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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.
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 …
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 …
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 …
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 …
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 …
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 …
39
votes
Describe a topic in one sentence.
Complex Analysis: Taylor series behave the way you want them to in real analysis.
59
votes
Accepted
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 …
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 …
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 …
13
votes
Justifying a theory by a seemingly unrelated example
If you're a combinatorialist and you want to know the asymptotics of a sequence $a_n$ with a nice generating function $A(z) = \sum_{n \ge 0} a_n z^n$, the very first thing you should do is find out if …
5
votes
Justifying a theory by a seemingly unrelated example
Suppose you are interested in random walks on an extremely structured graph such as a hypercube graph or a cycle graph. If your graph happens to be the Cayley graph of an abelian group $G$, as in bot …
9
votes
Generalizations of "standard" calculus
I can answer your last question, at least. The derivative acts as a shift operator on Taylor series, so the operator $\frac{d}{dx} - 1$ acts as the forward difference on Taylor series. So their eige …
- 118k
10
votes
Why are modular forms interesting?
Lots of good answers so far, and I hope that someone talks about Moonshine at some point. Let me just briefly mention an application which hasn't been mentioned so far to class field theory. Special …