Level of detail on a Phd application I am a graduate student planning to apply in Australia  for a PHd very soon.
I have observed that most of the universities require the applicant to provide a 100-words summary of the research they propose to undertake.
Now my graduate training was fairly general (MS doesn't really help you specialize). I am very clear on the field in which I want to pursue PHd (Mathematical Biology),
However the field is very wide in scope and I have not really narrowed down to a specific problem to which I want to dedicate the next four years of my life.
So my question is: what level of detail is being expected on such a "100-words summary"? Are recent graduate students really expected to have a project in mind they want to work on? (Am I finding this incredible only because I dont have one?)
Should I may be give an example of the kind of problems that excite me, while indicating my flexibility to be engaged in something new?
I vaguely understand that I should consider the area of work of the person I am writing to while preparing such an application, but more than that, I am not sure I understand.
Possibly related question
Research statement in PhD applications--how much is too much?
But I believe I am asking a different question.
Also, as further clarification to one of the answers given below, my MS did not involve writing a thesis. (If it did, that would certainly serve as a good starting point.)
 A: (I am in Australia.) To start with, don't make the opposite mistake: There is no guarantee you will end up working on the things you write in your research plan.  A good plan establishes your general interest area and can help to find a supervisor, but after you have done that and get admitted, expect the research plan to be forgotten. One thing we use the plan for is to help us identify potential students who are really interested in something and have a passion for research.  We know that someone with a passion for research can be guided into other research directions while maintaining the passion, so the precise details of the plan don't matter. The idea is to not give the impression that you are a clever person who is not especially interested in anything -- that's the student we don't want.  Give the impression that you just can't wait to start discovering things.
A: I'm on the Mathematics faculty of an Australian university. The way it seems to work at my university, and maybe at other Australian universities as well, is that a PhD applicant is accepted only if there is a faculty member committed to acting as supervisor. So it seems to me that the indicated strategy is, when applying to the program at University X, to look at the faculty at University X, see if anyone there is working on something you could imagine being interested in, quickly learn enough about that area to write a few sensible sentences about it, and proclaim that you would be happy to work on that topic or on such other topics in Math Biology as people at University X might want you to work on. 
It does seem strange to me that a beginning postgraduate would be expected to have a research proposal - it certainly didn't work that way when I was applying to PhD programs in the US in ancient days - but I guess you have to go with it. 
A: I am almost to submit a PhD thesis at an Australian university. My 100 word summary of what I planned to do is far different from what I actually did. I think that acceptance into the PhD almost entirely has to do with previous coursework results and having organised a supervisor, who can also tell you what to put for the 100 words. In mathematical biology two people at UQ come to mind to contact: Phil Pollet and Hugh Possingham. Fellow students I know have had good experiences.
A: I am in Sheffield, in the UK and this may well from country to country.  
We have an online application system that is shared by the whole university and not controlled by the maths department.  The form has a slot for a research plan, but in maths at least we ignore it - we have no expectation that beginning PhD students will have their own project in mind.  If you just told us that you wanted to do Mathematical Biology, that would be sufficient for the application form.  
A: This is related to the general question of how to write a proposal to do work that you haven't yet figured out how to do.  If you actually tell people what you are thinking of working on you are going sound naive, ignorant or shallow because you don't know how it is going to work out.  You could talk about the work you have in preparation (almost done but not quite ready to post or submit yet).  But chances are if you are on the job market you are rushing to get your best recent ideas out where people can see them (so they are already posted).  
What you are supposed to do in this kind of statement of research interests is first summarize the importance of your work (that means telling people why your thesis is non-trivial and important in a larger scheme).   Because this is based on stuff you have done you are going to sound knowledgeable, but more importantly you let people know why your work is not only technically good but important.  THEN  you  briefly mention classes of problems that are related, possible applications or large scale important problems that you might be able to make progress on that are similar or might be solved using the types tools you have developed in your thesis.  This part is brief because you don't know yet what you are going to do.
Sometimes people recommend that you look up the fields that people are working on at the institution you are applying to and suggest you might make progress in those or related fields or bring up connections.  That takes time but if you know somebody at the institution (and your application isn't in the blind paper bomb category so you have a chance of getting the job) then it might be work the effort.  Otherwise you might be recycling your research statement for many applications.  In this case a general statement of types of important problems that you might be able to make progress in is the way to go.
A: If it's really a 100-word summary, I don't really see how there can be any level of detail. But ultimately, the best thing to do and the one that will get you the most reliable answer is not to ask Math Overflow users, but would be to contact someone in Math in the university of your choice and ask them. Professors usually try to be helpful with their future recruits.
