Gap Year before Mathematics PhD Programs How does the gap year process work for math PhD admissions? For reference I am a student at a top 5 undergrad department in the US, applying to pure math programs. I would like to take a year off before attending a PhD program in mathematics to work in quantitative finance for a bit. My main interest is in Stochastic processes and mathematical analysis, but I doubt this matters for the question I ask. 

As far as I see it, I feel like I have two options if I want to take a gap year between now and graduate school.

The first: Apply now, get accepted somewhere, and tell them I am coming next year. (This is also contingent on whether the school even allows me to do this)

The second: apply next year, get accepted and attend like normal.

Now, there are a number of tradeoffs to evaluate. First, if I apply next year, I have more time to take more courses and increase my GPA, giving me a slightly more competitive application. Although I have already registered for the GRE, I would also have more time to study for it in case I don't do as well as I'd like. However, I won't be able to enjoy my senior year of college, because I'd need to not slack off and take demanding courses.

The only issue I have with the above is the uncertainty: I'd have to apply to grad school while out of school, whereas if I applied first and deferred later, I'd have a guaranteed spot, and be able to enjoy my gap year without worrying about applications and the GRE. Also, funding would be weird. I spoke to NSF GRFP people, who said that I can't apply for the fellowship and wait a year to start grad school. I'd instead have to apply the year immediately before I start.

Does anyone have any suggestions as to what I should do? My GPA is not the best (3.65, with mathematics GPA 3.73), but I have taken some very challenging classes, and my LORs will all be pretty strong.
As it turns out, some years ago I was the same situation you're in and took a gap year before enrolling in a PhD program in pure math. (Well, it wasn't exactly the same situation you're in. My gap year was because of military service rather than dabbling in quantitative finance.) Anyway, I decided on your second option and don't really have regrets about doing so. I was nervous of course, since I didn't have a guaranteed spot waiting for me, but I assume that had I applied in my senior year of college I would have been just as nervous anyway. And it was nice having the extra opportunity to retake the GRE math exam.

I can't tell you what to do, of course, but merely wanted to say that I find it unlikely that you're going to find too many math departments with information about guaranteed deferred admission posted to their website. (I'm by no means an authority on this. I'm a math professor at a small liberal arts college that does not offer graduate degrees in mathematics.) My rationale is that unlike undergraduate admissions where hundreds or thousands of students are admitted at once, the numbers in graduate math admissions are much smaller. For instance when I enrolled in Dartmouth's PhD program I was part of a class of only five students. Moreover, there are potential difficulties with funding in the event that some of the grad students are going to be supported primarily by grants. 

This isn't to say that deferred admission will likely be impossible. Just that I think you should send a message to someone working in admissions at each department you're thinking of applying to (there will be contact info for someone -- be it a professor or a departmental staff member -- on each department's website) explaining your situation and seeing what their policy is. My assumption is that you won't have too many problems getting a department to let you defer if you're able to convince them that your gap year is for a "good" reason, that you're likely to ultimately attend if they admit you and let you defer and that you're not planning on applying again next year and using their offer of deferred admission as your back up option. But you'd have to ask them to make sure.