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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 …
59
votes
Accepted
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
39
votes
Describe a topic in one sentence.
Complex Analysis: Taylor series behave the way you want them to in real analysis.
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
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 …
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 …
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 …
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 …
30
votes
Fundamental Examples
The Catalan numbers are definitely a fundamental example in combinatorics.
Answered by Qiaochu Yuan
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 …
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 …
20
votes
Thinking and Explaining
When I talk about determinants, I generally talk about something on the spectrum between "it measures how much volume scales" and "it's the induced action on the top exterior power." But the way I th …
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 …
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 …