Search Results

I find there is a world of difference between explaining things to a colleague, and explaining things to a close collaborator. With the latter, one really can communicate at the intuitive level, beca …

The surface area $|\partial S|$ of a (bounded, smooth) body $S$ is the derivative of the volume $|S_r|$ of the $r$-neighbourhoods $S_r$ of $S$ at $r=0$: $$ |\partial S| = \frac{d}{dr} |S_r| |_{r=0}.$ …

Here are some that came to mind: Equivalence. Basically, the idea that two things can be functionally equivalent (or close to equivalent) even if they look very different (and conversely, that two t …

This isn't an exact analogue to P != NP, in which two large classes exist and it is undecided whether they are equal or not; instead, two large "universes" exist, of which only one is the truth, with …

If $p$ is a prime, then every minor of the Fourier matrix $(e^{2\pi i jk/p})_{1 \leq j,k \leq p}$ is non-singular. This fact was proven by Chebotarev in 1927 (answering a question of Ostrowski), Dani …

In additive combinatorics, one often seeks to count patterns such as an arithmetic progression $a, a+r, \ldots, a+(k-1)r$. When doing so, one is naturally led to expressions such as $$ {\bf E}_{a,r \ …

Many of the standard abstract mathematical structures were first defined and studied "externally" (in terms of some sort of concrete representation) and only later defined "internally" (as abstract sp …

Van Vu and I coined the term in our book because there did not seem to be a widely adopted name for it previously. (Gowers, for instance, refers to "number of additive quadruples" rather than "additi …

It seems that certain problems seem to induce this sort of new thinking (cf. my article "What is good mathematics?"). You mentioned the Fourier-analytic proof of Roth's theorem; but in fact many of t …

I wanted to add one further point to the many good answers already given here: "black box" symbolic computation, in the absence of understanding the formal definitions, can work when everything goes r …

I believe that Fefferman's disproof in 1971 of the $L^p$ boundedness of disc multiplier for any $p \neq 2$ was considered a great surprise at the time; it showed that the classical result of Bochner a …

(Converted from a comment to an answer as requested.) Non-Euclidean geometry was initially developed in hopes of deriving the parallel postulate from the other axioms of Euclidean geometry, as can be …

This is almost certainly not the historical justification for the term "smooth", but I have found that the term happily coincides with my analytic intuition of smoothness. Namely, I think of a scalar …

After passing to a subsequence if necessary, a sequence of real numbers either (a) converges to a real number; (b) diverges to $+\infty$; or (c) diverges to $-\infty$. In a similar vein, a sequence …

Lie's third theorem - that every finite-dimensional Lie algebra (over the reals) is the tangent space of some Lie group - is quite difficult to establish without first establishing Ado's theorem that …