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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.
35
votes
Universal definition of tangent spaces (for schemes and manifolds)
Consider the real line $\mathbb R$ and $C^1_0$ , the ring of germs of continuously differentiable functions at zero.
Now take the ideal $M$ of germs vanishing at zero. The Zariski cotangent space $M/M …
- 2,821
14
votes
Fundamental Examples
In the theory of holomorphic functions of several variables, Hartogs's theorem that any holomorphic function on a punctured open set of $\mathbb C^n$ ($n\geqslant 2$) can holomorphically be continued …
- 6,210
9
votes
Are infinite dimensional constructions needed to prove finite dimensional results?
Serre and Cartan wrote a Comptes Rendus note Un Théorème de finitude concernant les variétés analytiques compactes where they proved that all cohomology complex vector spaces $H^q(X,\mathcal F)$ of an …
- 47.5k
44
votes
Theorems that are 'obvious' but hard to prove
That $\mathbb R^n$ has topological dimension $n$. In a similar vein that affine space $\mathbb A^n_k$ over a field $k$ has Zariski dimension $n$.
- 47.5k
16
votes
Justifying a theory by a seemingly unrelated example
Let us call "division algebra over $\mathbb R$" a finite-dimensional vector space $A$ equipped with a bilinear map $A \times A \to A: (a,b) \mapsto a \bullet b$ , such that $a\bullet b=0$ implies $a= …
- 47.5k
38
votes
Accepted
Justifying a theory by a seemingly unrelated example
[In front of a blackboard, in an office at Real College]
Skeptic: And why should I care about holomorphic functions?
Holomorphic enthusiast:$\;$ Can you compute $\quad$ $\sum_{n={-\infty}}^{\infty …
- 47.5k