New answers tagged

3 votes

Why do Grothendieck topologies used in algebraic geometry typically involve finiteness conditions?

To give an interesting data point, I can offer two examples of something that seems a bit more infinitary. If we consider the category of finite-dimensional manifolds, and covers being submersions (or ...
  • 31.8k
0 votes

Golden ratio in contemporary mathematics

Let $\rho$ be the binary substitution defined by: $$\rho(00)=\text{empty word}\quad\rho(01)=1\quad\rho(10)=0\quad\rho(11)=01.$$ Let $R$ be the self-map of $[0,1]$ associating to every $x=(0.w)_2$ the ...
1 vote

Golden ratio in contemporary mathematics

Golden ratio appeared in the recent breakthrough of Frankl's Union Closed Set conjecture by Justin Gilmer, as well as the subsequent optimization of the lower bound constant. It relies crucially on ...
13 votes

Why do Grothendieck topologies used in algebraic geometry typically involve finiteness conditions?

Finite presentation assumptions in algebraic geometry are usually there because of Noetherian approximation. Namely, if $f : X \to Y$ is a morphism of finite presentation and $Y = \operatorname{lim}_I ...
  • 2,708
2 votes

Domains that may require a good categorical background

Besides the CT-functional programming connection, in recent years a field of "applied category theory" (ACT) has emerged that seeks to apply category-theoretic ideas to fields beyond the ...
17 votes

Why do Grothendieck topologies used in algebraic geometry typically involve finiteness conditions?

I guess a conceptual explanation is that algebraic geometry deals with localizations in order to glue global from local data, but the functor $M \mapsto M[f^{-1}]$ only preserves finite limits. In ...
13 votes

Math papers where the only issue is that someone else could've done it but didn't

Let me take the liberty of rephrasing the question slightly. Does the mathematical community put undue emphasis on accomplishing something "difficult," and thereby undervalue certain highly ...
3 votes

Domains that may require a good categorical background

The blog by Bartosz Milewski comes to my mind. It focusses on the interplay between Haskell and category theory.
3 votes

Is there an alternative to the arXiv for uploading mathematical papers?

Aside the suggestion to send your preprint to a peer-review journal, here are some online repositories that I recently used together with (or as an alternative to) the arXiv (i.e., some of my ...
  • 313
1 vote

Has incorrect notation ever led to a mistaken proof?

I'm not entirely sure what the difference between wrong notation and wrong "underlying mathematics" is, so I'm going to present a few different examples, and hopefully this clarifies the ...
8 votes

Has incorrect notation ever led to a mistaken proof?

In module theory, there is a choice of which side the scalars acts on. Then, there is also the choice of which side the endomorphisms of the module act on. Let $M$ be a right module, with scalars ...
0 votes

Good "casual" advanced math books

Since no one has mentioned this I found the following trilogy by Ash and Gross fits your description: Fearless Symmetry , Elliptic Tales and Summing It Up. I personally had enjoyed reading them and ...
1 vote

Good "casual" advanced math books

I'm surprised that no one has yet mentioned The Symmetries of Things by John Conway, Heidi Burgiel and Chaim Goodman-Strauss. It's ostensibly an introduction to symmetry groups but touches into ...
3 votes

Has the mathematical content of Grothendieck's "Récoltes et Semailles" been used?

Yes, R&S proved to be influential in at least one sense, the mathematical work of Z. Mebkhout (part four is dedicated to him indeed: "À Zoghman Mebkhout l’ouvrier solitaire en témoignage de ...
0 votes

What are some very important papers published in non-top journals?

Graph minors. II. Algorithmic aspects of tree-width The seminal graph minors papers of Robertson and Seymour were mostly published in JCTB (one of the top discrete maths journals). But this paper, ...
1 vote

Shapes for category theory

A limit of an idempotent (viewed as a diagram whose shape is the "idempotent category") is a splitting of the idempotent. This cannot be expressed using directed graphs as shapes I think.

Top 50 recent answers are included