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Questions designed to get an overview of a specific subject or body of results or to understand the relations among similar definitions, techniques or concepts appearing in different sub-fields of mathematics. While such questions by their very nature sometimes cannot be made very narrow and focused, it can be helpful to keep in mind that the design of MathOverflow does not make it a good fit for questions that are too broad.
99
votes
Your favorite surprising connections in mathematics
From an essay of Arnol'd:
Jacobi noted, as mathematics' most fascinating property, that in it one and the same function controls both the presentations of a whole number as a sum of four squares and t …
- 21.3k
54
votes
30
answers
7k
views
What are examples of good toy models in mathematics?
This post is community wiki.
A comment on another question reminded me of this old post of Terence Tao's about toy models. I really like the idea of using toy models of a difficult object to underst …
- 1,362
19
votes
3
answers
5k
views
What is the relationship between algebraic geometry and quantum mechanics?
The basic relationship in algebraic geometry is between a variety and its ring of functions. Arguably a similarly basic relationship in quantum mechanics is between a state space and its algebra of o …
- 4,713
17
votes
2
answers
841
views
What is the conceptual significance of supercommutativity?
A $\mathbb{Z}/2\mathbb{Z}$-graded algebra is said to supercommute if $xy = (-1)^{|x| |y|} yx$; in other words, odd elements anticommute. Why is this the "right" definition of supercommutativity? (Pu …
- 35.5k
79
votes
12
answers
13k
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Is there a high-concept explanation for why characteristic 2 is special?
The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, b …
- 6,237
20
votes
Why is the definition of the higher homotopy groups the "right one"?
There are many things to say here. Here's one. Suppose you want to classify all spaces up to (weak) homotopy equivalence, or equivalently all CW complexes up to homotopy equivalence. The zeroth step i …
- 118k
32
votes
What's a groupoid? What's a good example of a groupoid?
Personally, the reason I'm interested in groupoids is something called groupoid cardinality and some other related ideas (the link contains a lot of other links). A motivating idea here is that certa …
- 13.9k
38
votes
Accepted
Linear algebra in terms of abstract nonsense?
To my mind there are two classes of interesting categorical facts here, loosely speaking "additive" facts and "multiplicative" facts. Some additive facts:
Finite-dimensional vector spaces over $k$ h …
- 118k
55
votes
14
answers
10k
views
Does any research mathematics involve solving functional equations?
This is a somewhat frivolous question, so I won't mind if it gets closed. One of the categories of Olympiad-style problems (e.g. at the IMO) is solving various functional equations, such as those giv …
- 19.4k
47
votes
Accepted
Grothendieck says: points are not mere points, but carry Galois group actions
Suppose $k$ is a field, not necessarily algebraically closed. $\text{Spec } k$ fails to behave like a point in many respects. Most basically, its "finite covers" (Specs of finite etale $k$-algebras) c …
- 118k
31
votes
Your favorite surprising connections in mathematics
It is possible to compute the Betti numbers of a smooth complex variety $X(\mathbb{C})$ by computing the cardinality of $X(\mathbb{F}_{p^n})$ for a prime $p$ with good reduction and a finite number of …
- 18.4k
9
votes
What's so special about $1$-categories?
You should increase your category level if you think it'll help you understand something you're thinking about. Otherwise, don't.
To the extent that there's something special about $1$-categories, it …
- 118k
36
votes
Accepted
Why are polynomials so useful in mathematics?
Polynomials are, essentially by definition, precisely the operations one can write down starting from addition and multiplication. More formally, polynomials with coefficients in a commutative ring $R …
- 118k
18
votes
12
answers
5k
views
What are some fundamental "sources" for the appearance of pi in mathematics?
I thought it might be fun to ask this question as a way of celebrating Pi Day. One way in which people popularize pi is that they say that even though it's defined in terms of properties of a circle, …
- 456
6
votes
Getting the story of Dynkin and Satake diagrams straight
2 is false. The smallest counterexample is $\mathfrak{sl}_2(\mathbb{R})$. A necessary and sufficient condition for a semisimple real Lie algebra to be the Lie algebra of a compact Lie group is that th …
- 118k