Do you know of any PhD programs that are highly flexible? I'm interested in either math or computer science programs, as I have done research in both areas and feel confident to apply to both.


With high flexibility I mean that there should be freedom for the PhD student to choose: 


1) who they work with (so that I can get to work with different researchers - I have already done a number of internships and have built a small network of collaborators, and I want to keep these collaborations alive, even if they can't be all brought the umbrella of one single PhD topic)


2) from where they work (this is tied to 1): it is very difficult at times to work through skype and having different collaborators entails the need to travel and do work at different places)


I have met only one single PhD student who was able to do that - he spent the first year at his home university in Sidney, then he was 6 months in another country, then 6 month again in Sydney and then again away for an internship. I was quite impressed by the network he build and his research output.


**EDIT** I'm from Europe and I'm in particular interested in PhD programs in (continental) Europe. Though I might consider the US, if it is known in advance that there's a large amount of flexibility. The domains that I'm interested is analysis/data science/topology, so probably in any of these I should be able to find a suitable topic and will make the choice mainly on how much flexibility I'll be offered.


**Second edit** Please ss me comments fo Jochen Glueck's answer which furthermore make it clear what I'm after.



Highly flexible PhD programs/school (with individual freedom)
This is not really an answer, but an extended comment.


I am repsonding to the following remarks of the OP:


* "I'd also be interested in names of PhD advisors in the domain of analysis/data science/topology that are known to be permissive and allowing their students to go abroead." [Quoted from the comments.]
* "The domains that I'm interested is analysis/data science/topology, so probably in any of these I should be able to find a suitable topic and will make the choice mainly on how much flexibility I'll be offered." [Quoted from the question.]


Admittedly, I have some severe doubts whether this approach is going to lead you where you would like to go. More precisely, I see the following problems: 


(i) The fields you specifiy are extremely large (at least analysis and data science), and there are most likely several thousand scientists in Europe which might be potential PhD advisors.


(ii) When you apply to a scientist that you do not yet know in person for a PhD position, they will most likely want to hear (or read) some explanation of why you want to do your PhD with them. "I'm interested in everything and you have a reputation of being permissive" will most likely not be what they want to hear.


(iii) "Known to be permissive" as a criterion is, in a sense, very unspecific. Whether your advisor gives you the freedom you wish is a more complex matter than just the question "May I go abroad?" Based on the information you give in your question it might, for instance, be important for you how much freedom you have to choose and to adjust the specific focus of your PhD topic, how much time you can spend on research projects which are not directly related to your PhD topic, how willing (and able) your advisor is to also give you support for such kind of topics (financially, morally and scientifically), how your teaching obligations (if any) can be aligned with your plans, whether your university or faculty provides specific support for PhD students which would like to go abroad for a while, whether such support is only granted under certain circumstances (for instance for special countries or partner universities), and so on. Those are questions which cannot simply be answered by hearsay on the internet.


**So my advise to you is as follows:**


I see essentially two options:


**Option A: Don't narrow down the field, but choose your advisor based on personal contacts.**


That is, have a look at your current university and at the scientists there that are working in analysis or topology or data science. Find out if some of them are open to your plans (for instance, by asking fellow students, PhD students, postdocs and of course by talking to the potential advisors directly). If somebody seems interested, it is important that you openly and clearly communicate your wishes to your potential advisor in advance, and that you request them to do the same - to make sure that the expectations of both of you are well aligned, and that things are also clarified on an organizational level.


If you don't find anybody at your home university, try to ask people (PhD students, postdocs, professors) at your university if they personally know somebody at another university who might be the right PhD advisor for your plans. Many scientists have a large personal network and might be able to help. Then, proceed as described before.


**Option B: Narrow down the field.**


If, for some reason, you do not want to (or you can't) proceed as described in Option A, there is the alternative to apply to potential advisors who you do not personally know (and who you did not find by means of personal contacts). In this case, however, I would strongly recommend to first narrow down the field you want to work in.


So, for Option B, I would suggest the following course of action:


1) You say that you have already done some work with several collaborators. This does by no means imply that you have to work on the same topics in your PhD. But it can serve both as an inspiration and as background knowledge for your PhD topic. Based on what you have already done, ask yourself:


"What techniques/results/theories did I find interesting? Could I imagine that it would be fun to use similar methods in another field?"


2) Based on this, try to narrow done potential fields that you would be interested in. This can be something closely related to what you have already done, or it can be another, but maybe somewhat nearby field.


Of course, you don't have to do this alone (and most likely, it would be very difficult if not impossible to do so alone). Instead, talk to your collaborators. Ask them whether they would advise you to do a PhD in exactly the field you have already worked in, or whether they know of a field where related techniques are used.


Also ask the staff members at your current university: bother teaching assistants, professors, PhD students and postdocs who you already know in person - it's their job to help you, and most of them will be very willing to do so.


3) After you have narrowed down the field, try to find out who is doing research in this field. You can, again, ask your collaborators and staff members at your university. You can also look for research articles in this specific field and see who authored them. This way, you can get a list of potential advisors which is much shorter than several thousand people.


4) Finally, approach (some of) those potential advisors (whether you immediately write an application or first approach them in a more informal way probably depends on the (academic) culture of the specific country). If they are interested, it is again important to ensure that their expectations are well aligned with your expectations. So openly and clearly communicate your wishes and plans in advance, and request your potential advisor to do the same.


Besides that, I also strongly suggest that you obtain some information of how your potential advisor deals with other PhD students - but not on the internet; rather by talking to people who personally know your potential advisor for some time.


If everything seems fine, you can start your PhD. If not, choose another advisor.


**Final Remark.**


Personally, I would suggest that Option A is better than Option B. Knowing your PhD advisor personally in advance of your PhD (for instance, if you have attended some of their lectures) is a very good starting point. But of course, people and circumstances tend to be very diverse, so I found it important to also list Option B (although I have to admit that I have no personal experience with the implementation of Option B).


