Yes, I saw a lot of really good students enter an applied math program when I did. I left with a Ph.D., but nearly all the rest didn't. The flunk out rate was

Yes, I saw a lot of really good students enter an applied math program when I did. I left with a Ph.D., but nearly all the rest didn't. The flunk out rate was so high that Navy Seal training or Army Ranger training look like fuzzy, bunny play time. The key to my getting my Ph.D. was research, research, and research. The first I did was on a little question that came up in a course. In two weeks I got a surprisingly good answer that was clearly publishable. Later I did publish, no problem, in JOTA. That work gave me a halo in the department and made the rest easier. For my dissertation, I had identified the problem and made good, intuitive progress on it before I entered grad school. In my first summer I turned the intuitive stuff into math, theorems and proofs, wrote an 80 page manuscript, and that was the research for my dissertation. So, I found my own problem, found my own solution, did the work with no direction, and submitted the final work. So, in no meaningful sense did I have any dissertation advisors. So, I didn't have to depend on an advisor.