In some math job advertisements, it says "the deadline is...", but some schools review applications before that. Does that mean we should submit applications as early as possible to be in a better position? (Because I normally apply when the date is close to the deadline). (I just want to make sure I understand things correctly). Thanks!
I attended a panel talk about this recently and the uniform opinion of the panelists was that one should submit job applications as soon as possible regardless of the deadline. Some schools start reviewing applications much earlier than the deadline (this includes my university). Of course, as long as you send in your application before the deadline, it will be looked at. The reason the panelists mentioned for early applications is that review committees tend to swamped just before and after the deadline. So if you get in your application early, there is more chance that you will get a more through look and if you are a candidate who might otherwise be on the fringe, this might work in your favor.
Also, here is a useful article in the AMS: (page 1024 answers your question)
I am quoting the relevant part for convenience:
When Do I Apply for Jobs? Most deadlines are between December 1 and January 15. For nonpostdoctoral positions, it is very helpful to send in your application by mid- December even if the deadline is much later, especially if you will be at the Joint Mathematics Meetings in January. For schools that conduct interviews at the JMM, you are better off if you have already applied and the schools have had the chance to look over your application. In mid-December, you may want to email the schools and remind them that you will be at the JMM and would be happy to meet with them there. If you are giving a talk, this also lets you send them the title, time, and location if you didn’t know those earlier. A short polite email, especially if you mention specifics about the individual school, may lead directly to an interview and is unlikely to bother anyone.
My only experience is applying for research postdocs last year, and a lot of schools started looking at applications shortly after Thanksgiving. Some of my friends had offers the first week of December. So I would say apply by mid-November to be safe. This might only apply to research postdocs though.
I don't like the blanket advice "apply as early as possible" because for many people, including me, the late summer and early fall of application season is a very productive time, and you might get a lot of research done. If you're hot on the trail of something good it's better to finish it off than apply early, especially if you're a grad student with not so many results. Of course if you're at a natural stopping point in your research (e.g. because you're essentially stuck) then it's better not to procrastinate and get the applications in early. Professors have access to your file on mathjobs as soon as you apply, and maybe someone in the department where you're applying is bored and browsing...