: Some things I've worked on since coming to CMU (excluding any research I do).
An Introduction to Analytic Number Theory
: I've found these notes on analytic number theory to be fairly gentle and easy to read, with lots of interesting examples and exercises. It culminates in a proof of the Prime Number Theorem, which is nice. A strong background in elementary number theory and analysis is advisable.
: An online ebook copy of Richard Diestel's Graph Theory textbook. As one of the popular graduate texts on graph theory, plenty of important ideas are thoroughly developed. A strong background in combinatorics (and some algebra) is advisable.
: An interesting result about modifications to the harmonic series that force convergence. I created a Mathematica demonstration visualizing this concept more generally here
A Mathematician's Lament
: A famous exposition by Paul Lockhart about the beauty of mathematics as well as challenges in how people interact with it.