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1. Advanced mathematics
Having only completed A-level maths not even further, but with a deep interest in pure maths I was wondering what I could study to be able to understand things like manifolds, and especially the poincare conjecture.
http://en.wikipedia.org/wiki/Poincar...cture#Solution
2. Re: Advanced mathematics
(Original post by kingkongjaffa)
Having only completed A-level maths not even further, but with a deep interest in pure maths I was wondering what I could study to be able to understand things like manifolds, and especially the poincare conjecture.
http://en.wikipedia.org/wiki/Poincar...cture#Solution
Just learn the definitions and never forget them. My only advice.
3. Re: Advanced mathematics
(Original post by kingkongjaffa)
Having only completed A-level maths not even further, but with a deep interest in pure maths I was wondering what I could study to be able to understand things like manifolds, and especially the poincare conjecture.
http://en.wikipedia.org/wiki/Poincar...cture#Solution
You can provided you do some self teaching!
4. Re: Advanced mathematics
(Original post by handsome7654)
You can provided you do some a lot of self teaching!
Read the wikipedia articles first, see what you think
5. Re: Advanced mathematics
(Original post by kingkongjaffa)
Having only completed A-level maths not even further, but with a deep interest in pure maths I was wondering what I could study to be able to understand things like manifolds, and especially the poincare conjecture.
http://en.wikipedia.org/wiki/Poincar...cture#Solution
you could
watch the 2010 research conference
wander through Stillwell's translation of analysis situs

I'm halfway through a degree and haven't really done manifolds but have a few bits of things about them that I can dig out later. It would be the blind leading the blind tho hope somebody more knowledgeable steps up.
6. Re: Advanced mathematics
...You want to understand the solution to the Poincare conjecture? There's only a few hundred or so mathematicians that have ever been able to understand that proof (and it took them a loong time to verify his proof was even correct, due to its complexity), since Perelman used a bunch of quite esoteric areas of mathematics and theoretical physics to prove it - a collection of knowledge not usually attained by a single person since it's very in depth across many disciplines. Might be the kind of thing a masters student specialising in differential geometry and topology would look at.
7. Re: Advanced mathematics
(Original post by FireGarden)
...You want to understand the solution to the Poincare conjecture? There's only a few hundred or so mathematicians that have ever been able to understand that proof (and it took them a loong time to verify his proof was even correct, due to its complexity), since Perelman used a bunch of quite esoteric areas of mathematics and theoretical physics to prove it - a collection of knowledge not usually attained by a single person since it's very in depth across many disciplines. Might be the kind of thing a masters student specialising in differential geometry and topology would look at.
you sound like you know what is required to understand it though.

I don't want to understand the solution per se just the style of maths that led to it in areas of topology?

Or is it just beyond the scope of comprehension unless you have an undergraduate degree in pure mathematics ?
8. Re: Advanced mathematics
(Original post by kingkongjaffa)
you sound like you know what is required to understand it though.

I don't want to understand the solution per se just the style of maths that led to it in areas of topology?

Or is it just beyond the scope of comprehension unless you have an undergraduate degree in pure mathematics ?
It is probably beyond the scope of comprehension even if you have an undergraduate degree.
9. Re: Advanced mathematics
An undergraduate degree wouldnt get you to the level required to understand the proof; generally you dont cover things like differential geometry/manifolds/algebraic topology until masters/phd level. However its possible to get a very rough idea about what sort of questions are being asked in modern geometry/topology/etc without going extremely deep into the mathematics, and some universities do teach very basic courses to final year undergraduates (either on their own, or as part of a class on general relativity)

Unless you are willing to devote several months/years of study, your best bet would probably be to try and find a popscience book that explains it in relatively nontechnical terms.ast Theorem, so I imagine you'll start to see more on the Poincare conjecture springing up in the next few years (if there arent any already) Ive never read it so I'm not recommending it specifically, but you want something like this: http://www.amazon.com/The-Poincare-C...are+conjecture

If you already had a Masters/MSci/PhD in pure mathematics then you might be able to approach it on a more technical level through this (which again I havent read): http://www.amazon.com/Ricci-Poincare...are+conjecture
Last edited by poohat; 29-06-2012 at 18:19.
10. Re: Advanced mathematics
(Original post by kingkongjaffa)
you sound like you know what is required to understand it though.

I don't want to understand the solution per se just the style of maths that led to it in areas of topology?

