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For context, I'm a 2nd year graduate student interested in geometric analysis. I'm at the stage where I'm starting to understand my field, at least being aware of some relevant names and their current interests. My advisor has explicitly stated that he doesn't want to give me a problem, and rather has encouraged me to develop my own taste and interests. Given that there are a million directions to go in, I wanted to poll the crowd for suggestions and general advice. I'm trying to be very deliberate in finding a direction both I and the people in my field care about since I want a job a few years down the line.

For those who consider themselves to have good taste, how did you go about developing it? How do you tell if a certain problem is worth pursuing? Did you read all the "hot papers" in your field, and let them influence you? Any relevant advice is welcome and appreciated.
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Essentially what your advisor is trying to convey to you is that if you want to become a good researcher you need to actually be curious about what you're researching. You can get one of those hip problems everyone is thinking about, but if you see it merely as work and scholarship material and have to force yourself to grind through it 8 hours a day you will be close to msierable because mathematics is 99% failure. Instead, find what really encites your curiosity as a mathematician. What do you really want to find out about more? Don't be afraid to look naive, naive questions lead often into the rabbit whole of wonderful and deep mathematics. It's time to explore and follow your intuition. Ask yourself, what really makes you curious in math?
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Honestly, that sounds a bit rough to me. Personally, part of how I gained "taste" was by *working on problems*. I think your advisor's strategy may work well on very strong students, but I have no idea how I would have come up with a thesis problem on my own.
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You can't not have your own taste, it's within you. Whatever you are motivated to pursue, that's your taste. The thing to develop is the confidence to pursue it.

I think that, by definition, what this means is going to be someone different for everyone. For me it meant actively working on problems that interested me and in a style that interested me, even when it seemed like the effort might not pay off.

I was in a similar position, although I was given a more specific topic at first. I was forced to follow my own interests because of my complete inability to do anything that does not sufficiently motivate me. It ended up working out well, which I do chalk up to my good taste haha.

I think a lot of it comes down to learning to tease apart what you want to do because you are driven to vs what you feel expected to do. For example, the way I like to do maths is different to the way a lot of people around me like to do maths, and that caused me to avoid doing it my way because I felt like it was wrong. Now I've learned to embrace my style more (it helps that I've got some good results now), although I still keep an eye out for ways I could improve.
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There isn't necessarily "good taste" and "bad taste", there's "your taste" and "not your taste". What papers have you read that made you want to dive deeper into the topic, and what papers have you left half-read? Discuss those papers with your advisor, they may be able to point you to more interesting papers.

There is another aspect which is "what problems are tractable?" For instance, if you're super jazzed about the Riemann Hypothesis, there's loads of ongoing work that's somewhat tangential, but it's unlikely you'll be able to solve the RH yourself and make that your thesis. It takes a *lot* of experience to be able to judge how hard a problem will be, and your thesis advisor is a great person to bounce ideas off of.

Your advisor has a great attitude: they should help you craft your thesis problem, not hand one to you on a platter.
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I asked a professor who studies topology at my university a similar question. he said that "it took me a thesis and a dissertation in analysis to understand that analysis is not my taste"
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I think a lot of mathematicians' tastes were the product of their peers and professors pushing a certain set of tastes onto them.  It's pure speculation, I don't know how you would even study this, but: I suspect this pressure probably accounts for people's taste more than a lot of mathematicians would care to admit.  

It's refreshing that your professor sounds like he's encouraging a greater degree of self-determination.  Kudos.

You might just want to read a bunch of literature, of your own interest.  Maybe especially stuff that also aligns with the interests of your peers, since those kinds of collaborations can pay dividends.  And from that, really just think to yourself what sounds interesting or promising to you, and take a *bunch* of ideas to your advisor and see how he receives them.  If he judges any of them harshly, maybe he's not as supportive as he currently sounds, and you can become strategic about that later.  But maybe he'll encourage you toward some and away from others, and it could be quite productive.
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