Interesting Topics

[Dave]

I'm coming up to a stage where I need to write an essay about some topic in Mathematics. Now, before you ask, the purpose of this thread is not to say "omg he's trying to get us to write his essay n00b fs". I'm trying to look at particularly interesting topics in mathematics to formulate some kind of essay title - and no, I don't want you to give me titles. Contrary to some common misconceptions about students, I want (and need) to do this by myself.

 

I've had a fair amount of mathematical experience now, but I'm struggling to find some topic I find interesting enough to write an essay on. If anyone has any suggestions, then that would be much appreciated.

[haggy]

What is the scope of the essay? Are you expected to give some sort of an overview of a particular field and its applications?

[matt grime]

First, what mathematics have you done? Standard topics would include RSA encryption if you've done basic ring theory, knots if you've done any topology, stability of models of physical situations if you've done analysis and differential equations. Quite a nice one, if you ask me, and you've done some metric space stuff, is to explain how the CONTRACTION MAPPING THEOREM implies that differential equations indeed have the solutions you think of and no others.

 

If the essay is not proof based, more expository, then how about the classification of all compact surfaces according to genus and orientability, or even the classification of finite simple groups.

 

What about some foundational questions: the implications of the (mis)use of the axiom of choice and other axioms in set theory (every set can be well ordered with these things and this leads to the banach tarski paradox - that a sphere can be "cut" into some pieces, 6 I think, and these pieces may be rotated and translated and reassembled into a shape of strictly smaller volume - as well as other philosophical questions such as defining a game which apparently both players win with probability 1).

 

A general discussion on the nature of mathematics might start off with any of the issues raised in these articles:

 

http://www.dpmms.cam.ac.uk/~wtg10

 

and look at his expository articles.

 

A good one to think about might be "the definition of the Real Numbers", or how maths is often shows that something exists (indeed many things exist) yet doesn't give any way of constructing them.

[Dave]

Thanks for all the suggestions

 

I'm in my second year, and have done a fair bit of analysis (just starting to formally define Fourier Series), just done Sylow's Theorem in Algebra and have just finished line integrals/curl/divergence/Stokes's theorem/etc in Vector Analysis.

 

The essay itself is just to write about a particular topic - as long as it involves a fair amount of maths, it's fine. I suppose therein lies the difficulty - there's just too much to choose from. Some of the suggested topics are things along the lines of "Sylow's theorem and it's applications", "Matrices in Robotics" and "Construction of the reals using ... " (can't remember what exactly). All kinds of stuff.

 

Thanks again.

[5614]

why dont you just pick the funnest thing (to you) you have done so far in uni.

 

that way it will be advance level and you (and hopefully others) will it find it interesting.

 

all that needs is for you to think back on your last two years and say "whats been fun recently" and then answer it!

[Dave]

Well, more or less none of the maths we've been doing is "fun" really. We're only just starting to touch on new stuff because we went back to the beginning and started over properly with sequences, series, etc.

 

I'm looking at doing something about curl/divergence of a vector field, so I'll get some books out tomorrow and start reading. Joy of joys

[5614]

"Matrices in Robotics" sounded fun to me... although maybe i just liked the word robotics!

 

maybe you could do numbers in computers... bringing in binary & hexidecimal etc, although again, thats because i like computers!

 

what do you enjoy outside of maths? bring maths into that!

[Dave]

I've been looking into using maths for video codec compression, but it's a little beyond me. There are various things I've been looking into (fractals being one) but I don't know yet. I shall ponder for a while.

[haggy]

Maybe you could consider looking at some Coding Theory. Any electronic communication is reliant on Codes, so it is a field that has direct applications.

[5614]

coding... or encryption of anything. doesnt have to be video codecs; could be anything.

[Dave]

I know, but video/image codecs interest me for some peculiar reason. My favourite topic of the moment is fractals and their applications, but I feel I have to research this a bit more. Hypergeometric series also look quite interesting.

[5614]

well its the things which you call interesting which are the things which you would do an interesting (to you) project on!!!

 

so maybe post #11 is the answer to post #1