Interesting advanced math topics?

Hi, I'm a senior in high school, and I'm in a class called math seminar. It's for kids who've finished all of the other math classes. Our teacher is very smart but also very oblivious. By this, I mean I feel like we don't really learn anything even though its supposed to contain interesting stuff. He Pretty much just rambles about vector space. Can someone Please mention a few advanced mathematics topics that could be taught in this class? At my school we have classes for things like Lin. Alg. And vector calc., so I don't need those. Also, I'm not asking for someone to just say quantum mechanics. I'm asking what are some areas where, in a 2 month period, a group of intelligent high school seniors who have all at least taken calc bc could learn really cool math stuff (through a teacher, assuming he knows A LOT about math). If you think the beginning of something like quantum mechanics would be worthwhile and can explain why then thats a different story.
 
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