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I love interesting and deep mathematical results, but on the other hand I cannot object when someone says that most likely all these complicating abstract theorem will not make a change to human kind (apart from more complicating math schedules ;) )

So I'm wondering what were mathematical results from the last decades that were put into real practical use - I mean there would be a noticeable difference without them (this most of the time can be translated to less productivity or money)???

(maybe excluding advances in numerics and algorithms which is easy to imagine. also proving theorems which give insights but yet dont change the world doesnt count.)

I heard about wavelets as one example. Maybe there is something in cryptography? What else? Is there something that seriously is likely to make a change during this century?

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There were MO-threads on this, e.g.: Applications of mathematics and Real-world applications of mathematics, by arxiv subject area? – t.b. Sep 5 '11 at 23:29
Thanks! I'll read through these. However most example do not fit this questions. They are cryptography, algorithms, Maths not from the last decades (complex variables etc, basic PDE) or some less applicable stuff like modelling Zebra fur patterns. There are some good finds, yet I'd like to know what all the maths areas are for where as a physicist I don't have the slightest idea what they are about... :) Wavelets was a good example I heard somewhere else... – Gerenuk Sep 6 '11 at 0:03
"maybe excluding advances in numerics and algorithms which is easy to imagine." - okay, now you've just made the question tougher to answer... :D – J. M. Sep 6 '11 at 1:27
I'm sure there are many applications ;) But I'm asking questions to find out things I don't know :) – Gerenuk Sep 6 '11 at 9:18
up vote 3 down vote accepted

There are serious applications of algebraic topology in medical imaging and in robotics. If you want to learn about applied topology, a good place to start is Robert Ghrist's webpage -- he's got some free textbooks and things on there.

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Thanks - thats a good mention. Medical imaging I can vaguely imagine, and about robotics I'm surprised to learn (also mentioned in the links above). Yet, are these tools from say the last 30 years? – Gerenuk Sep 6 '11 at 9:18

Perhaps compressive sensing qualifies. For accounts of recent progress in mathematics see the series What's Happening in the Mathematical Sciences . The chapter on compressive sensing is here.

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Take a look at this.

I feel some of the unsolved problems in math are world changing and might help humanity in one way or the other.Example the poincare conjecture which was solved recently by perelman might be used extensively to understand shapes of different kinds.some say it might be even used to understand the shape of the universe.

This is also interesting.

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Give me a break! A few of the millennium prize problems (for instance, the Navier-Stokes one and PvsNP) would have practical applications, but most of them don't. This includes the Poincare conjecture -- can you point to a single concrete place where it has played a role in applied mathematics? – Adam Smith Sep 6 '11 at 6:48
I had watched a video on you tube regarding poincare conjecture,for some reason i'am unable to find it.i'll post it soon once i find it.I remember he talks about the curvature of universe and the role played by the poincare conjecture. – alok Sep 6 '11 at 9:09
I knew about them and I agree with Adam. I don't know details, but strictly speaking even Navier-Stokes (in the form presented) or P-NP might not change anything apart from thinking?! These questions are not constructive proofs of new methods? I'm convinced advances in complex systems, turbulences, AI would make a hell of a difference, but I don't see anything going in the right direction in these areas. – Gerenuk Sep 6 '11 at 9:14
I'm not an expert in the field, but a proof of P=NP would be quite significant to operations research. A polynomial time solution to the traveling salesman problem and general integer programming problems would be nice. – Adam Saltz Sep 6 '11 at 12:24
Are you sure? Isn't a proof non-constructive? So you still wouldn't know how to travel in polynomial time. The the problem about conjectures. If you for a moment assume they are true and then nothing changes, so why should anything change if you rigorously proof they must be true? – Gerenuk Sep 6 '11 at 14:12

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