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 Apr 3 awarded Notable Question Oct 12 comment Research and exposition: how does writing “basic” books affect your “serious” research work? Respectfully, I find this answer and the fact that it has been accepted slightly strange. Perhaps there is something I am missing, but I find it hard to believe that it is agreed that the most important motivation for writing a basic textbook is to attempt to improve notation ?? Notation is just the actual symbols and stuff right? Oct 12 comment Why do so many textbooks have so much technical detail and so little enlightenment? (Of course its easy to say that one should "do math" in lectures rather than just "tell math" but probably very difficult. Having been taught by Tom Korner, I found that he had almost, after many, many years of teaching, got this down: After stating some lemma, he would successfully put on the act of thinking through the proof on the spot by saying aloud the thoughts that any analyst would have when asked to prove the lemma and then turning those thoughts into the proof, in a way that seemed natural. (This also relates to fedja's comment elsewhere on this page)) Oct 12 comment Why do so many textbooks have so much technical detail and so little enlightenment? Regarding a), I certainly agree that one of the major goals is to learn to do* mathematics, but regarding b) & c), I am led to remember Prof Tom Korner's essay "In Praise of Lectures". Roughly speaking, he argues that the real point of a lecture is for students to actually see mathematics be done . So a good lecturer actually does mathematics, rather than just lists what needs to be learnt. Thus e.g. the basketball example must be replaced by e.g reviewing videos of a basketball game, which can be instructive to someone who wants to play the sport well. (1/2) Jul 14 comment What dimension bound is known on the singular set of a linear combination of eigenfunctions of Laplacian? I am not bothered about making any stipulations about multiplicity or normalization. I guess I could really ask it for a general linear combination $f = a_1\phi_1 + \dots + a_m\phi_m$; $a_j \in \mathbb{R}$. Jul 14 revised What dimension bound is known on the singular set of a linear combination of eigenfunctions of Laplacian? added some relevant context Jul 14 asked What dimension bound is known on the singular set of a linear combination of eigenfunctions of Laplacian? Jun 28 awarded Necromancer May 1 awarded Notable Question Apr 8 asked Closed geodesics in free smooth loop space? Mar 18 revised Reference for higher order Campanato Lemmas, e.g. Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}' Edited after Willie Wong's answer to expand title Mar 18 comment Reference for higher order Campanato Lemmas, e.g. Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}' Thanks Willie, this is exactly it. Mar 18 accepted Reference for higher order Campanato Lemmas, e.g. Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}' Mar 18 asked Reference for higher order Campanato Lemmas, e.g. Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}' Feb 13 comment Must the Lebesgue measure of a $\rho$ - neighbourhood of an $(n-2)$ - dimensional set be at least $c\rho^2$? Thanks for the comment fedja, I see your point. I don't know what made me think it was true now. Feb 13 asked Must the Lebesgue measure of a $\rho$ - neighbourhood of an $(n-2)$ - dimensional set be at least $c\rho^2$? Oct 7 awarded Nice Question Oct 6 comment Why do roots of polynomials tend to have absolute value close to 1? I know very little about the math going on in the background but a colleague demystified this for me by saying that basically it is a special feature of the basis you have chosen, namely the monomials. If you choose a different basis or even just weight each monomial by a factor, the roots will tend to congregate on a different set. Basically although a) any polynomial can arise and b) you chose them randomly... They aren't as generic as as you think; they've in fact been chosen in a special way. Aug 31 awarded Yearling Aug 31 awarded Disciplined