What is so special about Arnold's Trivium?

Solution 1:

(Too long for a comment.)  Quoting author's motivation from the source (page 2 of the linked pdf):

In Feynman's words, these students understand nothing, but never ask questions, so that they appear to understand everything. [...] The students reach a state of "self-propagating pseudo-education" and can teach future generations in the same way. But all this activity is completely senseless, and in fact our output of specialists is to a significant extent a fraud, an illusion and a sham: these so-called specialists are not in a position to solve the simplest problems, and do not possess the rudiments of their trade.

Thus, to put an end to this spurious enhancement of the results, we must specify not a list of theorems, but a collection of problems which students should be able to solve. [...] The compilation of model problems is a laborious job, but I think it must be done. As an attempt I give below a list of one hundred problems forming a mathematical minimum for a physics student.

Solution 2:

There's nothing inherently special about the problems (and they're not even, say, math-contest-style problems); they're just intended to be reasonably comprehensive (with respect to an undergrad math education) and demonstrate the virtues of a written rather than oral examination. There's nothing there that anyone with a undergrad math degree should have any problem with, possibly modulo looking up a couple of specific formulas. He published a note after the problem set comparing it to other such examinations, if that's helpful to you.

That having been said, the subjects chosen aren't what I would expect. There's a lot of involved calculation, including some questions on numeric approximations. There are pages and pages and pages of questions about differential equations; in fact, the questions are heavily skewed to applied math in general. There are a couple of desultory questions about group theory and probability at the end, but that's about it; the rest is real and complex analysis (including a bunch of computations of specific integrals) and dozens of questions about differential equations. There's nothing about set theory, ring theory, commutative algebra, Galois theory, topology (beyond one question about Betti numbers and a couple about Riemann surfaces), Lie algebras (Problems #89 and #90 don't count), representation theory, any group theory that one wouldn't get in a physics class, etc. That would make more sense if Arnol'd was using it as a way to weed out prospective grad students working specifically under him, but it focuses on an extraordinarily narrow and applied curriculum. It doesn't strike me as a great way of demonstrating mathematical understanding.