How would you teach anything in an age when the "arcana" or guild secrets had been made public? Well, you would teach. And you would not ask questions that had answers that could be called "answers" ...

Asking for the moon, in my view. Here are 10 "heuristics" that try to place the theory. NB that many people stop at #1, as if this were enough. None of these points is particularly easy to track in ...

Quotation from Gauss: "...the greatest thing is purely mathematical thinking: this is worth much more than the application of mathematics." In conversation in 1854, a few months before his death, ...

Here are three points, and you'd have to care about at least one of them, I think. (1) A (co)homology class is better understood if it is represented geometrically in some way. This point really ...

Armand Borel's Bourbaki Seminar 121 Groupes algébriques is from 1955, and uses "drapeau" (page 7). (It's online at archive.numdam.org.) This may not be the earliest occurrence, but there is a good ...

Epsilon-delta represents a pair of quantifiers (for all ... there exists ...). Challenge-response. The discrete mathematics take ought to be "hey, this is like a game", because if you iterate the ...

The "classical" example is surely duality of abelian varieties. If you want this duality to work over finite fields (or in characteristic p generally), it becomes apparent that you can't work with ...

I feel the answer is obviously "yes", and indeed that much of 19th century mathematics was lost, in a serious sense, for much of the 20th century. I was struck recently by discovering that Henry Fox ...

Some heuristic remarks are helpful only to a subset of readers. (Maybe that's true of all heuristics, as a meta-heuristic - if everyone accepts a rough explanation, it's something rather more than ...

Actually there is a power of 2. It goes to show the power of binary arithmetic ... : write 2n in binary and write zeroes after the initial one.

The answer is "yes" - the order mod p of 2 is almost always as large as the square root of p (actually you get epsilon less than this in the exponent). If you take r multiplicatively independent ...

I'm going to stick my neck out, rather than recycle some well-known phrases in sheaf theory. I suggest trying to answer this as two possibly simpler questions: 1) What is the intuitive meaning of a ...

Recall C. L. Siegel's rant, about the modern theory of abelian functions not having any functions in it. From a point of view that would have made sense to Weierstrass, mathematics has "addition ...

I just don't think it's true, despite my own tastes in topics. Such formulations are substantially a matter of fashion. There is one basic axis, running from very detailed information at one end (...

Topics in Complex Function Theory, Abelian Functions and Modular Functions of Several Variables by C. L. Siegel is a standard reference using complex function theory. There are older works (e.g. H. F. ...

http://en.wikipedia.org/wiki/Hilbert%27s_fifth_problem is a decent survey. In general in the discussion of "status" of the Hilbert problems, there are at least two recognisable routes. Route A is the ...

It's a tendentious question, certainly. It might mean, if Bourbaki, let us say, had had more of an interest in lattice theory, that the French word for "lattice" of this kind would be more familiar at ...

I nominate the Chinese Remainder Theorem, in the form of an isomorphism of a ring of residues with a cartesian product ring. This isn't "profound" mathematics, but simply unpacking it (with ...

This is a classical Diophantine equation (Mordell, Diophantine Equations, p. 258). Apart from n = 0, 1, -1, there is only the solution n = 24. Proofs by G. N. Watson (1919), W. Ljunggren (1952). ...

I don't know whether this will be helpful, but here goes. There used to be things called the "Laws of Thought", and they used to be equated (tendentiously) with sort-of axioms for rationality, when "...

The grandfather of all examples is by Gauss: http://en.wikipedia.org/wiki/Weil_conjectures#Background_and_history Of course Gauss didn't mention finite fields other than the prime field. I think it ...

Your argument fails because the bracket can (sometimes) take pairs of elements into the centre. Therefore the direct sum as vector spaces isn't necessarily a direct sum of Lie algebras. For nilpotent ...

Iwasawa gave a characterisation, assuming you are given a subfield F, discrete and such that the quotient is compact. The other conditions are R a semisimple locally compact commutative topological ...

The motivation "because I'd like to know [...] if it is impossible to prove FLT using elementary methods" seems to require comment. It is much more likely (in my view) that it is true that FLT can be ...

There are perhaps three or four themes lurking in here. NB that "undergraduate linear algebra" is perhaps an artificial construct, and examples set to test whether students understand basic concepts ...

I think, tautologously, any method proving the Riemann Hypothesis (or even seriously improving our knowledge on the zeroes) becomes "number theory" immediately. That said, I know what the question ...

It seems clear enough to me that Grothendieck was (perhaps is) sui generis as a mathematician, something that can be said of a few other mathematicians in each of the 19th and 20th centuries (e.g. ...

Algebraic geometry has been an important part of mathematics since Descartes, who pretty much invented it. In other words it is part of 17th century mathematics, like calculus. It happens that there ...

As you say, projective modules correspond in a highly moral way to vector bundles. Bundles pull back but do not push forward, in topologists' terms. This might be a good point at which to start. ...

Bertrand's postulate (http://en.wikipedia.org/wiki/Bertrand%27s_postulate).