[graycat]

Yes, but he might stick around and take and pass the Ph.D. qualifying exams and then be well on his way to a Princeton degree: At least at one time, the Princeton math department Web site stated that, IIRC, "students are expected to prepare for the qualifying exams on their own and that no courses are offered to prepare students for the qualifying exams. Courses are introductions to research by experts in their fields."

So, look at the qualifying exams, see what might study, and while attending classes attend the ones that can help with the exams.

In addition, might have available some profs to answer questions.

In addition, likely one way to impress Princeton or any university is to publish, and one way to start to do that is to attend research seminars and see what some of the open questions are, also notice what some of the profs and grad students are working on. So, this way get some guidance on what might attack as a research problem.

I got a good pure/applied math Ph.D. Well over 50% of what I needed and used for courses, the qualifying exams, and my research was what I'd studied independently after my 4 year college degree and start of grad school.

Then a grad course in optimization gave a good introduction to the Kuhn-Tucker condition, maybe say Karush-Kuhn-Tucker. After the course I saw a tricky question about the constraint qualifications, didn't see an answer in the library, so signed up for a 'reading course' to 'investigate' the question. Two weeks later I had a nice, clean solution, wrote it up, and was done with the course -- two weeks. Later I published. There I'd noticed that my work also answered a question stated but not solved in the famous Arrow, Hurwicz, Uzawa paper applying the KKTC to economics. I published in JOTA.

So, a 'walk in' student at Princeton might have been able to have done much the same. With such research and passing the qualifying exams they would be on the way to a Princeton Ph.D.

[mcguire]

Many (Some?) schools operate that way---the first step in a PhD is a qualifying exam. Some students may be able to pass it immediately after their undergraduate studies, while others (most?) require some remediate studying---undergrad classes, I guess.

UT Austin, at the time I was there, used the other (and better, I think) method: breadth and depth graduate classes, typically more advanced versions of advanced undergrad classes. The breadth classes covered most of CS, while the depth classes were more introductions to specific areas of research.

Edit: Oral quals are just hazing, in my opinion.

[graycat]

On orals, I did some!!

Maybe the first was as a senior. I got Kelley, General Topology and once a week gave a lecture to a prof! One week a chapter, and the next some exercises! It was fun!

The next time was on a written qualifying exam: The exam had an error, and I wasted time trying to prove it! I asked for an oral. One guy tried to haze me, but the other profs were nice, and I walked out with a rare "High Pass".

The third time was for my oral defense of my Ph.D. dissertation.

I was out running, and when I got back my wife said I'd gotten a phone call from a prof, the Chair of the committee that was to approve my dissertation -- no pushover, the Chair and a majority of the committee had to be from outside my department. Well, the Chair called me. Still with sweat from running, on my back on the bed, I had the actual oral exam! He had a question about one paragraph, so I reworded it, had my word processing retype the thing, and he was happy.

Then the fourth oral exam was the dissertation defense -- it was for show since the real one had been over the phone. The Chair looked really serious and let me look really serious and good.

Maybe there was one more: I'd rushed ahead in freshman calculus and done the first year on my own. Then I asked to start on sophomore calculus and never take freshman calculus. So, the prof gave me in effect an oral exam on freshman calculus. He concluded I'd passed and let me in sophomore calculus. He did say that he couldn't give me course credit for freshman calculus, but fine with me -- I just wanted to get going, get on with calculus, and not repeat what I'd just studied well.

[hopler]

Princeton math (and other?) department specifically advertise that their research-only PhD program is unusual or unique.

https://www.math.princeton.edu/graduate

Generals/Quals require 1-2years of graduate level study, not remedial undergrad, if you look at the material covered and the syllabus and levels assigned to courses at various colleges. It just so happens that at a school like Princeton, 1st year PhD students have already taken graduate level classes in their field, or equivalent self study.