Reading circle in mathematics? How to learn about interesting topics in a small group of people?It seems very useful to broaden your mathematical background and get to know topics that are away from your field of specialization.
For instance, in philosophy, you can just gather some interested fellow students, pick a book, meet regularly and read some pages and discuss the content. Afterwards, you (hopefully) will have a reasonable knowledge of the arguments and theories sketched in the book. This is often done by people I know, and I have participated myself and enjoyed it a lot.
With mathematics this is more a problem, for many reasons. You often need more time for the same amount of text, it is hard to discuss proofs you don't really know yet, interesting examples require a lot of careful thinking, ...
Maybe there are some books or other resources that are well suited? Maybe some topics that are (in this way) better accessible than others? How should such a group be (or not be) organized? 

Two comments: Of course, one could prepare little presentations, read a lot in advance and so on. But then it is almost self-study, so that's not what I mean. Also, we may assume the members of this reading circle have good knowledge of undergraduate mathematics.
Thank you in advance.
 A: I don't know if the following is helpful. But anyway.
I agree that the "problem" with mathematics is that it is difficult to just sit around talking about it. We can't really just express our opinions about lower level math.
Another thing that makes it difficult is that if you meet with a group of people to talk about a topic, the topic usually assumes that you have a certain background level (as you also mention). If you want to talk about say abstract measure theory, then it is probably best if everyone in the group have seen integrals before.
However, I actually think that you can get away with meeting and going through a book together. You might not be able to do one book every week, but if you took the first chapters of a book and devote a full semester, then that might be successful. You can even find some good notes online so that you don't have to buy a book. I have participated in seminars where we have used notes by J.S.Milne. He has some great notes ranging from Group Theory to Class Field Theory. 
So as for organizing this: you might get a group of people together who are interested. Pick a book or some notes on a topic that you would like to learn. You should probably have some suggestions ready. Find a time that works for everyone, and then meet once (maybe twice) a week. Take turns giving a presentation over a section of the material and that way work you way through everything. Even though it might slow things down a bit, I do think that it is important to have everyone involved in giving presentations. One thing that I would recommend to do is to work out exercises. And don't get discouraged if/when you feel that you aren't making progress fast enough. As long as you keep learning stuff be patient. During the presentation you might (might not) adopt the policy that it is encourage to interrupt with questions (again to make sure that everyone is on board). This might also help to promote discussion.
If possible, you might also ask a professor at the math department to help you out with answering questions that you might have. Maybe the professor can also help suggest some topics to study.
Edit: Some topics that I have or know have been studied like this are


*

*category theory (great since many never have a class on this)

*homological algebra

*harmonic analysis

*basic representation theory

*basic number theory

*measure theory


Good luck!
