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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
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
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
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
27
votes
Why the triangle inequality?
The triangle inequality is natural. In any setting where the metric is related to some kind of optimization problem, for example if $d(a, b)$ measures the "length" of the "shortest path" between point …
- 118k
8
votes
What is the high-concept explanation on why real numbers are useful in number theory?
Someone once suggested on MO that this is because on the one hand Matiyasevich's theorem shows that no algorithm can solve Diophantine equations over $\mathbb{Z}$ (and the corresponding result is not …
- 118k
15
votes
Surprising and Useful Physical Intuition for Mathematical Objects
Kirillov's orbit method in representation theory establishes a correspondence (which is not exact in general) between irreducible unitary representations of a Lie group $G$ and orbits of the action of …
- 118k
8
votes
Surprising and Useful Physical Intuition for Mathematical Objects
There are several examples at the number theory and physics archive. To get you started let me mention the statistical-mechanical interpretation of the Riemann zeta function as the partition function …
- 118k
31
votes
Theorems that are 'obvious' but hard to prove
Subgroups of free groups are free. The plausible argument is that any relation satisfied in a subgroup must somehow translate to a relation satisfied in the larger group. Nowadays I guess most peopl …
- 118k
27
votes
What advanced area of mathematics can be delved into with only basic calculus and linear alg...
Stillwell's Naive Lie theory was essentially written as an answer to this question. I quote from the introduction:
It seems to have been decided that undergraduate mathematics today rests
on tw …
- 52.3k