I suspect there’s a huge opportunity in moving mathematics from the pure column to the applied column. There may be a lot of useful math that never sees application because the experts are unconcerned with or unaware of applications.

In particular I wonder what applications there may be of number theory, especially analytic number theory. I’m not thinking of the *results* of number theory but rather the elegant *machinery* developed to attack problems in number theory. I expect more of this machinery could be useful to problems outside of number theory.

I also wonder about category theory. The theory certainly finds uses within pure mathematics, but I’m not sure how useful it is in direct application to problems outside of mathematics. Many of the reported applications don’t seem like applications at all, but window dressing applied after-the-fact. On the other hand, there are also instances where categorical thinking led the way to a solution, but did its work behind the scenes; once a solution was in hand, it could be presented more directly without reference to categories. So it’s hard to say whether applications of category theory are over-reported or under-reported.

The mathematical literature can be misleading. When researchers say their work has various applications, they may be blowing smoke. At the same time, there may be real applications that are never mentioned in journals, either because the work is proprietary or because it is not deemed *original* in the academic sense of the word.

Arbitrage, eh? Were you by any chance inspired by yesterday’s xkcd?

Ken: No, I hadn’t seen that comic until you pointed it out. I thought about titling this post “Untapped math,” but then I thought “arbitrage” was more expressive.

One thing I am surprised by is how little applied mathematics is applied. Take operations research and the famous examples and of problems it solves. The number of mobile applications that could be in use to solve these is tiny

sorry that are in use versus could be in use

I don’t know much (really, anything) about category theory, but hasn’t it found application in certain programming languages? I hear it tossed around a lot when people speak of Haskell. Does that count as an application?

Category theory -> Haskell, about which at least one person gives a s****:

http://steve-yegge.blogspot.com/2010/12/haskell-researchers-announce-discovery.html

@Suresh: I remember the first time I read that I almost did a spit take. Thanks for reminding me of it.

@iamreddave

I too have been perplexed about applied math departments (which sometimes feel like PDE departments).

“Real” applied math is abound in engineering (and comp sci and machine learning, etc), which has had its under-appreciated share of sophisticated mathematicians! One of my favorite texts is Sontag’s Mathematical Control Theory, and some very nice uses of differential geometry and Lie theory are found in “Geometric Control” (see Sussman, Isidori, etc).

In my experience, “real” applied math brings a whole new level of difficulty that just can’t be appreciated without intimacy with noisy and messy data. In the words of neuroscientist Ramon y Cajal: “Unfortunately, nature seems unaware of our intellectual need for convenience and unity, and very often takes delight in complication and diversity.”

I agree with John in that there is untapped uses of pure mathematics. Maybe arbitrage is just around the corner… and maybe we just need more Von Neumanns or Poincares!

I’ve always been a fan of Robert Rosen’s introduction of Category Theory into biological modelling. Unfortunately he died before he could develop it much beyond the original motivating explanation. I think one of the main issues is that people are trained within individual disciplines and rarely take those skills and move to a new field, it requires a special sort of person to even want to do that. But slowly the discoveries do creep out.

I agree with you (and I guess this was Robert Ghrist’s viewpoint as well in the interview you posted with him a few years ago).

It does seem at times like “…and Applications” is like a tagline academics will put on their books because they know that, in some ways, their work is “supposed to” be productive.

[[I remember reading an obituary for Tim Cochran where they felt they had to work in a sentence with “applications” in it, I guess because people will ask “What’s the point?”. (To me, though, a remark about knotty DNA misses the real gold I think you’re alluding to: now that there are these wonderful tools, entirely new thoughts are possible. “Knoty DNA” feels like the remark the inventor of NMR made about “calibrating industrial magnets”. To me the value promise in something like knot theory is that it’s totally original, there was nothing like it in the 18th century for example. So the things that maybe could be made with it, might be something no one could have ever conceived of before.

On the one hand I think it’s very valid to say, for example “The Riemann–Hurwitz formula concerning (ramified) maps between Riemann surfaces or algebraic curves is a consequence of the Riemann–Roch theorem.” On the other hand, any book called “X in Theory and Practice” is guaranteed to be a book of theory.

BMeanwhile, the author is probably not meeting with heads of business or learning what patents have already been filed in an area or, really anything that looks at all like “industrial application”. Other than the aforementioned solving of mathematical problems, which I think does really count—it’s an application of a theory to a question, not an industrial application; the word “application” is so general it covers very different things.

Talking of ANT: I had a thought the other week that might (I hope) turn into something. Economists tend to reason in the continuous (with incentives) whereas reality happens in the discrete (for example human capital is bundled and it’s not clear, if you think about it carefully, that they look like ℝ^n). So an idea I want to explore is if you let the economists keep their continuous reasoning but have that happen in a smooth domain, but it will only show up in reality as, eg, some ideal (so the theoretical reasoning could work on like a functor-of-points’ domain, but the picture of our world.

* A simpler point is that you can’t buy a real number of anything. So yeah, understanding the techniques for solving the Weil conjectures could help in restating this in ℕ.

** Actually I have a lot of ideas like this. I’ll let you e-mail me if you want to hear more, rather than blabbing on in your comments. =)

Another “high” concept that I think a lot of normal people would want, or rather vaguely use rather than in this much “precise imprecision”, is that of a coarse space: «in general, X and Y correlate, but maybe not every little tick» is a feeling I’ve heard many normal people articulate.

About the connection of category theory to physics, have a look at this excellent review article written by John Baez.

http://arxiv.org/abs/0903.0340

Beyond the book by Abramowitz and Segun I was searching for detailed information on Airy functions. I made copies of a number of SciPy items I pulled up during my Internet PCH Search and recalled seeing that CRC had published a 2008 second addition concerning a 1143 Page tome on Special Functions. Now I cannot find or retrieve this information that I thought I copied; wherein I also was attempting to find out the ISBN # and the cost of this CRC 2008 book? I will be extremely grateful if you would be so kind as to e-mail anything you can find in the nature of this sought-after information.

Regards,

Frank L. Rees; Sunday, 07-05-2015.

Maybe this is what you have in mind: http://dlmf.nist.gov/