Most Popular
1500 questions
67
votes
7 answers
Open project: Let's compute the Fourier expansion of a non-solvable algebraic Maass form.
OK so let's see if I can use MO to explicitly compute an example of something, by getting other people to join in. Sort of "one level up"---often people answer questions here but I'm going to see if I can make people do a more substantial project.…
Kevin Buzzard
- 40,559
67
votes
4 answers
explicit big linearly independent sets
In the following, I use the word "explicit" in the following sense: No choices of bases (of vector spaces or field extensions), non-principal ultrafilters or alike which exist only by Zorn's Lemma (or AC) are needed. Feel free to use similar…
Martin Brandenburg
- 61,443
67
votes
3 answers
Is this differential identity known?
Recently I discovered the differential identity
$$ \frac{d^{k+1}}{dx^{k+1}} (1+x^2)^{k/2} = \frac{(1 \times 3 \times \dots \times k)^2}{(1+x^2)^{(k+2)/2}}$$
valid for any odd natural number $k$; for instance $\frac{d^6}{dx^6} (1+x^2)^{5/2} =…
Terry Tao
- 108,865
- 31
- 432
- 517
67
votes
39 answers
Results true in a dimension and false for higher dimensions
Some theorems are true in vector spaces or in manifolds for a given dimension $n$ but become false in higher dimensions.
Here are two examples:
A positive polynomial not reaching its infimum. Impossible in dimension $1$ and possible in dimension…
mathcounterexamples.net
- 1,355
67
votes
4 answers
The Arnold – Serre debate
I have read (but I cannot now find where) that V. I. Arnold & J.-P. Serre had a public debate on the value of Bourbaki. Does anyone have more details, or remember or know what was said?
Marius Kempe
- 843
67
votes
3 answers
Is there a 0-1 law for the theory of groups?
Several months ago, Dominik asked the question Is there a 0-1 law for the theory of groups? on mathstackexchange, but although his question received attention there is still no answer. By asking the question here, I hope to find some result solving…
Seirios
- 2,361
67
votes
1 answer
What if the Riemann Hypothesis were false?
There are lots of known and interesting consequences of the Riemann Hypothesis being true. Are there any known and interesting consequences of the Riemann Hypothesis being false?
Craig Feinstein
- 2,549
67
votes
33 answers
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 question, if legitimate at all, should have been…
Vesselin Dimitrov
- 13,703
67
votes
1 answer
What is the relationship between motivic cohomology and the theory of motives?
I will begin by giving a rough sketch of my understanding of motives.
In many expositions about motives (for example, http://www.jmilne.org/math/xnotes/MOT102.pdf), the category of motives is defined to be a category such that every Weil cohomology…
Makhalan Duff
- 5,819
66
votes
1 answer
(Approximately) bijective proof of $\zeta(2)=\pi^2/6$?
Given $A,B\in {\Bbb Z}^2$, write $A \leftrightarrows B$ if the
interior of the line segment $AB$ misses
${\Bbb Z}^2$.
For $r>0$, define
$S_r:=\{ \{A, B\} \mid A,B\in {\Bbb Z}^2,\|A\|
David Feldman
- 17,466
66
votes
9 answers
Taking "Zooming in on a point of a graph" seriously
In calculus classes it is sometimes said that the tangent line to a curve at a point is the line that we get by "zooming in" on that point with an infinitely powerful microscope. This explanation never really translates into a formal definition -…
Steven Gubkin
- 11,945
66
votes
0 answers
Constructing non-torsion rational points (over Q) on elliptic curves of rank > 1
Consider an elliptic curve $E$ defined over $\mathbb Q$. Assume that the rank of $E(\mathbb Q)$ is $\geq2$. (Assume the Birch-Swinnerton-Dyer conjecture if needed, so that analytic rank $=$ algebraic rank.) How do you construct a point of infinite…
H A Helfgott
- 19,290
66
votes
7 answers
How to picture $\mathbb{C}_p$?
I hope this is appropriate for mathoverflow. Understanding $\mathbb{C}_p$ has always been something of a stumbling block for me. A standard thing to do in number theory is to take the completion $\mathbb{Q}_p$ of the rationals with respect to a…
Phillip Williams
- 1,329
66
votes
1 answer
Behaviour of power series on their circle of convergence
I asked myself the following question while preparing a course on power series for 2nd year students. Let $F$ be the set of power series with convergence radius equal to $1$. What subsets $S$ of the unit circle $C$ can be realised as
$$
S:=\{x \in…
Piotr
- 663
66
votes
8 answers
Why are powers of $\exp(\pi\sqrt{163})$ almost integers?
I've been prodded to ask a question expanding this one on Ramanujan's constant $R=\exp(\pi\sqrt{163})$.
Recall that $R$ is very close to an integer; specifically $R=262537412640768744 - \epsilon$ where $\epsilon$ is about $0.75 \times 10^{-12}$. …
Michael Lugo
- 13,858