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1500 questions
79
votes
4 answers
Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?
Guillemin and Sternberg wrote the following in 1987 in a short article called "Some remarks on I.M. Gelfand's works" accompanying Gelfand's Collected Papers, Volume I:
The theory of commutative normed rings [i.e., (complex) Banach algebras],…
Jonas Meyer
- 7,279
79
votes
64 answers
Books you would like to see translated into English
I have recently been told of a proposal to produce an English translation
of Landau's Handbuch der Lehre von der Verteilung der Primzahlen, and
this prompts me to ask a more general question:
Which foreign-language
books would you most like to…
John Stillwell
- 12,258
79
votes
5 answers
Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem?
The Lawvere fixed point theorem asserts that if $X, Y$ are objects in a category with finite products such that the exponential $Y^X$ exists, and if $f : X \to Y^X$ is a morphism which is surjective on points in the sense that the induced map…
Qiaochu Yuan
- 114,941
79
votes
6 answers
Does Zhang's theorem generalize to $3$ or more primes in an interval of fixed length?
Let $p_n$ be the $n$-th prime number, as usual:
$p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $p_4 = 7$, etc.
For $k=1,2,3,\ldots$, define
$$
g_k = \liminf_{n \rightarrow \infty} (p_{n+k} - p_n).
$$
Thus the twin prime conjecture asserts $g_1 = 2$.
Zhang's…
Noam D. Elkies
- 77,218
78
votes
49 answers
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 typically presented, it leads not only to an absurd…
joro
- 24,174
78
votes
12 answers
Is there a high-concept explanation for why characteristic 2 is special?
The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, but an extra statement for $2$. So in these…
Qiaochu Yuan
- 114,941
78
votes
11 answers
How is it that you can guess if one of a pair of random numbers is larger with probability > 1/2?
My apologies if this is too elementary, but it's been years since I heard of this paradox and I've never heard a satisfactory explanation. I've already tried it on my fair share of math Ph.D.'s, and some of them postulate that something deep is…
Bill Thies
- 794
78
votes
9 answers
What is the significance of non-commutative geometry in mathematics?
This is a question that has been winding around my head for a long time and I have not found a convincing answer. The title says everything, but I am going to enrich my question by little more explanations.
As a layman, I have started searching for…
Ehsan M. Kermani
- 1,651
78
votes
6 answers
Rigidity of the category of schemes
Call a category $C$ rigid if every equivalence $C \to C$ is isomorphic to the identity. I don't know if this is standard terminology. Many of the usual algebraic categories are rigid, for example sets, commutative monoids, groups, abelian groups,…
Martin Brandenburg
- 61,443
78
votes
12 answers
Why aren't representations of monoids studied so much?
It seems to me like every book on representation theory leaps into groups right away, even though the underlying ideas, such as representations, convolution algebras, etc. don't really make explicit use of inverses. This leads to the fairly natural…
Mikola
- 2,352
78
votes
7 answers
Cubical vs. simplicial singular homology
Singular homology is usually defined via singular simplices, but Serre in his thesis uses singular cubes, which he claims are better adapted to the study of fibre spaces. This young man (25 years old at the time) seemed to know what he was talking…
Georges Elencwajg
- 46,833
78
votes
9 answers
Results that are widely accepted but no proof has appeared
The background of this question is the talk given by Kevin Buzzard.
I could not find the slides of that talk. The slides of another talk given by Kevin Buzzard along the same theme are available here.
One of the points in the talk is that, people…
Praphulla Koushik
- 5,921
78
votes
9 answers
Mathematical conjectures on which applications depend
What are some examples of mathematical conjectures that applied mathematicians assume to be true in applications, despite it being unknown whether or not they are true?
Christopher King
- 5,715
78
votes
3 answers
Norms of commutators
If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant $\lambda=\lambda(n)$ so that you can always choose $B$…
Bill Johnson
- 31,077
78
votes
15 answers
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 particularly enlightening, and would love to have a list…
Daniel Moskovich
- 21,887