Here is a topic in the vein of  [Describe a topic in one sentence](https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence "Describe  a topic in one sentence") and  [Fundamental examples](https://mathoverflow.net/questions/4994/fundamental-examples)  : imagine that you are trying to explain and justify a mathematical theory T to a skeptical mathematician who thinks T is just some sort of abstract nonsense for its own sake. The ideal solution consists
of a problem P which can be stated and understood without knowing anything about T, but which is difficult (or impossible, even better) to solve without T, and easier (or almost-trivial, even better) to solve with the help of T. What should be avoided is an example where T is "superimposed", e.g. when T is a model for some physical phenomenon, because there is always something arbitrary about the choice of a specific model. 


A classical example is Galois theory for solving polynomial equations. 


Any examples for homological algebra ? For Fourier analysis ? For category theory ?



Justifying a theory by a seemingly unrelated example
[In front of a blackboard, in an office at Real College] 


Skeptic: And why should I care about holomorphic functions?


Holomorphic enthusiast:$\;$ Can you compute $\quad$ $\sum\_{n={-\infty}}^{\infty} \frac{1}{(a+n)^2}$ ? Here $a$ is one of your cherished real numbers, but not an integer.


Skeptic: Well, hm...


Holomorphic enthusiast, nonchalantly: Oh, you just get


$$\sum\_{n={-\infty}}^{\infty} \frac{1}{(a+n)^2}=\pi^2 cosec^2 \pi a $$


It's easy using residues.


Skeptic: Well, maybe I should have a look at these "residues".


Holomorphic enthusiast (generously): Let me lend you this introduction to Complex Analysis by Remmert, this one by Lang and this oldie by Titchmarsh. As Hadamard said: "Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe".You can look for a translation at Mathoverflow. They have a nice list of mathematical quotations, following a question there.


Skeptic: Mathoverflow ??


Holomorphic enthusiast (looking a bit depressed) : I think we should have a nice long walk together now.
[Exeunt] 


