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Mathematics Essay

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    I have to write an 'extended essay', although its only about 2000/2500 words, for a sort of school project and i was looking for some advice. It can be on absolutely any topic but since i want to study mathematics at university, that's probably my topic. I want something that i will potentially be able to speak about and impress in an interview (hopefully applying to oxford) but it also needs to be interesting for a reader, who wouldn't know much about the topic.

    I was thinking possibly something about the Riemann hypothesis, a brief history and perhaps whether its reasonable to assume. Or maybe an introduction into group theory but i wasn't sure if this would be particularly interesting. It needs to be sufficiently mathematical but also enjoyable for a reader. Another idea was an essay on mathematical proofs of implausible ideas and mention the banach-tarski paradox and others but this might be too general and i think it would be better to concentrate on a small area of maths and learn a lot about it.

    So please, i'm open to all help and ideas. What my teachers did stress is the essay should have a point, and what would help with this would be phrasing the title as a question. But this sort of implied that you should have a sort of argument, but i thought a main point of maths was that there were no points of view, its proven fact... Help please And thank you.
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    If you want it both mathsy and accessable I would avoid group theory and the riemann hypothesis. You could consider something more abstract like game theory, something people can easily relate to and picture.

    Who is going to be reading it? The readers level of mathematics knowledge/apltitude will affect what topics you can do easily.
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    what about that guy (sorry - can`t remember his name - went to cambridge i think) who proved fermat`s last theorem with galois theory?
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    Since this project is basically going to be used to help with your uni interviews, here are my tips:

    - Don't aim to make it accessible. Only like 2 people will read it in full (sad but true), and at least one of them will be a marker. If they're too incompetent to understand what you've written, that's their loss, and you could complain if they then gave you a bad mark. If you make it accessible, you risk dumbing down the mathematics too much. My guess is that 1250 words of solid maths translates into about 2500 words of dumbed down maths. It'd just be less useful for your uni applications - you'd have solid maths to talk about. Dumbing down has its place - it means the ideas are much easier for you to articulate, in say, an interview - but it shouldn't be done at the expense of actual content. Go above the word limit if you need to.

    - Don't pick maths that is too hard. If you write about the Riemann Hypothesis, or Fermat's Last Theorem, or somesuch advanced mathematical idea that's famous simply for being easy to understand and difficult to reason about, you won't be able to go into much detail. You won't be making any progress, you'll just be restating ideas you read in books. For most interviews, it's no better than simply stating on your PS that you read the books. It'll make for a very short project, and to be perfectly honest, I bet about three quarters of the people who take an extended essay to their maths interview have looked at one of Fermat's Last Theorem, the Riemann Hypothesis or game theory. They're stale.

    - Try to pick something that can be split into a nice three sentence summary. For example 'I identified that Rubik's cube can be solved using aspects of group theory. I then used the knowledge obtained in order to derive an algorithm for solving Rubik's cube - I then used the same principles to extend this algorithm to a 4-dimensional Rubik hypercube.' Now, I'm not suggesting you do that - it's probably a lot harder than it sounds - but it's the sort of thing that could impress an interviewer - far more so than 'I condensed Simon Singh's book 'Fermat's Last Theorem' into five pages'.

    - Do something different, but not too different. There are loads of blogs on the internet from famous mathematicians, and many of these identify really interesting problems or open problems that could be interesting avenues for your work. I particularly recommend 'Godel's Lost Letter' - it's a compsci blog but pretty much all it talks about is open questions in mathematics. If you pick something too different, you'd risk your interviewer not knowing what you are talking about and needing further explanation - and that's a massive can of worms that I experienced more than once.

    I did an extended project last year, and I'm studying CompSci+Maths next year so YMMV, but it came up in all of my interviews, including my Cambridge one. Over the applications cycle, I was probably in about two hour's worth of interviews, and I probably spent about a quarter of that time talking about my extended project. The academics seemed very impressed with what I'd done. On the other hand, people at my school who did pretty basic projects barely talked about them at all.
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Updated: April 26, 2013
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