Linked Questions

182 votes
33 answers
32k views

What should be learned in a first serious schemes course?

I've just finished teaching a year-long "foundations of algebraic geometry" class. It was my third time teaching it, and my notes are gradually converging. I've enjoyed it for a number of reasons (...
103 votes
13 answers
37k views

How misleading is it to regard $\frac{dy}{dx}$ as a fraction?

I am teaching Calc I, for the first time, and I haven't seriously revisited the subject in quite some time. An interesting pedagogy question came up: How misleading is it to regard $\frac{dy}{dx}$ as ...
26 votes
12 answers
2k views

Examples of improved notation that impacted research?

The intention of this question is to find practical examples of improved mathematical notation that enabled actual progress in someone's research work. I am aware that there is a related post ...
24 votes
5 answers
31k views

What is the standard notation for group action

Please let me know what is the standard notation for group action. I saw the following three notations for group action. (All the images obtained as G\acts X for ...
16 votes
5 answers
5k views

When did the abuse of notation $y=y(x)$ start?

It's quite common nowadays to name a function and the application of the function to its input with the same letter. (Possibly more so in applied areas. Certainly many calculus textbooks do this.) ...
Michael Bächtold's user avatar
20 votes
3 answers
1k views

central extensions of Diff(S^1) and of the semigroup of annuli

$\mathit{Diff}(S^1)$ refers to the group of orientation preserving diffeomorphisms of the circle. The semigroup of annuli $\mathcal A$ is its "complexification": the elements of $\mathcal A$ are ...
André Henriques's user avatar
9 votes
2 answers
1k views

Notation for the set of all injections from $A$ into $B$

Is there a common notation for the set of all injections from $A$ into $B$? Some set-theorists use $B^{(A)}$, e.g., A. Levy in his book Basic Set Theory. But some combinatorists use $B^{\underline{A}...
Guozhen Shen's user avatar
  • 1,782
41 votes
0 answers
1k views

Homotopy type of TOP(4)/PL(4)

It is known (e.g. the Kirby-Siebenmann book) that $\mathrm{TOP}(n)/\mathrm{PL}(n)\simeq K({\mathbb Z}/2,3)$ for $n>4$. I believe it is also known (Freedman-Quinn) that $\mathrm{TOP}(4)/\mathrm{PL}(...
Ricardo Andrade's user avatar
2 votes
1 answer
400 views

References for the categories: DIFF PL LIP TOP

Is there any survey paper focusing on the study of DIFF PL LIP TOP categories?
J. GE's user avatar
  • 1,101
3 votes
0 answers
238 views

How to denote a partial derivative?

This question is related to Was Jacobi the first to notice the ambiguity in the partial derivatives notation? And did anyone object to his fix? and Suggestions for good notation . When there are two ...
Alex Gavrilov's user avatar