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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.
1
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Generalized notions of solutions in various areas of mathematics
One of the most fruitful notion of generalized solution in optimization and combinatorics is linear programming relaxation. Quoting from the wikipedia article: In mathematics, the linear programming r …
1
vote
Generalized notions of solutions in various areas of mathematics
A form of "generalized solution" which I saw in various areas like for combinatorial optimization problems, for diophanine equations, for computational complexity purposes, and others is "statistical …
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 …
10
votes
Examples of major theorems with very hard proofs that have not dramatically improved over time
The decomposition theorem for intersection homology
The decomposition theorem for (middle perversity) intersection homology (for algebraic varieties) was proved in 1982 by Beilinson-Bernstein-Deligne …
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 …
12
votes
Your favorite surprising connections in mathematics
There are several surprises regarding convex polytopes:
A) There are combinatorial types of polytopes that cannot be realized with rational coordinates (first discovered by Perles). This is not the c …
13
votes
Proposals for polymath projects
Update (Aug 26, 2016), see Ofir's comment to this posting: Ofir Ofir Gorodetsky and Ron Peled have proved the identity!
Update 2 (Sept 27, 2016) In Guo-Niu HAN's 2000 paper "Generalisation de l’id …
6
votes
Proposals for polymath projects
The $3^d$ conjecture
Here is a problem of mine from 1989 that could be the basis of a good polymath project.
The $3^d$ conjecture: Let $P$ be a centrally symmetric $d$-dimensional polytope. Then $P$ h …
17
votes
Proposals for polymath projects
Here is a (known) open question that I heard from Peter Sarnak.
Show that $2^n+5$ is composite for almost all positive integers $n$. (Namely, for sets of integers of density 1.)
Since $\prod\l …
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 …
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 …
3
votes
Accepted
Classifying two-faces of four-polytopes
I dont know the anser to the specific question. It seems that for the study of hyperbolic Coxeter polytopes even if using some properties of general simple 4-polytope one needs to use the very restric …
- 24.7k
1
vote
What precisely Is "Categorification"?
Let us use this answer to bring additional relevant links.
Here is a blog post Categorification step I by Peter Cameron based on a lecture by Igor Frenkel.
I myself heard an amazing lecture by David …
- 24.7k
2
votes
What precisely Is "Categorification"?
Here is a very nice lecture about categorification by Jacob Lurie: Categorification of Fourier Theory. (I thank Arye Deutsch for telling me.)
(Update June 5, 2021): Here are four lectures by Catharina …
- 24.7k
17
votes
Interesting mathematical topics arising from biology
Mathematical biology is a huge area which is not so young.
Statistics is a major research tool in biology (as in most other areas of natural and social science) so biology questions rely and have led …