Last term I started a weekly seminar like thing. The basic idea was this — At the beginning of the term each person will choose a topic that they don't know about and isn't usually covered in the standard undergraduate math curriculum. They'll work on it throughout the term and present it at some point during the term. Overall it was a success as most people involved got exposed to mathematics that they wouldn't have otherwise.
Last term I suggested topics to each person one by one based on their preferences and experiences. For example some of the talks we had were on: "How to have fair elections – Cryptography and Voting theory", "Quandles – The algebra of knots", "Hypercomplex Numbers", "Basic Elliptic Curves", "Ultrafilters and Non-standard analysis".
However this term I am struggling to come up with topics for the 20 or so people that have signed up to speak. So I'd like to ask if you have any suggestions for topics that are interesting and accessible to 2nd, 3rd year undergraduates but that are also not covered in a standard undergraduate curriculum.
PS So far for this term I have "P-adic Analysis" (the book by Katok is great), "Bernoulli Numbers and their Application in Number Theory", "The mathematics of Bitcoins", and "Penrose Tilings". I also understand that any book in the Student Mathematical Library will work.
Best Answer
You might try pilfering a few topics from summer schools (particularly ones your students don't have access to, for whatever reason). For example, the LMS Summer School page has links to what was talked about in previous years, and for some you even get free lecture notes.
Among the topics listed there, I would recommend (mostly out of my own interests):
In particular, a set of notes for each of these topics can be found on the LMS pages linked above, which include references to other useful sources.