The University of Pennsylvania had a successful grad-student-organized seminar series called the "Pizza Seminar" when I was there (2005-2010).  It appears to [still be active][1].  Some people talked about their research, others  about just math they found interesting.    Grad-student-only audience, with the occasional faculty speaker.  At the time it probably had more pure talks than applied just because of the department population; it looks like this is still true.


  [1]: https://web.sas.upenn.edu/pizza-seminar/