With the recent cut-backs in mathematics and statistics programmes around Australia, we are now being asked to rationalise our offerings and provide less choice for our undergraduates. So what are the essential threshold concepts that someone intending to major in pure mathematics ought to know? For example, “linear independence” is such a concept, but “group theory” is not a good answer as there are extraneous areas of group theory which are not essental (e.g., Grigorchuk groups). Nor is “linear algebra” a good answer as it is debatable whether everyone should learn what a tensor product is (but you are allowed to disagree with me on this one). So should every pure-maths graduate know…
(1) that every integer has a unique factorisation into primes?
(2) the epsilon-delta definition of a limit?
(3) the Prime Number Theorem?
(4) proof by induction?
(5) an axiomatic notion (e.g., vector space, group, ring)?
(6) finite Galois fields?
(7) a theorem by Euler?
(8) the basic ideas of cardinalities (e.g., rationals are countable)?
So please contribute so we can cut out courses!