Every time I meet Peter Sarnak, he mentions his deep chagrin at being unable to blog, a sad state of affair which he attributes to his inability to type (straight quote). However, as he points out, he does the next best thing: he writes letters to various correspondents to explain a variety of topics, and then puts them on the web (together with some other papers, notes for lectures, and a few papers). And all this is well-worth looking at for inspiration and information. And since the address doesn’t seem to be as well-known as it could, here is the link.
Highlights of these are (to my mind at least):
* The Schur lectures on Arithmetic Quantum Chaos (1993); they have been published, but are not so easy to find. There has been a lot of recent progress (which I discussed briefly here; Sarnak also has a short letter on this topic, and is readying a more complete version as he prepares a lecture next week at IAS).
* The letter to Lagarias on integral Appolonian packings, a very beautiful topic where spectral theory of infinite covolume discrete subgroups meets number theory and geometry.
* The Notes on the Generalized Ramanujan conjectures, a great educational read for people who (like me) have been mostly comfortable with classical (GL(2)) modular forms, but want to see and learn what happens in higher rank.
* The letter to Morawetz on the supremum norms of eigenfunctions; again, it’s a topic where new phenomena occur when one moves beyond the classical case of hyperbolic surfaces (which still contains many mysteries, of course).