A killer app is an program so useful that people will buy a larger system just to use it. For example, the VisiCalc spreadsheet was a killer app for the Apple II, maybe the first program to be called a “killer app.” People would buy an Apple II just so they could run VisiCalc. Microsoft Office is a killer app for Windows: many people run Windows so they can run MS Office.
I was thinking about the mathematical analog of killer apps. For example, finding maxima is a killer app for calculus. If that’s all you could do with calculus, we’d still teach calculus. After one semester of calculus, students can easily solve optimization problems that would be virtually impossible otherwise.
Mechanical vibrations are a killer app for differential equations. (Non-mechanical systems such as LRC circuits follow the same equations.) If an engineer applies anything from a differential equation class, this is probably it.
Contour integration is a killer app for complex analysis. There are other applications of complex analysis, but contour integration is certainly one of the big ones.
Lack of killer apps
Some areas of math don’t have a killer app that I can think of. They may have practical application, but there isn’t a particular application that stands out, not one that many people would agree on . If you did a Family Feud-style poll on applications of calculus, differential equations, and complex analysis, I image the examples above would be on the board if not the top result.
Category theory can be useful, but its applications are scattered. If category theory has a killer application, I doubt there’s a consensus of what that application would be.
A killer application is different from a key theorem. I imagine a lot of people would say that the Yoneda lemma is the most important theorem in an introductory course in category theory, but I wouldn’t call it a killer app. My idea of a killer app is something that fills in the blank “You should take a course in X if for no other reason than that you’ll be able to ______.” For instance, many people take a course in statistics just so they can do linear regression.
If you have ideas about what killer apps would be in various areas of math, please share them in the comments below.
 I’m reminded of someone’s description of G. K. Chesterton as a master who left no masterpiece. That is, he wrote a lot of great lines, but no great book.
8 thoughts on “Mathematical killer apps”
Fourier analysis is a worthwhile endeavor, if only to learn and apply the Fast Fourier Transform, the backbone of digital signal processing and, of course, JPEGs.
I agree on Fourier transform (not only FFT, but all the insight from the full mathematical theory), but I think the killer app is Magnetic Resonance Imaging (MRI).
The Radon transform is indispensable in medical imaging, but I don’t know of any other uses unless they’re analogous to medical imaging.
Encryption is the killer app of number theory.
What a great way to look at it. I would have said that the killer app for category theory was the Seifert-Van Kampen theorem, but that might be because I was pretty deep into algebraic topology at the time. I’m pretty sure that category theory did arise, historically, from formalizing alg. top., so I guess that’s not surprising.
That said, it’s more likely that one will pick up the language of category theory in the course of doing alg. top., as opposed to learning CT in order to do alg. top.
I did surprise myself one time when I was trying to figure out a version of the standard computer “topological sort” utility and ended up using CT to figure out what to do. I actually found that what I wanted was a map that was minimal among a set of maps that created a commuting diagram. I’d never have expected CT to be useful in such a context.
I would say linear programming is a “killer app” in two respects. It is arguably the most useful and motivating takeaway from convex analysis (which is where I learned it) or discrete math. It was a “killer app” in the sense that (according to things I read a while back) the first commercial mainframe computers (post-WW II) were bought in large part by oil companies, to let them optimize refinery operations by solving LPs using the new simplex algorithm from George Dantzig.
When I was a child, my mother was convinced that Office should be bundled with Windows, otherwise what’s the point of Windows? Just to take away your money and give you something you can’t do anything with.
Monte Carlo simulation is the killer app for pseudorandom number generation.
Agree with the previous suggestion that linear programming is the killer app — for many things, including linear algebra.