Coolest Topic in Math?

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discgolf
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Coolest Topic in Math?

Postby discgolf » Wed Jan 20, 2010 1:47 am UTC

I just recently discovered that there is a math/science research competition in which I would love to participate. I really enjoy math as do my team members, but we're a little at a loss for relatively interesting and new topics in math that would be good fodder for research. So, what are the most interesting unsolved or researchable topics in math that we would be able to research?

Details: We have about 7 months to research; our current math knowledge is limited to the level of AP calculus at high school. One of the team members has suggested fractals and self-similarity; would it be plausible to conduct research into those subjects?

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t0rajir0u
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Re: Coolest Topic in Math?

Postby t0rajir0u » Wed Jan 20, 2010 3:58 am UTC

This is a hard question to answer impersonally. Your best bet is to find a mentor, say a professor from a nearby college, who can suggest some things to get you started and help you explore your interests. I'm a little worried, though; you might not have enough time if none of you have been exposed to any serious mathematics outside of high school. Research is much harder than it sounds, and at the high school level it almost never produces anything useful or interesting, even if the researcher has considerable background. (That's not to say you shouldn't enthusiastically pursue your interests - but it's good to have realistic expectations.)

(Disclaimer: I entered such a competition in high school with research I produced at the Research Science Institute. It was not useful or interesting.)

Edit: Actually, let me just say the following. I don't think you should attempt this at all given that you don't already have exposure to proof-based mathematics. What you should do, instead, is expose yourself to proof-based mathematics.

DavCrav
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Re: Coolest Topic in Math?

Postby DavCrav » Wed Jan 20, 2010 7:47 pm UTC

I agree with t0rajir0u. Even third-year undergraduate research projects (such as in my institution) are generally on uninteresting and unimportant areas, with very few exceptions (and generally the person involved was far beyond the average third-year level).

There are some really interesting areas of mathematics, but in general the low-hanging fruit has been plucked by generations of eager undergraduates, I'm afraid. Of course, if you can find something, good luck, but (and I see this with PhD students as well) getting all eager to do research too early often leads to disappointment when reality sets in.

achan1058
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Re: Coolest Topic in Math?

Postby achan1058 » Wed Jan 20, 2010 8:24 pm UTC

It depends on who the target audience is, as well as what your fellow competitors are. If it was for/to high school students, things like fractals is a nice thing to do. They are visually appealing, at the very least. You are also not really expected to do original research, but rather research upon the known results. On the other hand, if this is for a university or higher level, I agree with the above posters on not to enter it at your stage.

discgolf
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Re: Coolest Topic in Math?

Postby discgolf » Wed Jan 20, 2010 11:07 pm UTC

Thanks for the input everyone! After discussing it with a teacher and my team, we've decided to try to do something in a science-field. Still, I looked up some higher level math on wikipedia to see what it was like, and that's pretty incredible stuff.

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t0rajir0u
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Re: Coolest Topic in Math?

Postby t0rajir0u » Wed Jan 20, 2010 11:18 pm UTC

Science is good. The nice thing about trying out science research is that, even if you don't find what you were looking for, you at least have data that you can talk about. Unsuccessful mathematical research, on the other hand, gives you nothing to talk about. This is a problem sometimes encountered at RSI.

If you want to get some sense of what mathematical research is like, I highly recommend you read some math blogs. My favorite are Terence Tao's and Tim Gowers', both Fields medalists. While some of their posts are highly technical, they are both very clear writers, and many posts are quite accessible. The discussions that go on on these blogs are a great way to peek inside the minds of real mathematicians. (Make sure you start at the beginning of both blogs; the recent posts in both are somewhat technical.)

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effective_
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Re: Coolest Topic in Math?

Postby effective_ » Wed Jan 20, 2010 11:41 pm UTC

It would be if you created a short story in pure mathematics/physics. You might not be exposed to all of the formulas but if you studied a bit you guys could get it. For example, start with an apple hanging from a tree. Calculate it's potential energy as it hangs from the tree. The apple falls and then builds kenetic energy. Tell us how fast the apple is falling and then the force it generates when it hits the ground. The apple tree was on a hill so then the apple beings to rolls down the hill. How fast is it traveling down the hill.. how much friction is working against it? Oh no, there's a lake.. but it's winter.. so the apple slides across the lake. How far does it slide? It hits a weak point in the ice, is it going to break through? How much mass would the apple need to break through the surface of 1/2 centimeter of ice.. LOL it's kind of stupid but it would be cool in my opinion. Plus it's not too far past your level. Have all of the formulas from the apples journey laid out. Then you can tell the story with English while people stier at your formulas.
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