What are some good ways to figure out one's interests in mathematics, before applying to PhD programs? Are there mathematics seminars to attend in the summer, or do they typically only occur during the academic year?  I am referring to American universities.

An idea I had last night was to YouTube some math seminars to see what talks I can watch quickly - talks are about an hour long.  I found a good PDEs talk from a Princeton seminar.  I hope to find more talks that are available online.  

Is this not advisable?  These are talks given by experts, and as a lowly master's student, I naturally do not understand about 90% of the mathematics being talked about and written on the chalkboard.  I am mainly just trying to get some sense of purpose of, say, studying PDEs, its difficulties, etc.

I am also making progress on a master's thesis, but my gut feeling is that this problem is too narrowly focused and in the end may not help me figure out better my research interests in a PhD program.

What else could I do?
Are there mathematics seminars to attend in the summer, or do they typically only occur during the academic year?


I think none of the answers thus far actually addressed this.  At most US universities, seminars don't happen, at least not regularly, during the summer.  One reason is many faculty will be traveling.  However, sometimes some people, possibly grad students, will organize an informal seminar to read through some books or papers.  If other people aren't doing it, you can ask around to see if people are interested and organize it yourself.


  What else could I do? 


In place of seminars in the summer, there are usually lots of conferences and workshops as well as summer schools specifically geared towards young people.  Look around for what's available (you can try asking some professors if they have suggestions) and go to what you can.  Many of these have funding for students, but unfortunately most of the funding may have already been given away by now.

Edit: By the way, it's not expected that you know what you want to research when you enter PhD programs (in math in the US, say).  Many people who start aren't even sure if research is for them.  Most programs are designed so that you can use the first couple of years figuring out what you want to do.  (And even later, research interests may change---I may be doing something completely different in a year or two from what I'm doing now.)