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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.
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 …
6
votes
The use of computers leading to major mathematical advances II
At ICM 2022 Kevin Buzzard gave a talk "The rise of formalism in mathematics" about computer verification of mathematical proofs (item E). It is about writing large chunks of mathematics in "lean" (or …
7
votes
The use of computers leading to major mathematical advances II
The paper Constructions in combinatorics via neural networks by Adam Zsolt Wagner uses reinforcement learning algorithm and the deep cross-entropy method, to find explicit constructions and counterexa …
16
votes
The use of computers leading to major mathematical advances II
An example of type C. DeepMind launched an endeavor for using machine learning (and deep learning in particular) for finding conjectures based on data. Two recent outcomes are toward the Dyer-Lusztig …
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
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 …
5
votes
Experimental mathematics leading to major advances
In March 2019, Andrew Booker in the paper Cracking the problem with 33 showed that 33 is a sum of 3 cubes in $\mathbf Z$. This is a striking example of a case where checking the solution is very easy …
3
votes
Experimental mathematics leading to major advances
Another famous example is Harald Helfgott's 2013 proof that every odd integers greater than five is the sum of three primes. "The ternary Goldbach conjecture is true". arXiv:1312.774
4
votes
Experimental mathematics leading to major advances
Boolean Pythagorean triples Theorem (Heule, Kullmann, Oliver 2015) — The set $\{1, \dots , 7824\}$ can be partitioned into two parts, such that no part contains a Pythagorean triple, while this is im …
8
votes
Experimental mathematics leading to major advances
A direct search on CDC 6600 led to a ctrx counterexample for a conjecture of Euler. Following an answer by Will Brian to a question about tweetable mathematics.
5
votes
Experimental mathematics leading to major advances
$R(5,5) \le 48$
Is a title of a remarkable recent paper by Vigleik Angeltveit and Brendan D. McKay. More than 20 years ago McKay and Radziszowski proved that $R(5,5) \le 49$ and in this paper t …
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 …
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 …
1
vote
Expert, Intuitive, Organizing Analogies
Here is an example: Sullivan's analogy (or dictionary) between complex dynamics and Kleinian groups.
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