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1500 questions
59
votes
3 answers
Relaxed Collatz 3x+1 conjecture
The Collatz $3x+1$ conjecture claims that any positive integer can eventually be reduced to $1$ by iterative application of the maps $x \mapsto 3x+1$ whenever $x$ is odd and $x \mapsto x/2$ whenever $x$ is even.
While the Collatz conjecture is still…
Max Alekseyev
- 30,425
59
votes
1 answer
Which region in the plane with a given area has the most domino tilings?
I just finished teaching a class in combinatorics in which I included a fairly easy upper bound on the number of domino tilings of a region in the plane as a function of its area. So this led to this question, which I suspect is an open problem:…
Greg Kuperberg
- 56,146
59
votes
2 answers
"Gross-Zagier" formulae outside of number theory
The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the derivative of the $L$-function of an elliptic curve…
Bruno Joyal
- 3,880
59
votes
4 answers
Has Fermat's Last Theorem per se been used?
There is a long tradition of mathematicians remarking that FLT in itself is a rather isolated claim, attractive only because of its simplicity. And people often note a great thing about current proofs of FLT is their use of the modularity thesis…
Colin McLarty
- 10,817
59
votes
7 answers
How closed-form conjectures are made?
Recently I posted a conjecture at Math.SE:
$$\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$
where $J_\mu(x)$ and $Y_\mu(x)$ are the Bessel function of the first and second…
Vladimir Reshetnikov
- 6,709
59
votes
34 answers
Are there any books that take a 'theorems as problems' approach?
Are there any books that present theorems as problems? To be more specific, a book on elementary group theory might have written: "Theorem: Each group has exactly one identity" and then show a proof or leave it as an exercise. The type of book that…
aspy591
- 41
59
votes
7 answers
Mathematician trying to learn string theory
I'm a mathematician. I want to be able to read recent ArXiv postings on high energy physics theory (String theory) (and perhaps be able to do research). I want to understand compactifications, Dualities, D-branes, M-branes etc. What's the easiest…
quark
- 173
59
votes
3 answers
Operations via Morse Theory
I am interested in seeing if and how Morse Theory can "do everything". Some core things are handle decomposition, Bott periodicity, and Euler characteristic. But what do the normal (co)homology operations look like from Morse Homology?
Poincare…
Chris Gerig
- 17,130
58
votes
3 answers
Is "semisimple" a dense condition among Lie algebras?
The "Motivation" section is a cute story, and may be skipped; the "Definitions" section establishes notation and background results; my question is in "My Question", and in brief in the title. Some of my statements go wrong in non-zero…
Theo Johnson-Freyd
- 52,873
58
votes
9 answers
Learning Class Field Theory: Local or Global First?
I've noticed that there seem to be two approaches to learning class field theory. The first is to first learn about local fields and local class field theory, and then prove the basic theorems about global class field theory from the corresponding…
David Corwin
- 15,078
58
votes
8 answers
How true are theorems proved by Coq?
Less tongue in cheek, is it known what the relative consistency is for theorems proved with an automatic theorem prover? Of course this depends somewhat on what assumptions one makes with respect to predicativity and so on. But for concreteness take…
David Roberts
- 33,851
58
votes
9 answers
Motivation for and history of pseudo-differential operators
Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds). Which prominent example problems lead you to work with pseudo-differential operators?
I would appreciate any good examples, as…
shuhalo
- 4,736
58
votes
22 answers
How to respond to "I was never much good at maths at school."
We've all heard it. I even got it in Norwegian recently. It's number 1 on the list of responses to the statement "I'm a mathematician.". Does anyone have any good comebacks? What other responses have you heard?
Standard community wiki rules (not…
Andrew Stacey
- 26,373
58
votes
3 answers
Global character of ABC/Szpiro inequalities
In [M24] it is asserted that "considering $abc$ triples of the form $(1,p^n,1+p^n)$ for a prime number $p$ and an arbitrarily large positive integer $n$, one can verify that ABC/Szpiro inequalities can never be obtained as a result of summing up…
Jon23
- 717
58
votes
6 answers
Torsion in homology or fundamental group of subsets of Euclidean 3-space
Here's a problem I've found entertaining.
Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups is not torsion-free?
Context: The analogous question has…
Ryan Budney
- 43,013