Statement of purpose addressing expected contribution and outcomes

Question

I am writing the statement of purpose for a PhD application (applied math, in the US) and wondering about how to address these two questions:

• What are your expected outcomes from the program?
• What do you expect to contribute as a student and later?

The university I am applying to expects answer for these questions in the SoP.

I am confused because at this point, I do not know which professor I will work with. And more importantly, what the professor (And I) will work on during PhD. Of course, it is possible to write a general statements such as I will conduct original research and expect to gain deeper understanding of the matter being investigated, and I will acquire skills of publishing a scientific article, etc. However, I do not think these sentences add any value to SoP.

Can someone comment on how to address these questions?

Edit:

I have a clear idea about the professors I will work with and the area I will work on. However, it is difficult to decide on a single professor because of many reasons. For instance, the said professor may not be able to accept any new student etc. Looking at the group's website, I can get ideas what the professor is working on, in general. But, one cannot say what projects will be worked on in future.

Answer 1

Note: I am not in the field of Mathematics. The below reflects how I would address this based on my experience elsewhere.

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It sounds like you have a very clear answer to both of these.

What is your expected outcome?

You're looking to perform original research, become more familiar with the process and responsibilities of the academic Mathematics field, and contribute to the broader community. You say as much in your post; write that in your letter! Furthermore, you added later that you know which sub-field you're interested in... write that specifically! "I'm interested in studying <field>, and specifically investigating <subfield>." That makes you sound even more promising, because you're coming in with an understanding of the literature and existing research.

The fact that you haven't identified an advisor is something they are unlikely to penalize you on, as its expected that you're still working on that. You should focus on convincing them that you have done your homework with respect to understanding what academic Mathematics is.

What do you expect to contribute?

You stated that as well; you want to do original research, publish, and gain expertise in a specific subfield of math. That's likely all they're looking for. They do say "...and later"; it may be worth mentioning that you're looking for a long-term academic position, as opposed to, say, banking. That will likely reflect well on you.