GASARCH over at Computational Complexity asks:
- What should everyone know?
- What does everyone know?
The questions are supposed to be in the context of mathematics, that is, particular theorems and results that you learn about in your academic career far later than you should have. If someone has a Ph.D. in mathematics, there are definitely certain “simple” things they should know. For example, elementary real analysis (cantor set), how to prove the irrationality of sqrt(2) using contradiction, and other basic results that produce hundreds of thousands of google search results.
When it comes to research, one big problem is that people often come across neat math results, but are unfamiliar with terminology/notation that already exists for the result. This has probably happened to every mathematician who has done some serious research. It’s happened to me, and also to my supervisors where certain math concepts have different terminology depending on the field.
Regarding personal experience, the University I am at hired an academic as a professor position who believes there are square matrices that don’t have any eigenvalues.