All Questions
Tagged with big-list gm.general-mathematics
60 questions
18
votes
13
answers
2k
views
When is 4 qualitatively different than $n\leq 3$?
Inspired by When is 2 qualitatively different from 3?
Also similar to Are there mathematical concepts that exist in dimension 4, but not in dimension 3? (Math SE), but with the restriction of being ...
61
votes
71
answers
9k
views
When is 2 qualitatively different from 3?
I'd like to get a list of instances in mathematics where a problem with two parameters (or some parameter set to $2$) is qualitatively different from the instance of that problem with the value set to ...
2
votes
1
answer
526
views
What are some (popular) references on variants of the classical gambler's ruin problem that exists in literature?
It is fascinating that the gambler's ruin problem which is so ubiquitous in modern probability theory (cf. the Levin-Peres text on Markov chain and Mixing Times) actually dates back to a letter from ...
- 850
51
votes
9
answers
3k
views
Examples of theorems where numerical bounds on $\pi$ played a role
This is a whimsical question, motivated purely by curiosity rather than for any application.
We are all familiar with countless mathematical results which use Archimedes' constant $\pi$ either in ...
- 114k
6
votes
1
answer
242
views
Results with a flavor “every automorphism of automorphisms is inner”
It seems that there are a number of results which take more or less the following form: let $X$ be some (specific) kind of structure, let $Y$ be the group of automorphisms of $X$ or perhaps ring of ...
53
votes
1
answer
9k
views
What mathematical problems can be attacked using DeepMind's recent mathematical breakthroughs?
I am a research mathematician at a university in the United States. My training is in pure mathematics (geometry). However, for the past couple of months, I have been supervising some computer science ...
- 765
5
votes
0
answers
568
views
What sets are known to have cardinality equal to $\mathbb{N}$ or $\mathbb{R}$ but open as to which?
A long time ago a similar question was asked on math.stackexchange.
There are many sets which we know to be either finite or infinitely countable but do not know which cardinality specifically.
An ...
12
votes
7
answers
3k
views
Books containing new results
In Endless controversy about the correctness of significant papers, Denis Serre writes:
The research community is able to point out incorrect statements, at least among those which have some ...
49
votes
2
answers
5k
views
Well known theorems that have not been proved
I believe that there are numerous challenging theorems in mathematics for which only a sketch of a proof exists. To meet the standards of rigor, a complete proof of these theorems has yet to be ...
4
votes
1
answer
645
views
Novel examples, proofs or results in mathematics from arithmetic billiards
The goal of the post is get a repository of mathematical results, proofs or examples by users of the site, arising from arithmetic billiards in number theory, analysis, geometry,….
Wikipedia has an ...
1
vote
2
answers
734
views
What is the most "informative" Yes/No math question you know? [closed]
Imagine that alien civilization contacted you and offered to answer one math question. This should be a Yes/No question (so, you cannot ask for a million-digit binary string encoding the answers to a ...
15
votes
4
answers
974
views
What are some examples of understanding a space by studying the functions on this space?
In Quantum theory, groups and representations, Peter Woit writes:
A fundamental principle of modern mathematics is that the way to
understand a space $M$, given as some set of points, is to look at $...
6
votes
0
answers
287
views
Mathematical questions or areas amenable to AI [duplicate]
This question regards the new paper "Advancing mathematics by guiding human intuition with AI" by Davies et al. (Nature, 2021) (DOI link in open access) in which researchers at Deepmind ...
4
votes
2
answers
573
views
Theorems with finite sets of exceptions
Exceptions are interesting. Sometimes, they're also important. If a theorem with exceptions is important for a subject, there are liable to be many corollaries of the form "either this is true... ...
44
votes
26
answers
9k
views
Theorems with many distinct proofs
I was told that whenever one learns a new technique, it is a good idea to see if one can prove a well-known theorem using the new technique as an exercise. I am hoping to build a list of such theorems ...
67
votes
21
answers
9k
views
Situations where “naturally occurring” mathematical objects behave very differently from “typical” ones
I am looking for examples of the following situation in mathematics:
every object of type $X$ encountered in the mathematical literature, except when specifically attempting to construct ...
