phd - What could/do applicants usually ask professors, and how does this reflect in a statement of purpose?

When applying for PhD programs (or MPhil or MS) that require only a statement of purpose and not a research proposal, I was advised to first check if some faculty members (or is one good enough?) in the university have the same research interests as I do. I remember reading or hearing something like, it may be helpful to read up on the research that those faculty members have done and possibly contact some of those faculty members, and then incorporate the reading and contacting in the statement of purpose.

1. So what could I ask those faculty members? What I think:

Hello Prof/Dr ___ .I'm interested in applying in your univ and am interested in X. Can I ask more about what you research? This is my background (I'm from blah blah blah. I have a master's, but it's in applied not pure math. I've done some self-study blah blah blah). Do you think I am a good fit for the university? Would you consider taking me on as a PhD student? Do you think I am better suited for MS/MPhil than a PhD?

2. And then how do I incorporate the e-mails in my statement of purpose?

Answer

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      Can I ask more about what you research?

      
      
      

      
      

      Don’t be so vague. You can find out for yourself what a professor is researching by reading some of their most recent publications. Think about the papers carefully and try and ask a pertinent question, for example:

      
      

      
      

      I was reading your paper X and have a question about Theorem Y. Is it related to Lemma Z as described in paper Q?

      
      

      
      
   
   
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      Do you think I am a good fit for the university?

      
      

      
      
      

      Instead of asking what the professor thinks, I would emphasise why I know I’m a good fit for that university. If you don’t like the look of the department, why bother contacting them in the first place?

      
      
   
   
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      Would you consider taking me on as a PhD student?

      
      

      
      

      I prefer “Will you be taking on a PhD student next year?”, although this may be just a cultural/ politeness thing.

      
      
   
   
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      Do you think I am better suited for MS/MPhil than a PhD?

      
      

      
      

      Again, I feel this is something you need to decide before contacting them. Look closely at the program requirements and think critically about your own skills and knowledge and where you will fit best. If you’re still in doubt, contact the admissions tutor with this question, not a potential supervisor.

      
      
   
   
   
   


2. 
   

   Many of the PhDs I’ve been applying for lately strongly encourage inclusion of a potential supervisor's name in the statement of purpose, so they can direct the application to the relevant person. You could include a mention of the professor you contacted in the following way:

   
   

   
   

   My research interests lie in the field of pure maths, and more specifically how Lemma Z can be applied to Theorem Y. I would like to work with Professor A in research group B.

   
   
   

   
   

All of this is based on my own experience of applying to UK physics PhDs (which I am still in the process of doing).