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I have a major in Mathematics and Mathematical Physics and a masters in Theoretical Physics. I have enrolled the PhD course and must pick an advisor/theme until june.

I have three main big fields of interest and for the PhD I wanted to work in the intersection of these themes. I then got interested in one particular formalism which is related to all three themes. This particular paper is written by an expert in the field and seems to be part of the people who initiated this research effort.

I did talk to a professor in the department that has knowledge in one of the fields and has a lot of interest in the others (despite being no expert there) and which is usually open to new things.

He, however, said that I should come up with a concrete research objective inside this field.

Now, I must admit I can't do it. I look at what has already been done and I admit I have no idea what is still open and what can specifically be done. I just know I want to work with this and start a career with it.

So I considered the option of writing one email to the author of the paper that got me interested into this and ask for advice on what can be done.

The point of the question is that I'm unsure if this is acceptable in the academic world and if this would be well received.

So my question is: would be it ok to write an email to the author and ask what for advice in what is still open and can be done for a PhD research given my interests? Or is it discourteous and shouldn't be done? Or is there some specific way in which this should be done to not be discourteous?

Yes, you can ask and you might get a reply, or not. If you demonstrate that you "have done your homework" and have some knowledge and not just interest then you will stand a better chance of success. What (or who) have you read? What have you explored? What ideas have you tried and failed with?

You can certainly state that you are looking for a research problem suitable for a dissertation and that you are coming up short. You can ask for guidance on where to search (additional papers to be read, people to contact...).

However, it is unlikely that a problem will be dropped in your lap. If you were able to attend a seminar with the author's group you might be able to do better. And be aware that a "problem" quickly thrown out may not be suitable as it may end up trivial or impossible. Such things are often approached only gradually.

But it isn't discourteous to ask as long as you can show that your search isn't just a random one.

But in general, also be prepared to work on a few problems before you come up with the one that leads to success. Even an unsuccessful attempt can teach you how to be more successful with the next one.