How well should one understand a PhD's topic when applying for a PhD?

Question

I'm looking into applying for a PhD in math in Europe (I've seen positions in Belgium, Netherlands, Germany, Sweden and Norway). A concern that I've encountered is regarding PhD advertisements that are recruiting people for an already established topic/area in mathematics. How well should I understand the topic in these cases? Take cohomology as an example, is it enough to know that it's within group theory and topology, which are areas that I like, or do I need to go through literature and be more thorough?

Personally, I don't really understand most, if not all, of these PhD pre-established topics, so it's hard to be and feel confident about sending an application and the/my future. Is this common and I should just trust that it works out at the end, or should the fact that I don't know about the topic already indicate I'm not suitable and should focus instead on applications with undecided topics?

Answer 1

In most places in Europe (especially the places mentioned by the OP). It is expected that you have completed a master's degree before starting your PhD. The PhD program itself typically contains very little course work (or none at all). The expectation is that you will have followed some graduate courses that would prepare your for the PhD. Following the OPs example, if a PhD with a subject in cohomology is advertised, one would typically expect candidates to have completed at least a course in algebraic topology (possibly more). If you haven't, you will be at a serious disadvantage compared to other candidates. Not only in prerequisite knowledge (things can be learned), but more importantly in that you don't know the subject well enough to know if you will be passionate about it. This will make it harder to convince your potential advisor of your passion for the subject.

On the other hand, you would not be expected to perfectly understand subject of the PhD before hand (this is what the PhD for!). It is much more important that you are able to show a great aptitude at understand new material, and some ability to do mathematical research.

Two of the biggest risk factors in hiring a PhD that looks great on paper are:

1. That they won't be able to convert their promise as an excellent student to become an excellent researcher.
2. That they realize some time into their PhD that they are not interested in the chosen subject and become demotivated.

Anything that you can provide that can alleviate these risks, will likely trump any deficiencies in prior knowledge. However, as said above, the prior knowledge itself can play an important part in alleviating these risks.

Answer 2

There are no moral or ethical issues in applying as long as you are honest.

But how much you need to know depends on the system. In the US, not much prior specific knowledge is needed. In (parts, as least, of) Europe it is different. Many places require a masters degree to apply, assuming deeper knowledge. Some will look for research experience, though not necessarily specifically related to the topic at hand.

But in many places (Germany...) it may be necessary to apply to a specific advisor for a position in their working group. In such a case it depends on their needs and wishes and we can't predict what that will be. But, I'll guess that general knowledge in the field is usually sufficient. In general you don't earn a PhD to prove that you don't need one. It is still a learning process into which you grow. Most (probably not all) potential advisors understand that, but you will need to convince someone that you are ready to begin the task. (Note "begin the task")

Answer 3

You might never know what you will actually need. On paper, I did my PhD in Complex analysis and some algebraic geometry. However, all the actual work was done by various combinatorial methods. The problems I studied did not really sound combinatorial at the start, it just happened that the main tool always ended up being some type of combinatorics.

So you do not know if you really are prepared or not (of course, it is good if you know enough mathematics to understand the problem). Also, some programming experience will always be very helpful.