Most Popular
1500 questions
44
votes
1 answer
Infinitely many solutions of a diophantine equation
If $P(x,y,...,z)$ is a polynomial with integer coefficients then every integer solution of $P=0$ corresponds to a homomorphism from $\mathbb{Z}[x,y,...,z]/(P)$ to $\mathbb{Z}$. So there are infinitely many solutions iff there are infinitely many…
user6976
44
votes
1 answer
Pach's "Animals": What if the genus is positive?
Janos Pach asked a deep question 23 years ago (1988) that remains unsolved today:
Can every animal—a topological ball in $\mathbb{R^3}$ composed of unit cubes glued face-to-face—be reduced to a single unit cube by adding and deleting cubes, while…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
44
votes
3 answers
Smooth functions for which $f(x)$ is rational if and only if $x$ is rational
A friend of mine introduced me to the following question: Does there exist a smooth function $f: \mathbb{R} \to \mathbb{R}$, ($f \in C^\infty$), such that $f$ maps rationals to rationals and irrationals to irrationals and is nonlinear?
I posed this…
J. J.
- 543
44
votes
7 answers
Does $SL_3(R)$ embed in $SL_2(R)$?
Is there any non-trivial ring such that $SL_{3}(R)$ is isomorphic to a subgroup of $SL_{2}(R)$?
$SL_{3}(\mathbb{Z})$ is not an amalgam, and has the wrong number of order $2$ elements to be a subgroup of $SL_{2}(\mathbb{Z})$.
Is there any…
Jonathan Kiehlmann
- 1,016
44
votes
10 answers
Fourier transform for dummies
So ... what is the Fourier transform? What does it do? Why is it useful (both in math and in engineering, physics, etc)?
(Answers at any level of sophistication are welcome.)
Kevin H. Lin
- 20,738
44
votes
2 answers
Is multiplication implicitly definable from successor?
A relation $R$ is implicitly definable in a structure $M$ if there is a formula $\varphi(\dot R)$ in the first-order language of $M$ expanded to include relation $R$, such that $M\models\varphi(\dot R)$ only when $\dot R$ is interpreted as $R$ and…
Joel David Hamkins
- 224,022
44
votes
2 answers
Langlands in dimension 2: the Yoshida conjecture
Background:
One prominent part of the Langlands program is the conjecture that
all motives are automorphic.
It is of interest to consider special cases that are more precise, if less
sweeping. The idea is to formulate generalized modularity…
Laie
- 1,694
- 1
- 13
- 14
44
votes
42 answers
What should be offered in undergraduate mathematics that's currently not (or isn't usually)?
What's one class that mathematics that should be offered to undergraduates that isn't usually? One answer per post.
Ex: Just to throw some ideas out there
Mathematical Physics (for math students, not for physics students)
Complexity Theory
Michael Hoffman
- 1,785
44
votes
4 answers
Are there motives which do not, or should not, show up in the cohomology of any Shimura variety?
Let $F$ be a real quadratic field and let $E/F$ be an elliptic curve with conductor 1 (i.e. with good reduction everywhere; these things can and do exist) (perhaps also I should assume E has no CM, even over F-bar, just to avoid some counterexamples…
Kevin Buzzard
- 40,559
44
votes
2 answers
Are rigid-analytic spaces obsolete, since adic spaces exist?
Recently in a seminar the following question was raised and, despite my familiarity with theory, I couldn't come up with a good answer:
Are there any good reasons to use Tate's theory of rigid-analytic spaces, given that Huber's theory of adic…
Wojowu
- 27,379
44
votes
3 answers
Is there an elementary proof that distal maps are invertible?
Let $T: X \to X$ be a continuous map on a compact metric space $X$. We say $T$ is distal if $\inf_n d(T^n x, T^n y) = 0$ implies $x = y$.
Then it is true that $T$ is bijective.
Question: Is there an elementary proof of this fact? (Injectivity…
Nate River
- 4,832
44
votes
6 answers
Best tablet computer for mathematics
I'm not sure if this is completely appropriate, but I thought I'd ask here.
I'm in the market for a tablet computer. Unfortunately, my (mathematical) needs are very different from the needs of the sorts of people who usually review these things. …
A grad student
- 271
44
votes
3 answers
Why is the Vandermonde determinant harmonic?
It can be checked that the Vandermonde determinant defined as
$$V(\alpha_1, \cdots, \alpha_n) = \prod_{1 \le i < j \le n}(\alpha_i-\alpha_j) $$
is a harmonic function, that is $\Delta V = 0$ where $\Delta$ is the Laplace operator. Is there a deeper…
Sandeep Silwal
- 733
44
votes
5 answers
An "analytic continuation" of power series coefficients
Cauchy residue theorem tells us that for a function
$$f(z) = \sum_{k \in \mathbb{Z}} a(k) z^k,$$
the coefficient $a(k)$ can be extracted by an integral formula
$$a(k) = \frac{1}{2\pi i}\oint f(z) z^{-k-1},$$
with a contour around zero. Now, there is…
MCH
- 1,304
44
votes
1 answer
Existence and uniqueness of Haar measure on compacta; a cohomological approach
I am trying to use a modification of group cohomology to prove the existence and uniqueness of Haar measure on a compact Hausdorff group.
I think the best way of introducing the idea I am pursuing is via analogy.
Let $G$ be a finite group and let…
