Would Math professors ace PhD qualifying exams?

[hutchphd]

A series of odd events led to my teaching the undergraduate quantum course (Gasiorowicz) as I finished up my thesis research. I set a very good (if I do say so myself) 90 minute final exam which progressively increased in difficulty: I expected no one to complete it and said so. One student produced an exam paper which was not only entirely correct but was clearer, cleaner, and more lucid than my carefully posted solution set. To this day I cannot describe my ambivalence !

 

[atyy]

Folklore sets Hilbert as the last mathematician who was well versed in all areas. I do not know for sure.

According to the Wikipedia on Terry Tao, Timothy Gowers wrote "Tao's mathematical knowledge has an extraordinary combination of breadth and depth: he can write confidently and authoritatively on topics as diverse as partial differential equations, analytic number theory, the geometry of 3-manifolds, nonstandard analysis, group theory, model theory, quantum mechanics, probability, ergodic theory, combinatorics, harmonic analysis, image processing, functional analysis, and many others. Some of these are areas to which he has made fundamental contributions. Others are areas that he appears to understand at the deep intuitive level of an expert despite officially not working in those areas. How he does all this, as well as writing papers and books at a prodigious rate, is a complete mystery. It has been said that David Hilbert was the last person to know all of mathematics, but it is not easy to find gaps in Tao's knowledge, and if you do then you may well find that the gaps have been filled a year later."

 

[mathwonk]

Re:post #30. That is impressive, as apparently only 5 perfect scores on the putnam have occurred in its first 82 years (through 2019).

In these discussions of Hilbert, I am reminded of the famous saying that he apparently did not know the definition of a Hilbert space, which does occur on some exams. So it is easy to ask a question in a way that will be challenging. Galois probably didn't know what a Galois goup was either. And Euclid probably couldn't define a "Euclidean domain", etc...

Re: post #29: I don't think it would be hard to find people who know a lot of those areas. I myself know "something" about 20 or 30 of the 60 pure math areas, and I am just an average low level retired mathematician. I have colleagues from my own school who know far more, and people like David Mumford or Curt McMullen, or Robin Hartshorne, or Yuri Manin, or John Morgan, or Jean Pierre Serre, know essentially infinitely more. Given the opportunity, and 50 or 60 years devoted to it, one can learn a lot. Some of us spent our lives teaching calculus over and over, but some people at elite places spent decades pushing the boundaries.