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Results tagged with big-picture
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user 1532
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.
66
votes
What notions are used but not clearly defined in modern mathematics?
I have three (somewhat related) examples:
The notion of explicit construction. Seeking explicit constructions to replace non-constructive existence proofs is an old endeavor. Computational complexity …
63
votes
Fundamental Examples
The Petersen Graph in graph theory.
61
votes
Accepted
How to find ICM talks?
Update: (Oct. 2018) For the first time, all ICM 2018 lectures (plenary, invited and special) as well as panels and special events are presented (by good-quality videos) on the ICM 2018 You tube channe …
- 24.7k
55
votes
Fundamental Examples
Brownian motion has a central role in the theory of stochastic processes
46
votes
Fundamental Examples
The Prisoner's Dilemma in Game Theory.
42
votes
Fundamental Examples
In complex dynamics: The Mandelbrot set
42
votes
Accepted
Why is the current math community not contributing to machine learning much?
This is a interesting question but, in my opinion, are several misguided or at least questionable conceptions:
1) The future is what matters.
Scientists should concentrate now on what will have mo …
36
votes
Fundamental Examples
Russell's paradox
Showing that Frege's set theory leads to contradiction.
The example can be described as the set of all sets $A$ such that $A \notin A$.
It is related in spirit to Cantor's proof …
33
votes
What is an important mathematical question?
The question what makes a mathematical problem worth studying and even important is itself an important meta question about mathematics. Here are a few points (at time subjective) one can consider
Di …
32
votes
Fundamental Examples
1The Fano plane in finite geometry
http://en.wikipedia.org/wiki/Fano_plane
27
votes
Fundamental Examples
The quaternions
Probably the natural numbers, real numbers, and complex numbers are "too fundamental" to count here. But the field of quaternions discovered in the mid 19 century qualifies.
27
votes
Examples of major theorems with very hard proofs that have not dramatically improved over time
The Graph-Minor Theorem.
A graph $H$ is a minor of a graph $G$ if it can be obtained from $G$ by a sequence of deletion and contraction edges. Roberton and Seymour's graph-minor theorem asserts that …
26
votes
Fundamental Examples
In geometry, group theory and other areas: The Leech Lattice:
22
votes
Experimental mathematics leading to major advances
I have one personal experience with experimental mathematics. It started with a computer assisted simple proof of a result that every five dimensional polytope contains a 2-dimensional face which is a …
21
votes
Examples of major theorems with very hard proofs that have not dramatically improved over time
The classification of finite simple groups
This theorem describes completely all finite simple groups: A finite simple group is either cyclic groups of prime order, alternating groups, groups of Lie …