Does Using Ajax eliminate need for sessions?

I was just thinking, the way most web application are designed if you log in
you need to do session tracking to preserve state from page to page
since the web is stateless.

The problem of handling sessions is a well solved problem.

But if you are using xmlhttpreq to transfer data, and dom to rewrite
your pages on the fly. You are staying on the same page so you could,
especially for simple applications get rid of the need for session
management altogether.

This is all coming from basic principles. And it very early in the
morning so I don't know if I am firing on all thrusters but if you can
use Ajax to build your application on one page you can eliminate the
need for session management.

Am I missing something ?