I'm writing a statement of purpose for my PhD applications and in some cases I'd be very happy if any person from a certain group (say the Algebra Group) would take me as his/her student. Could this be considered a weakness in my application? How should I handle this?
I've been researching people and groups for over 4 months and I know (as much as I can right now) what I want to do and who does it. It's just that some universities and programs have some amazing faculty.
I would include which certain aspects of the faculty would support your intentions and why, especially if you can include meaningful references to their previously published work.
For example
"I believe that University X would be a great place to undertake my research into topic Y because it has a great reputation in this field. For example books A and B published by person C are well known texts in the field and I believe the research of person D in the area of E would compliment mine."
You should definitely mention your research group preferences. At my university, PhD applicants are required to do so. This implicitly forces the applicants to learn more about the research groups that are offering PhD positions. If your application get through the selection process, you will be interviewed by the professors of your choices. It would be a big plus if you can make an impression that you have general understanding of their research work. It shows that you have passion for the topics and also increases your chance of being accepted.
If none of the professors think that you are a match for them, then let it be and choose different PhD Programme. You should not do your PhD in a research group blindfolded, just because you are accepted there. If you don't like the topics, you could run into the risks of not being able to finish your PhD and wasting few years of your life. Academia is all about passion, without it your chance of success is very small.
I think this is a high-risk high-reward scenario. I did this; I called out one particular professor, and I nailed it because he accepted me, but this same tactic failed on another graduate school application. At least where I went, when an application letter mentions a specific professor, internally that application gets sent to that professor for review. If you are a match for them, you have just won a spot on the team of your choice. But if you call out this professor and they are not hiring or do not like your application, you will probably alienate others. So, I think it is risky due to potentially alienating others who may have seen your application if you didn't specify, but can be rewarding because it can land your application on their desk.
The sad, but true, situation is that admissions committees are often aware of the needs and funding status of the faculty they are serving. If the people you list are actively looking for students, you might have an easier time with admissions, but if the people you list are in funding trouble, or their labs are full, that might be a bit of a negative.
In the long run, though, programs are looking for good people. The impact I describe is more of a nudging effect than the big decision maker.
I'd recommend mentioning your interest, but also mentioning that you're flexible. It shows good planning.
