I am an undergraduate student of mathematics. I am interested to know the conjectures that are proved in the last, say 5 or 10 years. Or any other development. Preferably in pure mathematics. One of the motives to ask this question is to get a feel for the cutting edge research areas. (If you feel this is too broad, please suggest me in comments to improve this question.)
EDIT: I am interested to understand (not just know the statement of) the theorems. In your answers, you could just state the theorems, with references where I can research. If proofs are also provided, that would of course be acknowledged with deeper gratitude.