53
votes
10
answers
7k
views
Changes forced by the pandemic
The Covid-19 pandemic has changed our work-lives in ways few of us could have anticipated. These exceptional circumstances have forced each one of us and each one of our institutions to adapt, ...
92
votes
11
answers
15k
views
What are possible applications of deep learning to research mathematics?
With no doubt everyone here has heard of deep learning, even if they don't know what it is or what it is good for. I myself am a former mathematician turned data scientist who is quite interested in ...
45
votes
4
answers
4k
views
How to invoke constants badly
In a nice and witty lecture titled "how to write mathematics badly" (available on YouTube at https://www.youtube.com/watch?v=ECQyFzzBHlo&t=23s), Jean-Pierre Serre describes various ways ...
23
votes
14
answers
4k
views
Math talk for all ages
I've been asked to give a talk to the winners of a recent math competition. The talk can be entirely congratulatory, or it can contain a bit of actual mathematics. I'd prefer the latter. I'd also ...
85
votes
19
answers
15k
views
Each mathematician has only a few tricks
The question "Every mathematician has only a few tricks" originally had approximately the title of my question here, but originally admitted an interpretation asking for a small collection ...
170
votes
47
answers
34k
views
Every mathematician has only a few tricks
In Gian-Carlo Rota's "Ten lessons I wish I had been taught" he has a section, "Every mathematician has only a few tricks", where he asserts that even mathematicians like Hilbert ...
26
votes
12
answers
2k
views
Examples of improved notation that impacted research?
The intention of this question is to find practical examples of improved mathematical notation that enabled actual progress in someone's research work.
I am aware that there is a related post ...
3
votes
1
answer
428
views
Improvements to one's own theorems
What are some notable (famous?) instances where the following has occurred.
A particular author proves:
Every P which satisfies Q has property Z.
A few years later (roughly speaking) the same author ...
9
votes
0
answers
299
views
List of modern points of view simplifying or clarifying classical topics
There are many modern mathematical achievements which greatly clarify or (and) simplify classical important topics. I believe a list of such achievements, among other benefits, would be a big help for ...
93
votes
20
answers
10k
views
Short papers for undergraduate course on reading scholarly math
(I know this is perhaps only tangentially related to mathematics research, but I'm hoping it is worthy of consideration as a community wiki question.)
Today, I was reminded of the existence of this ...
39
votes
5
answers
6k
views
How to improve writing mathematics?
My first language is not English. How can I improve my mathematical writing. I feel like the only things I can write down are numbers and equations. Is there any good suggestion for improving writing, ...
174
votes
53
answers
57k
views
17 camels trick
The following popular mathematical parable is well known:
A father left 17 camels to his three sons and, according to the will, the eldest son should be given a half of all camels, the middle son ...
18
votes
8
answers
2k
views
Computationally challenging integer sequences
I wonder what are the examples of integer sequences, where only few elements are known and the researchers are still actively looking for the new terms. I think this discussion might be a good ...
- 1,625
10
votes
8
answers
4k
views
Most important mathematical results in last 30 years [closed]
Which results from the last 30 years, in any area of mathematics, do you think are the most important ones?
Specifically, which are the ones that will have more impact across all math and/or settle ...
31
votes
4
answers
2k
views
Expert, Intuitive, Organizing Analogies
In learning a new area it is very helpful to have high-level intuitive analogies that keep track of the various parts of an important argument or strategy. Experts have a store of such things, and ...
42
votes
26
answers
8k
views
Where can square roots come from when they are not distances?
In a recent survey "Supergeometry in Mathematics and Physics", Kapranov points out cases in which observable quantities of immediate interest are represented as bilinear combinations of more ...
5
votes
23
answers
5k
views
A search for theorems which appear to have very few, if any hypotheses [closed]
I'm interested in theorems which appear to have very few, if any hypotheses. Essentially a search for unexpected regularity or pattern in a relatively unstructured situation.
By "few hypotheses" I ...
