I just submitted my PhD applications. What do I do now?

Is there anything I can do during the wait to help my application?

I'll answer assuming you are applying to programs that work like most U.S. math graduate programs. In other words, programs where admissions are decided by a departmental committee, rather than by individual professors, and where there are no interviews or other direct involvement of the applicant after the application is submitted. For other sorts of systems, I'm not competent to offer any advice.

To a first approximation, there's nothing you can do during the wait to help your application. It's best to find a distraction and try not to worry about it. That being said, there are several things to keep in mind:

1. You should make sure your letter writers actually submit their letters. Every year, a few letter writers are many weeks late, and this can cause real damage. Even once you have submitted everything you are responsible for, the application is not complete until enough letters are received.

2. Getting in touch with potential advisors might help, but it should be done exceedingly sparingly. If the content of your e-mail amounts to "please look at my application" or "see how great I am", it will only annoy people. "Please estimate my chances of admission" is even worse. I'd recommend restricting pre-admission e-mails to cases where you have something genuinely substantive to say. For example, if you have intellectually serious comments or ideas related to the person's research, then that's always worth an e-mail. Same thing if you would like to bring their attention to a paper of yours that is closely connected with their work. A good test is whether you would still send the e-mail even if you weren't applying to their university. I don't recommend manufacturing a contrived reason to send e-mail, but if you never got in touch with someone whose research you extended, now would be a good time to do so.

3. It's usually possible to add information to your file if something important comes up (e.g., a paper acceptance or a prize). This is not worth even trying unless it's a real improvement in your file, and you certainly shouldn't make repeated changes in an attempt to draw attention to your file (any additional attention will be negative). However, if you get good news you should consider sharing it. The best news is often financial: if you get a fellowship that would pay for one or more years of grad school, then you should definitely add this information to your file. It's both a serious vote of confidence from another organization and a real savings to the department.