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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.
24
votes
How does "modern" number theory contribute to further understanding of $\mathbb{N}$?
Let me illustrate the point with a specific and well-known example: the congruent number problem, perhaps the oldest open problem in mathematics. An integers $n>0$ is said to be a congruent number if …
- 18.7k
5
votes
Equality vs. isomorphism vs. specific isomorphism
Hi Minhyong,
I don't know whether this example throws some light on your question, but here it is :
Let $K$ be a finite extension of ${\bf Q}_p$, and suppose you want to count the number of degree-$ …
- 18.7k
15
votes
Collection of equivalent forms of Riemann Hypothesis
Li's criterion ?
9
votes
Collection of equivalent forms of Riemann Hypothesis
See M. Balazard, Un siècle et demi de recherches sur l'hypothèse de Riemann.
34
votes
Accepted
Algebraic number theory: building and simplifying
A major simplification in algebraic number theory occurred in the beginning of the 20th century when Hensel explicitly introduced his $\mathfrak{p}$-adic numbers. Compare the original cumbersome defi …
- 18.7k
4
votes
Mazur's torsion theorem on elliptic curves and its generalisations
The general case over arbitrary number fields has been treated by Loïc Merel. A good place to start would be Bas Edixhoven's Bourbaki exposé Rational torsion points on elliptic curves over number fie …
41
votes
current status of crystalline cohomology?
This is a "big-picture" question, but allow me to illustrate some recent progress by taking a small example close to my heart.
Let us adjoin to the field $\mathbb{Q}_p$ a primitive $l$-th root of $1$ …
- 18.7k
4
votes
Doing geometry using Feynman Path Integral?
If you read French, Henniart's survey Les inégalités de Morse. Séminaire Bourbaki, 26 (1983--1984), Exposé No. 617, 19 p. might be a good place to start. He explains Witten's analytic proof of the Mo …
- 18.7k
10
votes
What are interesting families of subsets of a given set?
The notion of a uniformity on a set was introduced by Weil in 1937 to formalise the notion of a uniformly continuous function. See Bourbaki's General Topology, Chapter II, Uniform structures.
- 18.7k