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1500 questions
85
votes
12 answers
Is Euclid dead?
Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonné, "A bas Euclide! Mort aux triangles!" (see King of Infinite Space: Donald Coxeter, the Man Who…
smyrlis
- 2,873
85
votes
8 answers
What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?
Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader knows what the "1970s version of the local Langlands…
Kevin Buzzard
- 40,559
85
votes
17 answers
Important open problems that have already been reduced to a finite but infeasible amount of computation
Most open problems, when formalized, naturally involve quantification over infinite sets, thereby obviating the possibility, even in principle, of "just putting it on a computer."
Some questions (e.g. the existence of a projective plane of order 12)…
David Feldman
- 17,466
84
votes
34 answers
books well-motivated with explicit examples
It is ultimately a matter of personal taste, but I prefer to see a long explicit example, before jumping into the usual definition-theorem path (hopefully I am not the only one here). My problem is that a lot of math books lack the motivating…
Rado
- 1,023
84
votes
2 answers
A little number theoretic game
I came up with this little two player game:
The players take turns naming a positive integer. When one player says the number n, the other player can only reply in two different ways: They can either respond with n+1 or they can divide n by a prime…
Leif Sabellek
- 771
84
votes
11 answers
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, where somehow "all of the information" about a…
Ned
- 111
84
votes
9 answers
What's wrong with the surreals?
Of all the constructions of the reals, the construction via the surreals seems the most elegant to me.
It seems to immediately capture the total ordering and precision of Dedekind cuts at a fundamental level since the definition of a number is based…
user2498
- 1,823
84
votes
38 answers
What are some correct results discovered with incorrect (or no) proofs?
Many famous results were discovered through non-rigorous proofs, with
correct proofs being found only later and with greater difficulty. One that is well
known is Euler's 1737 proof that
$1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots…
John Stillwell
- 12,258
84
votes
1 answer
Is there a complex surface into which every Riemann surface embeds?
This question was previously asked on Math SE.
Every Riemann surface can be embedded in some complex projective space. In fact, every Riemann surface $\Sigma$ admits an embedding $\varphi : \Sigma \to \mathbb{CP}^3$. It follows from the…
Michael Albanese
- 18,817
84
votes
23 answers
Solving algebraic problems with topology
Often, topologists reduce a problem which is - in some sense - of geometric nature, into an algebraic question that is then (partiallly) solved to give back some understanding of the original problem.
Sometimes, it goes the other way, i.e. algebraic…
Jens Reinhold
- 11,914
84
votes
1 answer
A hard integral identity on MathSE
The following identity on MathSE
$$\int_0^{1}\arctan\left(\frac{\mathrm{arctanh}\ x-\arctan{x}}{\pi+\mathrm{arctanh}\ x-\arctan{x}}\right)\frac{dx}{x}=\frac{\pi}{8}\log\frac{\pi^2}{8}$$
seems to be very difficult to prove.
Question: I worked on this…
Y. Zhao
- 3,317
83
votes
59 answers
Blackbox Theorems
By a blackbox theorem I mean a theorem that is often applied but whose proof is understood in detail by relatively few of those who use it. A prototypical example is the Classification of Finite Simple Groups (assuming the proof is complete). I…
Benjamin Steinberg
- 37,275
83
votes
4 answers
Do we still need model categories?
One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak equivalences; better, if you have a simplicial model…
Akhil Mathew
- 25,291
83
votes
28 answers
What could be some potentially useful mathematical databases?
This is a soft question but it's not meant as a big-list question. I have recently been asked whether I want to provide feedback at the pre-beta stage on a forthcoming website that will provide a platform for data sharing, and rather than giving…
gowers
- 28,729
83
votes
8 answers
The inverse Galois problem, what is it good for?
Several years ago I attended a colloquium talk of an expert in Galois theory. He motivated some of his work on its relation with the inverse Galois problem. During the talk, a guy from the audience asked: "why should I, as a number theorist, should…
Lior Bary-Soroker
- 3,282