What is the greatest mathematician in the world today?

I came in here to suggest Tao as well. Someone else who came to mind was László Lovász – even though he wouldn’t make any such list, some of his proofs in graph theory are incredibly elegant.

Probably a human. Or a breadbox.

Exactly what does it mean to “drop out early?” What are the most common reasons for this?

Is it a lack of interest, interest being replaced by motivation to do something else, disillusionment (as it seems might be the case in Pereleman’s case) or just an inability to perform and develop in mathematics at a similar level as during their prior career?

Read the Wikipedia page for Grigori Perelman at the link given in post #12 and the page for Alexander Grothendieck given below to see what’s meant by dropping out:

Well, Perelman and Grothendieck are very different, so I doubt you’d have the same answer for both… Perelman seems to just feel a little rejected by/disillusioned with with the mathematics community, while Grothendieck… well, he’s the epitome of the stereotype of the brilliant mathematician who is also bordering on insanity (not a slur, just, he goes beyond eccentric).

Disclaimer: what do I know? I’ve never met them.