When I was an undergraduate 35 years ago, I made the mistake of asking some of my mathematics professors what well-known open problems they liked to think about. I got the message that this was something one did not discuss in polite society. Perhaps the culture has changed?
This question asks mathematicians to "out" technical trends which (only) experts understand as lines of attack against well-known outstanding problems.
I suggest that answers take the form of a citation or two from the literature plus a statement or mere naming of the target open problem. A few words additional about the philosophy underlying a given approach might also help, but I suggest allowing users to submit further questions if they want copious details.
Please be careful not to "out" individual mathematicians, unless you are one. Lots of mathematicians are really uncomfortable with their private tentative thoughts being made public. The goal here is to "out" technical trends that are in the public record, but not widely known. Alternately, if you are an expert in a famous open problem and want to out your own methods, feel free to do so.