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I'm currently in the stage of my PhD (in math) where I need to start discussing with my advisor about my PhD project and I'm wondering what kind of things am I allowed to say and which ones should I avoid. I want to know how should I manage my communications with them.

More concretely, to which extent can I decline or object about certain projects they may suggest? Is it appropriate for me to say that a project doesn't appeal me, or that I'd rather do something more [concrete, abstract, with more applications, etc.]? Is it reasonable to bring up the projects of my peers?

I've heard about people having their advisers proposing multiple projects from which they may decide, however, up until this point that doesn't seem to be my case. Otherwise I wouldn't be looking for advise. In any case, I understand they're experts in their area and their suggestions are very well thought (relative to me and the research area) and on-point.

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You're going to be stuck with your adviser and the chosen topic for several years -- so you better be honest in your conversation and say what you really think. I don't think there are any off-limit topics. The worst that may happen is that the areas you are interested in have no overlap with those your adviser has knowledge of/is interested in, and which he might decline to advise -- but that would be good to know as well.

When I have these conversations with my students, I actually do not offer any topics. I ask my students to come up with areas they are interested in, and to think about what they want to do before we make decisions about PhD topics. There is nothing worse -- for both sides -- if a student is stuck with a topic they are not interested in, and so I let my students choose what they want to do, and only steer things a bit to the left or right given my knowledge of what the relevant topics today are.

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I agree with Wolfgang's answer, be open and honest about what you want. But I want to add some practical advice...

Coming up with a concrete project will take multiple meetings with your advisor and one hurdle is getting your PI on the same page as you each time you meet. What I mean is that they are very busy with classes, projects, and collaborations that span a variety of specific topics. So often, it takes a minute for both ppl to wrap their heads around a particular topic.

I like to prepare a few powerpoint slides before meeting with my PI. Usually one methods slide that pictorially describes the procedure for the relevant project, and then either predicted results or actual data if I have it. I don't know exactly how that translates into your field, but sometimes a picture really is worth a thousand words!

The other thing that can be frustrating is that it's hard to remember all the discussion later and so I asked my PI if I could record our meetings. Might seem a little weird but I only record certain parts like when we're really getting into the theory. I want to be engaged in the discussion not rapidly taking notes that end up being insufficient later. My PI was a little awkward the first time I did this (obviously I asked and PI said OK) but now my PI has come to appreciate it because I'm taking more away from our meetings.

In short, prepare a few powerpoint slides on proposed projects and record conversations when discussing complex ideas.

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  • Your advice is mostly on-target. One thing: math (at least, pure math) is a highly non-powerpoint culture. I've never seen a mathematician prepare power point slides for anything less formal than a seminar talk. If a student wanted to present me with a powerpoint presentation, the first thing I'd have to do is figure out where to go to project them, since I don't have a projector in my office. (I should say that a few of my colleagues do -- this is quite recent.) Like almost all of my colleagues, I have a blackboard in my office, and I would expect a student to use it.
    – Pete L. Clark
    Jan 27, 2017 at 6:08
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    Indeed, I got the sense it may not be applicable to math, but still good for the SE archive. Because I run experiments, describing the design of the project is the best way to remind my PI what we're up to--I don't know what would be the equivalent in math. But, its worth noting, I don't project these slides in a meeting with my PI, we just look at them on my laptop or monitor :)
    – kindredChords
    Jan 27, 2017 at 7:09

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