[ravi-delia]

Math has problems where everyone suspects the proof will open doors and give valuable insight- I'm mostly plugged into topology where there are a lot of those. There are also problems that aren't interesting except in their difficulty, which drives the creation of new techniques and tools- Fermat's Last Theorem isn't particularly useful, but the effort to prove it created a vast body of spinoff work. But you also see problems that are more passive, waiting for someone to approach them with new firepower. Tilings are more like that- a testing ground for new techniques, and a way for mathematicians to keep their wits sharp.

Also, they do have some inherent beauty. I mean an aperiodic tiling is crazy right? And with one tile?

[doomrobo]

Out of curiosity, could you share the exciting unproven theorems in topology you referred to?

[ravi-delia]

One big name is the Borel Conjecture- a very productive aspect of topology is developing finer and finer tools to detect differences between spaces. The Borel Conjecture essentially states that for a certain class of spaces, a well used and loved tool is equivalent to an extremely strong tool.

I was thinking more historically though. The development of those tools was driven by specific problems- classifying the behavior of higher dimensional spheres, determining if genus uniquely classifies spaces (not at all, but I believe people were once hopeful), even knot theory is an outgrowth of this kind of research.