I would like to understand at least one of the several existing approaches to algebraic geometry over $\mathbb{F}_1$ (the field with one element). Is there an example of an "interesting" theorem that can be formulated purely in the language of ordinary schemes, but which can be proved using algebraic geometry over $\mathbb{F}_1$?
Of course, the interpretation of the word "interesting" is entirely up to your own taste. An example in which the theorem cannot be proved using "classical" methods would be most desirable, but examples where (one of) the theories of schemes over $\mathbb{F}_1$ gives an alternative proof of an already known result would also be very much appreciated.
[On a related note, perhaps there should be a tag "naive-question" for situations like this one.]
Best Answer
I'm confident that the answer to the original question is no. There are hardly any theorems at all in the subject, much less ones with external applications! In other words, if no further progress is ever made in any of the directions people have pursued, everything will likely be forgotten (which would not make it so unusual an area). What attracts people to these things is not a track record of existing applications but the possibility of exciting future ones. So investing time in the subject is something of a gamble---it might pay off if you're good at divining the future (or if you have insider information), or you might end up wasting a lot of time.
I don't mean to be too pessimistic. I for one have high hopes for certain directions (!), but I think it's best to see clearly what you'd be getting into.