The standard route towards PhD in the DAMTP Applied and Computational Analysis group is through the Cambridge Centre for Analysis. Having said so, before you apply to CCA, it is a good policy to touch base with a prospective PhD supervisor.
To save my time and yours, please bear in mind the following:
Your email is likely to be the only source of information I have about you. What you tell me about yourself – and what you avoid telling – tells me all I know about you. This includes not just the explicit stuff you choose to tell me but also implicit points, e.g. your command of the language, clarity of exposition, maturity of approach and the ability of doing some basic homework (and doing it well: read on).
Let us be realistic: I will not change the course of my own research because of your mathematical interests. If you wish to be my student, you agree implicitly to work in one of the areas within my range of current interests. Not because they are in any sense more `important' but because this is what I can and am potentially willing to do.
Please have a look at my recent papers (the web is such a wonderful source of information!) and think what there interests, indeed excites you. This is what I have meant by "doing your homework".
It is not good enough that you want to work in a subject, an important judgement call is how well are you prepared to work in that subject? Has the trajectory of your mathematical education prepared you sufficiently well to grapple with it? It is not sufficient that you consider yourself prepared, I will want evidence and the final judgement is mine and mine alone.
A major reason why we want our students to go through the CCA is that we wish them to have broad and rounded mathematical education, in particular throughout mathematical analysis. Unless you already have broad background and unless you rejoice in cool mathematics just because it is cool then, probably, the CCA is not for you and I am not the right PhD supervisor.
It is not just about you working with me, it is also about me working with you. Have a look at my website and ask yourself "do I want to spend the next four years of my life working with that guy?". If you answer in the affirmative, fine, we can talk further.
I have no delusions of grandeur and fully expect you to approach other prospective PhD supervisors as well. It is your life, after all.