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 math GPA is not the best, but I have taken some very challenging classes, and my LORs will all be pretty strong.