Best summer plans for Math PhD student?

[dqaupop]

I am a current Mathematics Ph.D. student in my first year. In undergrad, by far the best way to spend a summer was doing research, and there were no shortage of programs to apply to. Now that I'm in grad school though, I don't even know where to start looking. What sort of research programs should I be looking for in order to advance my career? Are there any in particular you would recommend? I mostly specialize in Algebra and Combinatorics, and am currently at a northwestern US school.

[Stat Assistant Professor]

Will you be finished with all your qualifying exams before next summer? Most PhD students don't start research until they have completed all their qualifying exams. The first summer after their first year, most PhD students in math/statistics teach a summer class and/or prepare for their qualifying exams.

If you want a "headstart" on research though, I guess you could start reading some papers to get accustomed with the academic jargon. But I would definitely prioritize studying for your qualifying exams over anything else.

[bayessays]

I would say prep for quals and go on a vacation so you're not part of the 50% of people that will fail out or quit the program before finishing. Statistically speaking, that might be the best thing to do for your career. 

[dqaupop]

I passed one qual on the way in, and am pretty confident about the other one, given how I've been doing in the corresponding class so far.

[Stat Assistant Professor]

I second the suggestion of going on vacation. Or teach a class if you need extra money. You could also ask your graduate coordinator for summer suggestions.

In math and statistics, most people don't start research until after the qualifying exams and the bulk of coursework (minus a few electives possibly) are finished. As long as you meet these milestones, you should be able to graduate "on time."