If you're a PhD student and want to give a talk, there are plenty of opportunities. (If you are a Part III student and want to give a talk, see the Part III Seminar Series). Gaining experience by presenting to friendly audiences can be useful, and the people below would be very pleased to hear from you:
Directions in research series — this runs most Mondays at 4pm during Michaelmas and Lent terms. It's an opportunity to tell your fellow PhD students about your current research, and it also aims to give Part III students a flavour of the work that goes on in DAMTP and DPMMS. Contact Alexander Shannon (adjs2) to take part.
Archimedeans seminars for undergraduates — more general maths talks. Contact Dean (publicity officer for the Archimedeans, dar53) if you'd be interested in giving one of these.
If you took Part III and want to stay in touch with friends from your courses, you can organise a return conference. These are informal conferences run by and for students. They are an opportunity for research students who took the Part III course but are now studying elsewhere to come back to Cambridge and share their experiences. Each conference is arranged by a group of students, some of them still in Cambridge and some at other universities. Anyone who is interested is welcome to attend and everyone gives a talk. So far five return conferences have been organised, get in touch with the committee if you'd like to hold the sixth.
A massive return conference was held in April 2009 — Beyond Part III. Another is already planned for 2010. We'd really like all Cambridge PhD students to be involved. If you're a CUGMS member interested in chairing a session then the Beyond Part III committee would very much like to hear from you.
In the summer of 2009 a group of research students from CMS went to Linyi Normal University in Shandong province to run a mathematics summer school for advanced mathematics students (final year and Masters level students) from Shandong province. A 2010 summer school is now in the planning stages.
See this page for details of how PhD students can help support Part III students.