Categories of “Interesting” Mathematics

‘Interesting’ is a subjective notion. One might distinguish several different categories of ‘interesting’ mathematical statements or problems. This article talks about a few of these categories with examples.

Many problems are interesting because they invoke a nifty, or surprising solution, a ‘trick’. It is enough to see and understand the solution to be satisfied. These are pedagogically frustrating to less mathematically mature students who can’t see these tricks as quickly.

An example might be a classification of a representation of a lie algebra whose proof is particularly nice, and involves a clever argument and no brute force calculations.

Some problems are not particularly difficult or consequential, but feel very solvable and compel us as mathematicians to want to solve them ourselves, for fun or to prove we can do it.

For example, recently I was wondering when the following equation has (positive) integer valued solutions of the form $(x,y)$ $\dfrac{\Gamma(x+1)}{\Gamma(x-y+1) \Gamma(y+1)} - k = 0$

Where $k \in \mathbb{N}$.

This doesn’t seem difficult, I’ll just need to sit down and think about it for a bit. I’m not so curious that I feel the need to look up the answer, but I am curious about the process of solving the problem.

Another example that requires more work is the problem of calculating derived quantities in the video game Kerbal Space Program. Such as the amount of ‘change in velocity’ needed to escape the atmosphere. Or the optimal trajectory for a rocket to ascend. This is one of the more inspiring applications of differential equations in my mind.

Some mathematics is interesting because it comes to a surprising conclusion. Examples might be the banach tarski ‘paradox’, or the limit of ratios of the look and say sequence. Or even more simply that there is a cross product on $\mathbb{R}, \mathbb{R}^3$ and $\mathbb{R}^7$. But not for any other $n$.

It is inherently surprising that mathematics has nice results like this at all. It indicates some profundity in the discipline that motivates its study in a universal way. Having said that, please don’t talk to me about universal properties.