics. Calculus, Logic, Geometry, they're all good. So long as I'm dealing with numbers.

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Re: Favourite Field of Math?

Postby Hedos » Thu Nov 29, 2007 12:13 am UTC

That pool is perfectly invalid because you did not make the distinction between analysis (which is advanced math and fits in the list) and calculus, which does not fit in with the rest.

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Re: Favourite Field of Math?

Postby Buttons » Thu Nov 29, 2007 12:47 am UTC

Hedos wrote:That pool is perfectly invalid because you did not make the distinction between analysis (which is advanced math and fits in the list) and calculus, which does not fit in with the rest.

A compromise! If you like calculus because it's useful, say "Applied Mathematics/Physics." If you like calculus because it's beautiful, say "Analysis." If you like it because it's the only math you've really taken, go take more math; you're not allowed to vote.

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Re: Favourite Field of Math?

Postby adlaiff6 » Thu Nov 29, 2007 4:04 am UTC

Algebra, because I like more than just Number Theory.
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Re: Favourite Field of Math?

Postby LeftHandOfTheDevil » Thu Nov 29, 2007 4:05 pm UTC

I liked Discrete Mathematics last year, mainly because my Precalculus teacher, who also taught some Discrete Mathematics, had us mostly teach ourselves that.

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Re: Favourite Field of Math?

Postby Robin S » Thu Nov 29, 2007 5:59 pm UTC

Well, draughts is now a solved game :D
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