High School Looking for examples of unexpected applications of math concepts

[Tiger Blood]

Sometimes I run into this claim

There are of course many other cases where mathematical concepts were first developed with no particular intended application, and then later found application in an unexpected physical domain.

But what are some examples? Is it perhaps the discovery of binary numbers and then their much later application to computers?

Or what about this?

In yet other cases, many mathematical concepts have been developed from their own merit and find essentially no application in describing the laws of nature.

 

[Paul Colby]

But what are some examples?

Group Theory.

 

[pbuk]

Sometimes I run into this claim

As you seem to be quoting a particular source you should provide a reference.

But what are some examples?

For me two stand-out examples are:

Manifold theory which underpins general relativity, the best model we have for the universe.

Prime numbers and their application in public key cryptography which underpins secure internet communications.

Or what about this?
"In yet other cases, many mathematical concepts have been developed from their own merit and find essentially no application in describing the laws of nature."

I think it would be better to say that "some mathematical concepts have not yet found a non-mathematical application".

Group Theory.

Very important to manifold theory and therefore general relativity (if spacetime is not closed things could get interesting).

 

[mfb]

Complex numbers found applications in quantum mechanics, electronic circuits and elsewhere.

 

[fresh_42]

I have two paperbacks which are titled "Applied Pure Algebra". The authors list 36 topics, and I'm really too lazy to type them.

 

[pbuk]

I have two paperbacks which are titled "Applied Pure Algebra". The authors list 36 topics, and I'm really too lazy to type them.

I think I had something similar as an undergraduate, however I seem to remember that all of the applications were simply to other sub-fields of algebra. Is there an interesting topic among the 36?

 

[robphy]

I have two paperbacks which are titled "Applied Pure Algebra". The authors list 36 topics, and I'm really too lazy to type them.

Well, at least you provided the title...
https://www.google.com/search?q="Applied+Pure+Algebra"
https://www.google.com/search?q="Applied+Pure+Algebra"&tbm=bks

 

[fresh_42]

I think I had something similar as an undergraduate, however I seem to remember that all of the applications were simply to other sub-fields of algebra. Is there an interesting topic among the 36?

I liked the genetic algebras, but more popular are error correcting codes, cryptography (e.g. DFT), crystallography.

I'd say until the 19th century mathematics was developed along real numeric problems, including Riemannian manifolds and most of calculus. Maybe you thought about AdS and de-Sitter spaces as you answered manifolds. Even the complex numbers listed here had been developed to serve a goal. It wasn't originally abstract mathematics which suddenly found an application. The investigations were parallel. The first area I can think of which was primarily mathematics and found its applications later was topology at the beginning of last century.

In the meantime I have the impression that physicists loot mathematics in the vague hope to find the key for GUT and be the first. Nothing seems more safe. E.g. one could answer the OP's question by graded Lie algebras, which were certainly known prior to string theory. But as nobody really cared about them, it's still difficult to decide whether they count as an answer. The most popular and most cited answer is probably number theory. Primes experienced a reincarnation since electronic communication came into life.

 

[fresh_42]

Well, at least you provided the title...
https://www.google.com/search?q="Applied+Pure+Algebra"
https://www.google.com/search?q="Applied+Pure+Algebra"&tbm=bks

This is not the one. And I only did not provide more than the title, because they are not in English.

 

[etotheipi]

How about RSA (BTW some good notes, for bedtime reading )? Even GH Hardy I think called number theory "useless".

Edit: Whoops, I see this has already been mentioned...

 

[fresh_42]

How about RSA (BTW some good notes, for bedtime reading )? Even GH Hardy I think called number theory "useless".

However, he had found an application!
https://www.physicsforums.com/threads/micromass-big-integral-challenge.867904/page-2#post-5457903

 

[Paul Colby]

Very important to manifold theory and therefore general relativity (if spacetime is not closed things could get interesting).

I spent some time last night trying to think of a field of physics which doesn't include significant applications of group theory. Biophysics? I only suggest this cause I don't know much biophysics beyond don't eat too much mac and cheese.

 

[fresh_42]

Biophysics?

Free halfgroups.

 

[pbuk]

I'd say until the 19th century mathematics was developed along real numeric problems, including Riemannian manifolds and most of calculus. Maybe you thought about AdS and de-Sitter spaces as you answered manifolds.

Not really, I probably didn't think about this clearly enough. For sure Riemann's work was grounded in reality, that is clear from his remarkable lecture of 1854 published in 1868: "The answer to these questions [regarding the metric nature of space] can only be got by starting from the conception of phenomena which has hitherto been justified by experience". Lorentz corresponded with Einstein on GR so this can hardly be seen as an "unexpected application", and AdS/de-Sitter spaces come later.

So you are right, manifold theory is as much a tool built to 'do' GR as Newton's and Leibniz's calculus were to 'do' classical mechanics.

We all seem to be agreed on number theory and cryptography though

 

[DrClaude]

Boolean algebra.

 

[hutchphd]

I don't really know the history but what about Fourier synthesis and X-ray diffraction?

The Radon Transform and all the various tomographic methods?