<p>I’d second the comments by newmassdad. There are a number of options available, and as mentioned it is very important to have a plan (and backups) moving forward regardless of what options or fellowships he decides to purse. In terms of math, from an academic standpoint, the Part III math course in Cambridge is probably the most famous of its kind in the world and doing well there would certainly be a boost to any math career. However, that’s only a year long course so he would need a plan for the second year or do a PhD (although there are plenty of math options for a second year). The original comment sort of suggests that your son is considering doing his PhD at Princeton. I don’t know about math, but in most fields in the US it is generally frowned upon to get a PhD and undergraduate degree from the same University, especially if it’s all done in one shot. One professor I know said it’s like ‘academic inbreeding.’ At the very least, if he wanted to stay at Princeton, it would be a good idea to take a year or two of study elsewhere first. </p>
<p>I think a lot of time students focus on the issue of timeframe (e.g. if I take a year or two off to do something else first I’ll be ‘behind’) however, these worries are almost always unwarranted. For one, in the course of ones whole career a year or two is insignificant. Furthermore, I think most people that pursue time abroad through one of these programs (even if it means putting other plans temporarily on hold) would say that the time was well worth it and the education and experience they received (academic, cultural and otherwise) was invaluable and worth the time spent many times over.</p>