I have been an organizer and frequently speak at the Undergraduate Math Club and the Graduate Number Theory Seminar at UMass. These talks are expository and not on my own research. It should be emphasized that slides below are intended to be a companion to the talk, not to include every word that will be said. Therefore, some of the arguments in the slides are deliberately oversimplified or incomplete.
Fermat's Last Theorem for Regular Primes (adapted from an undergraduate senior project done with Samantha Chen, Marietta Geist, and Alex Kiral).
Supersingular primes for elliptic curves over Q (content is from Elkies' paper).
Adelic proofs of the finiteness of the class number and Dirichlet's unit theorem (adapted from Cassels and Frolich).
Mazur's theorem for pointless genus one curves (content is from Dwarshuis et al. paper).