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Results tagged with big-picture
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user 4362
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.
16
votes
Integrals from a non-analytic point of view
As the others have mentioned, integration over a connected oriented smooth manifold $M$ can be characterized (modulo some technicalities) according to the fact that it fits into an exact sequence:
$\ …
- 29.2k
13
votes
What are some slogans that express mathematical tricks?
There are two interesting tricks in K-theory / operator algebras / homotopy theory - one attached to an amusing slogan and the other with an amusing name - that I think foot the bill.
The first is "u …
60
votes
Accepted
What is the significance of non-commutative geometry in mathematics?
$\DeclareMathOperator\coker{coker}$I think I'm in a pretty good position to answer this question because I am a graduate student working in noncommutative geometry who entered the subject a little bit …
- 29.2k
9
votes
What is an important mathematical question?
Words like "important" and "natural" are obviously subjective, and that's OK! It shouldn't surprise you to learn that other people have opinions about the value of certain mathematical ideas, just as …
35
votes
1
answer
3k
views
Is there a common genesis for ADE classifications?
Recall that a certain type of object admits an ADE classification if there is a notion of equivalence relative to which equivalence classes of objects of the given type can be placed in one-to-one cor …
- 29.2k
19
votes
Theorems that are 'obvious' but hard to prove
I think that the ergodic theorem is a good example of this. In down-to-earth terms it says that if you have a box full of gas then the average velocity of all of the gas particles at a given time (th …
13
votes
The resolution of which conjecture/problem would advance Mathematics the most?
I am going to submit the Baum-Connes conjecture because probably nobody else will and I believe its importance is quite understated. The conjecture asserts that the assembly map from the K-homology o …
20
votes
Why are polynomials so useful in mathematics?
At risk of being overly bold, allow me to suggest:
Polynomials are useful because quadratic polynomials are useful.
If we can all agree that linear algebra is an indispensable tool in mathematic …
10
votes
What is the interface between functional analysis and algebraic geometry?
Have a look at how the Hirzebruch-Riemann-Roch can be deduced as a special case of the Atiyah-Singer index theorem. The idea is to consider the hodge operator $\overline{\partial} + \overline{\partia …
- 29.2k
20
votes
Accepted
Atiyah-Singer theorem-a big picture
I agree with @coudy's answer that the best approach is to first understand the theorem's special cases / applications / generalizations. That can help highlight some of the key pain points in the var …
- 29.2k