All Questions
51 questions
8
votes
1
answer
388
views
Formalisation of intuitive concepts in the language leading to mathematical progress
In his work, Albert Lautman thinks the genesis of some mathematical works as a dialectic that takes place between opposite notions, like between global and local. He argues that while those notions, ...
- 531
47
votes
7
answers
8k
views
Swimming against the tide in the past century: remarkable achievements that arose in contrast to the general view of mathematicians
I would like to ask a question inspired by the title of a book by Sir Roger Penrose ([1]). The germ of this is to ask about the role, if any, of the fashion in research of pure and applied mathematics....
52
votes
14
answers
9k
views
Modern results that are widely known, yet which at the time were ignored, not accepted or criticized
What is your favorite example of a celebrated mathematical fact that had a hard time to become accepted by the community, but after overcoming some initial "resistance" quickly took on?
It ...
6
votes
3
answers
558
views
Anomalous phenomena [closed]
What are examples of strikingly anomalous phenomena in mathematics, where just one or a rather small number of cases stand out because they don't fit a general pattern?
This is most interesting when ...
14
votes
29
answers
7k
views
Which great mathematicians had great political commitments? [closed]
Some mathematicians claim that their field has nothing to do with political concerns; others are deeply involved in political life.
Are there many great mathematicians with great political commitments?...
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 ...
43
votes
9
answers
6k
views
What are some examples of theorem requiring highly subtle hypothesis?
I would like you to expose and explain briefly some examples of theorems having some hypothesis that are (as far as we know) actually necessary in their proofs but whose uses in the arguments are ...
79
votes
13
answers
21k
views
Nontrivially fillable gaps in published proofs of major theorems
Prelude: In 1998, Robert Solovay wrote an email to John Nash to communicate an error that he detected in the proof of the Nash embedding theorem, as presented in Nash's well-known paper "The Imbedding ...
64
votes
68
answers
16k
views
Mathematicians with both “very abstract” and “very applied” achievements
Gödel had a cosmological model. Hamel, primarily a mechanician, gave any vector space a basis. Plücker, best known for line geometry, spent years on magnetism. What other mathematicians had so distant ...
84
votes
11
answers
12k
views
What are examples of (collections of) papers which "close" a field?
There is sometimes talk of fields of mathematics being "closed", "ended", or "completed" by a paper or collection of papers. It seems as though this could happen in two ways:
A total characterisation,...
38
votes
17
answers
8k
views
Examples of "unsuccessful" theories with afterlives
I am looking for examples of mathematical theories which were introduced with a certain goal in mind, and which failed to achieved this goal, but which nevertheless developed on their own and ...
35
votes
30
answers
6k
views
Examples of simultaneous independent breakthroughs
I'm looking for examples where, after a long time with little progress, a simultaneous mathematical discovery, solution, or breakthrough was made independently by at least two different people/groups. ...
53
votes
11
answers
6k
views
What definitions were crucial to further understanding?
Often the most difficult part of venturing into a field as a researcher is to come up with an appropriate definition. Sometimes definitions suggest themselves very naturally, as when you solve a ...
56
votes
3
answers
11k
views
Work of plenary speakers at ICM 2018
The next International Congress of Mathematicians (ICM) will be next year in Rio de Janeiro, Brazil. The present question is the 2018 version of similar questions from 2014 and 2010. Can you, please, ...
63
votes
7
answers
8k
views
Theorems demoted back to conjectures
Many mathematicians know the Four Color Theorem and its history: there were two alleged proofs in 1879 and 1880 both of which stood unchallenged for 11 years before flaws were discovered.
I am ...
10
votes
5
answers
919
views
Important results with one or more than one proof [closed]
Can you give examples of deep, important results that have only one known proof, and not just because the first proof is fairly recent, or because not many people really cared to think about it? How ...
122
votes
41
answers
29k
views
What are some very important papers published in non-top journals?
There has already been a question about important papers that were initially rejected. Many of the answers were very interesting. The question is here.
My concern in this question is slightly ...
3
votes
1
answer
3k
views
Famous examples of PhD advisors younger than their student [closed]
What are the most famous examples of PhD advisors in mathematics, younger than their student?
(if possible put the date of birth and/or the difference in age).
29
votes
25
answers
7k
views
Mathematicians who made important contributions outside their own field? [closed]
It is often said that scientists who cross disciplinary borders can make unexpected discoveries thanks to their fresh view of the problems at hand.
I am looking for mathematicians who did just that. ...
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 ...
110
votes
10
answers
15k
views
Analogues of P vs. NP in the history of mathematics
Recently I wrote a blog post entitled "The Scientific Case for P≠NP". The argument I tried to articulate there is that there seems to be an "invisible electric fence" separating the problems in P ...
76
votes
19
answers
18k
views
What are some deep theorems, and why are they considered deep?
All mathematicians are used to thinking that certain theorems are deep, and we would probably all point to examples such as Dirichlet's theorem on primes in arithmetic progressions, the prime number ...
89
votes
27
answers
12k
views
Modern Mathematical Achievements Accessible to Undergraduates
While there is tremendous progress happening in mathematics, most of it is just accessible to specialists. In many cases, the proofs of great results are both long and use difficult techniques. Even ...
35
votes
13
answers
5k
views
Great mathematics books by pre-modern authors
Last summer, I read Euclid's Elements, and it was an eye-opening experience; I had assumed that three thousand years' difference would make the notation incomprehensible and the reasoning alien, but ...
