By the way, it’s totally off topic (I’d email but you don’t have an email listed in your HN profile), but on the subject of new mathematical technologies with practical application, I’ve really been enjoying learning about this thing for the last month or two:
http://www.chebfun.org
Here are a couple of hour-long lecture videos
https://www.youtube.com/watch?v=cQp-kB9EOQs
http://downloads.sms.cam.ac.uk/1160831/1160835.mp4
and the first 6 chapters of a book explaining the basics of the underlying theory are freely available online (the rest of the book is great too) http://www.chebfun.org/ATAP/
Basic concept is to approximate functions as high-degree polynomials, to within machine precision, and then do operations on those approximations instead of on the original functions, at each step chucking out any coefficients which are smaller than machine precision. This in many cases provides a dramatic speed boost over existing methods for taking integrals/derivatives, finding roots, solving differential equations, convolution, etc. etc., while staying extremely accurate, much better than discrete solutions.
It’s not universally applicable to all functions (for obvious reasons) or all problems, but the team responsible has really done an incredible amount of solid work on the project.
Apparently it’s 10+ years old now, but nobody I’ve asked has ever heard of it before. Which is kind of surprising, because it seems amazing to me so far.
Unfortunately stuck as a Matlab library for the moment (there are partial ports to NumPy and Julia but they are very feature poor compared to the Matlab version).