Or is it just beyond the scope of comprehension unless you have an undergraduate degree in pure mathematics ?
It depends what level of comprehension you want. You can take a lot of the results for granted and try to get to the nub of it (the conjecture rather than the proof) it uses bits of different areas of mathematics that might be confusing without context though.
11. Re: Advanced mathematics
(Original post by poohat)
An undergraduate degree wouldnt get you to the level required to understand the proof; generally you dont cover things like differential geometry/manifolds/algebraic topology until masters/phd level.
Differential Geometry to some extent used to be on the MEI Additional Further Maths syllabus, in P6! Also, most good universities would cover such topics in the third year.

(Original post by kingkongjaffa)
you sound like you know what is required to understand it though.

I don't want to understand the solution per se just the style of maths that led to it in areas of topology?

Or is it just beyond the scope of comprehension unless you have an undergraduate degree in pure mathematics ?
I know what is required, but I don't know what is required.. if that makes sense. I have a very rough, qualitative 'understanding' of topology, and as far as the Poincare conjecture goes, I could potentially explain with only slight contempt from a more experienced mathematician what the theorem means. Definitely for it's lower-dimensional analogue for the 2-sphere! I've never done any proper differential geometry outside of that P6 unit I mentioned, either (I've only just finished my first year!); but again I have some (and this time, better) understanding of some of the ideas from studying physics, in this case Special Relativity, where the setting of Minkowski space is a 'Pseudo-Riemannian' manifold, which is just a slight generalisation of normal Riemannian Manifolds, a concept studied in DG.

In all, you would definitely learn the "style of maths" that led to such things with an undergraduate degree, since IIRC most universities offer some form of a course in topology. To understand the proof of the theorem on the other hand, you'd likely need to go in depth for it in a masters.
12. Re: Advanced mathematics
(Original post by FireGarden)
Differential Geometry to some extent used to be on the MEI Additional Further Maths syllabus, in P6! Also, most good universities would cover such topics in the third year.
not mine

I know what is required, but I don't know what is required.. if that makes sense. I have a very rough, qualitative 'understanding' of topology, and as far as the Poincare conjecture goes, I could potentially explain with only slight contempt from a more experienced mathematician what the theorem means. Definitely for it's lower-dimensional analogue for the 2-sphere! I've never done any proper differential geometry outside of that P6 unit I mentioned, either (I've only just finished my first year!); but again I have some (and this time, better) understanding of some of the ideas from studying physics, in this case Special Relativity, where the setting of Minkowski space is a 'Pseudo-Riemannian' manifold, which is just a slight generalisation of normal Riemannian Manifolds, a concept studied in DG.

In all, you would definitely learn the "style of maths" that led to such things with an undergraduate degree, since IIRC most universities offer some form of a course in topology. To understand the proof of the theorem on the other hand, you'd likely need to go in depth for it in a masters.
it's not my thread but any kind of 2-sphere runthrough would be awesome
13. Re: Advanced mathematics
(Original post by sputum)
it's not my thread but any kind of 2-sphere runthrough would be awesome
As far as weird abstract maths goes, the problem for 2D surfaces actually isn't very hard to understand in a heuristic sense, but for 3D you'd have problems visualising it I would think.

So imagine you have a surface without bound - so that if you carry on in one direction forever, you stay on the surface; maybe at a point infinitely far, or maybe back where you started, depending on the shape of the surface, all we care about is that you won't find some line that you can cross to leave the surface. Now, imagine making a loop on this surface. Can you continuously shrink the loop until it becomes just a point? Well, on a flat surface you obviously can and you can on a sphere, too; and that's the idea - for every surface you can do this on, it is then said to be homeomorphic to the 2-sphere, or in other words, has the same kind of connectedness between the points on the surface, as a regular sphere has.

An example of a surface you cannot do this to, and so is not homeomorphic to the sphere, is the torus. Clearly, since it has a hole, and it will prevent some loops from being able to deform to a point. To visualise this is easiest by thinking about tying a string through the hole of a donut - you can't tighten the string into a knot without slicing through the donut. On a plane however, it has no trouble going to a knot, and on a sphere (pretend it's the earth), the loop would go from being the equator of the earth to being one of the poles, as a point. If you're interested, a good reason to enjoy this relation for the case between a 2D plane and a sphere, is it gives a solid basis to understand the Riemann Sphere, the geometrical representation of the extended complex numbers.

Now the Poincare conjecture is the same deal, with 3D spaces rather than 2D surfaces; whatever this idea even means for a 3D space..
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