67
votes
19
answers
14k
views
Mathematicians whose works were criticized by contemporaries but became widely accepted later
Gauss famously discarded Abel's proof that an algebraic equation of degree five or more cannot have a general solution (Abel himself had rejected divergent series as the work of the devil). Cantor's ...
71
votes
34
answers
12k
views
Trichotomies in mathematics
Added. Thanks to all who participated! Let me humbly apologize to those who were annoyed (quite understandably) by this thread, deeming it nothing more than an exercise in futility. If you thought the ...
23
votes
7
answers
7k
views
What are some Applications of Teichmüller Theory?
I'm trying to collect some specific examples of applications of Teichmüller Theory. Here are some things I have collected thus far:
No-wandering-domain Theorem (Sullivan)
Theorems of Thurston (...
9
votes
9
answers
1k
views
Examples where adding complexity made a problem simpler
I can think of a few situations in math where a problem becomes easier or an object becomes simpler when some complexity is added. Examples:
$S^n$ is never contractible, but $S^{\infty}$ is.
The ...
40
votes
29
answers
8k
views
Autobiographies of mathematicians
According to Wikipedia, an autobiography is an account of the life of a person, written by that person sometimes with a collaborator.
An autobiography offers the author the ability to recreate history....
36
votes
65
answers
13k
views
Fiction books about mathematicians? [closed]
What are some fiction books about mathematicians?
It seems to me rather difficult for writers to create good books on this subject.
Some years ago I thought there were no such books at all.
There ...
36
votes
35
answers
11k
views
Titles composed entirely of math symbols
I apologize for burdening MO with such a vapid, nonresearch question, but
I have been curious ever since
Suvrit's popular October 2010
Most memorable titles MO question
if there were any "$E=mc^2$...
27
votes
4
answers
3k
views
What are examples of theorems which were once "valid", then became "invalid" as standard definitions shifted?
That is, results established by correct proofs within some framework, yet the manner in which their author or the general mathematical community at the time would describe these results would, in ...
75
votes
29
answers
13k
views
(Preferably rare) Audio/Video recordings of famous mathematicians?
Terence Tao's homepage has a link to a collection of quotes, and one among them was Hilbert's famous "We must know, we will know" quote. This quote also had an audio link to it. Now although I'm not ...
5
votes
3
answers
923
views
Examples of results which were surprising but later shown to be natural. [closed]
After Ramanujan formulated his conjectures on the Tau-function, and after the importance of the function was realized, it took the development of the theory of Modular forms for the complete ...
30
votes
11
answers
5k
views
New proofs to major theorems leading to new insights and results?
I am wondering, historically, when has a new proof of an old theorem been particularly fruitful. A few examples I have in mind (all number theoretic) are:
First example is classical... which is Euler'...
424
votes
93
answers
149k
views
Video lectures of mathematics courses available online for free
It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...
67
votes
16
answers
9k
views
What do named "tricks" share?
There are a number of theorems or lemmas or mathematical ideas that come to be known as eponymous
tricks, a term which in this context is in no sense derogatory.
Here is a list of 11 such tricks (the ...
92
votes
74
answers
27k
views
Pseudonyms of famous mathematicians
Many mathematicians know that Lewis Carroll was quite a good mathematician, who wrote about logic (paradoxes) and determinants. He found an expansion formula, which bears his real name (Charles ...
185
votes
127
answers
65k
views
Most memorable titles
Given the vast number of new papers / preprints that hit the internet everyday, one factor that may help papers stand out for a broader, though possibly more casual, audience is their title. This view ...
399
votes
23
answers
69k
views
Thinking and Explaining
How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words differ, ...
401
votes
53
answers
151k
views
Widely accepted mathematical results that were later shown to be wrong?
Are there any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time later, possibly ...
104
votes
19
answers
14k
views
Can a mathematical definition be wrong?
This question originates from a bit of history. In the first paper on quantum Turing machines, the authors left a key uniformity condition out of their definition. Three mathematicians subsequently ...
11
votes
3
answers
3k
views
The definition of "proof" throughout the history of mathematics
It is widely believed that mathematicians have a uniform standard of what constitutes a correct proof. However, this standard has, at minimum, changed over time. What are some striking examples where ...
41
votes
17
answers
6k
views
What are some examples of narrowly missed discoveries in the history of mathematics?
What are the examples of some mathematicians coming very close to a very promising theory or a correct proof of a big conjecture but not making or missing the last step?
32
votes
21
answers
16k
views
What are some applications of other fields to mathematics?
It is commonplace to consider applications of mathematics to other fields, especially the exact sciences. But what I would like to know about is the converse topic, namely:
What are some applications ...
137
votes
26
answers
29k
views
What are some famous rejections of correct mathematics?
Dick Lipton has a blog post that motivated this question. He recalled the Stark-Heegner
Theorem: There are only a finite
number of imaginary quadratic fields
that have unique factorization. They
are $...
1
vote
1
answer
2k
views
What are examples of theorems get extensions based on simple observation?
Here are some examples illustrate what I meant:
Bonnet-Myers:Bonnet in 1855 proved n=2 case, Myers in 1941 extended to any dimension using the same idea.
Bishop-Gromov Volume comparison: Bishop knew ...
197
votes
94
answers
107k
views
Famous mathematical quotes [closed]
Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of?
Standard community wiki rules apply: one ...
35
votes
26
answers
5k
views
Examples of mathematics motivated by technological considerations
I would like examples of technological advances that were made possible only by the creation of new mathematics. I'm talking about technology that was desired in some period of history but for which ...