Statement of purpose addressing expected contribution and outcomes
Note: I am not in the field of Mathematics. The below reflects how I would address this based on my experience elsewhere.



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 &lt;field&gt;, and specifically investigating &lt;subfield&gt;." 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.