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Results tagged with soft-question
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user 290
Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.
23
votes
0
answers
651
views
Which proofs of the fundamental theorem of algebra are "essentially the same" vs. "essential...
The classic MO thread Ways to prove the fundamental theorem of algebra contains $60$ proofs of FTA, and I'm sure there are many more in the literature. It would be nice to have some way to organize th …
- 118k
10
votes
Commutative rings : Topoi = Fields :?
This is a long comment. I would prefer to say that (Grothendieck) topoi are "(some) affine schemes over $\text{Spec } \text{Set}$." Here is my preferred version of the table, sprinkle $\infty$s accord …
- 118k
39
votes
Mathematical conjectures on which applications depend
The use of RSA for public-key encryption is widely believed to rely on the assumption that factoring is hard. Actually it relies on a stronger assumption than this, namely that the RSA problem is hard …
12
votes
Accepted
Semantics of derivations as derivatives
In all of these contexts, derivations are infinitesimal automorphisms, in the sense that $D$ is a derivation on $A$ (an algebra, a Lie algebra, etc.) iff $\exp(Dt)$ is an automorphism of $A \otimes k[ …
- 118k
8
votes
Can ETCC/ETCS talk about 'size issues'?
We can talk about a family of sets $Y_x$ parameterized by the elements $x \in X$ of another set using a bundle of sets, namely a map
$$Y = \bigsqcup_{x \in X} Y_x \mapsto X$$
whose fibers are the fa …
- 118k
28
votes
What is a field [Körper] really?
Fields are the simple (no nontrivial quotients) commutative rings. Grothendieck told us to work in nice categories with nasty objects rather than nasty categories with nice objects; fields are the nic …
- 118k
8
votes
Why do people say DG-algebras behave badly in positive characteristic?
In characteristic zero, for example, it's possible to find cdgas describing the rational cohomology $H^{\bullet}(X, \mathbb{Q})$ of a (say simply connected) topological space; this is the starting poi …
- 118k
4
votes
Need examples of homotopy orbit and fixed points
Homotopy fixed points and orbits make sense in a much wider (but not necessarily less elementary!) context than for spaces and spectra. In particular, they make sense for categories. Some examples:
…
- 118k
8
votes
Heuristics for 2-morphisms of (algebraic) stacks
Whatever a stack is, it should in particular be a functor $\text{CRing} \to \text{Gpd}$ from commutative rings to groupoids (a prestack). Groupoids naturally form a $2$-category and so such functors a …
- 118k
16
votes
Accepted
Examples of Kan extensions, adjunctions, and (co)monads in analysis, Lie theory, and differe...
The following is really an adjunction between $2$-categories but I am going to ignore that subtlety. This blog post discusses everything in more detail and with a few more examples.
Consider on the o …
13
votes
Math books for advanced high school students
The list I give undergraduates and strong high schoolers is here.
26
votes
Why higher category theory?
Re: David Corfield's comment, this answer will mostly address "higher as in $(\infty, 1)$-categories" rather than "higher as in $n$-categories."
I also do understand you need the notion of abelia …
87
votes
What non-categorical applications are there of homotopical algebra?
As a student, I'm always looking for organizing principles in mathematics to help me keep track of all of the mathematics I learn. It's easy to get lost in a deluge of definitions unless I organize th …
4
votes
What are trivial objects, in general?
I think nearly everything I call trivial is the initial or terminal object in some category (but not necessarily a zero object, as many familiar categories don't have zero objects). Let's go through a …
138
votes
Accepted
What is entropy, really?
Here is a simple story one can tell about the entropy
$$H = -\sum_{i=1}^n p_i \log p_i$$
of a discrete probability distribution. Suppose you wanted to describe how surprised you are upon learning …