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Results tagged with big-picture
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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.
7
votes
7
answers
635
views
Given a sequence defined on the positive integers, how should it be extended to be defined a...
This question is inspired by a lecture Bjorn Poonen gave at MIT last year. I have ideas of my own, but I'm interested in what other people have to say, so I'll make this community wiki and post my ow …
8
votes
3
answers
406
views
How does one identify properties of objects with good "inheritance"?
When you are dealing with a very general object like a topological space or a ring, usually you impose an additional condition (such as compact Hausdorff or Noetherian) with the property that the subs …
- 118k
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 …
- 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 …
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.
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, …
59
votes
Accepted
55
votes
14
answers
10k
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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 …
- 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 …
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 …