79
votes
15
answers
9k
views
Sophisticated treatments of topics in school mathematics
Sophisticated mathematical concepts typically shed light on sophisticated mathematics. But in a few cases they also apply to elementary mathematics in an interesting way. I find such examples ...
111
votes
32
answers
14k
views
Special rational numbers that appear as answers to natural questions
Motivation:
Many interesting irrational numbers (or numbers believed to be irrational) appear as answers to natural questions in mathematics. Famous examples are $e$, $\pi$, $\log 2$, $\zeta(3)$ etc. ...
34
votes
3
answers
8k
views
What are the applications of operator algebras to other areas?
Question: What are the applications of operator algebras to other areas?
More precisely, I would like to know the results in mathematical areas outside of operator algebras which were proved by ...
103
votes
15
answers
17k
views
Have you solved problems in your sleep?
I have hit upon major (for me—relative to my trivial accomplishments)
insights in my research
in various sleep-deprived altered states of consciousness,
e.g., long solo car-drives extending through ...
50
votes
37
answers
6k
views
Structures that turn out to exhibit a symmetry even though their definition doesn't
Sometimes (often?) a structure depending on several parameters turns out to be symmetric w.r.t. interchanging two of the parameters, even though the definition gives a priori no clue of that symmetry. ...
21
votes
6
answers
2k
views
Online high quality colloquium talks
In my department we're thinking about showing online lectures one day per week at lunch, as sort of a virtual colloquium appropriate to mathematics undergraduates as well as faculty. To start with we'...
42
votes
4
answers
6k
views
Famous vacuously true statements
I am interested to know other examples vacuously true statements that are non-trivial. My starting example is Turan's result in regards to the Riemann hypothesis, which states
Suppose that for each $...
6
votes
6
answers
464
views
Procedure-based (as opposed to definition-based) concepts
Euler's work on divergent series was guided by computational procedures, rather than any definition of the "value" of such a series. E.g., he was happy to have half a dozen procedures that ...
35
votes
15
answers
2k
views
Objects which can't be defined without making choices but which end up independent of the choice
It happens a lot of times that when one defines a new object (ring, module, space, group, algebra, morphism, whatever) out of given data, one first chooses some additional structure. And sometimes (...
81
votes
22
answers
15k
views
Are there proofs that you feel you did not "understand" for a long time?
Perhaps the "proofs" of ABC conjecture or newly released weak version of twin prime conjecture or alike readily come to your mind. These are not the proofs I am looking for. Indeed my question was ...
7
votes
4
answers
1k
views
On similar concepts in mathematics whose similarity is a non-trivial fact.
Recently, while undertaking a study of commutative algebra, I learned three concepts: (i) a local ring, (ii) a regular local ring and (iii) a regular ring.
At the end, I found myself asking this ...
30
votes
15
answers
17k
views
Useless math that became useful
I'm writing an article on Lychrel numbers and some people pointed out that this is completely useless.
My idea is to amend my article with some theories that seemed useless when they are created but ...
8
votes
8
answers
4k
views
Beautiful theorems with short proof [closed]
I like to ask for beautiful mathematical theorems with short proof. A proof is short in my sense if it is at most one page assuming basic notations and very basic results a second year student will ...
78
votes
49
answers
45k
views
Examples of interesting false proofs
According to Wikipedia False proof
For example the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as ...
14
votes
9
answers
2k
views
math circles video lectures for school children?
Hello,
I am from India. I find the mathoverflow amazing.
I have a question: Are there any good quality video lectures on school math topics?
There are a lot of high quality lectures available on ...
16
votes
7
answers
2k
views
Unexpected applications of the fact that nth degree polynomials are determined by n+1 points
I had a funny idea for proving an identity in Euclidean geometry. While it didn't end up being a very nice proof strategy in my case, I would still like to collect nice examples of where the proof ...
7
votes
10
answers
1k
views
Examples of "Unusual" Classifications
When one says "classification" in math, usually one of a handful of examples springs to mind:
-Classification of Finite Simple Groups with 18 infinite families and 26 sporadic examples (assuming one ...