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1500 questions
78
votes
0 answers
The exponent of Ш of $y^2 = x^3 + px$, where $p$ is a Fermat prime
For $d$ a non-zero integer, let $E_d$ be the elliptic curve
$$
E_d : y^2 = x^3+dx.
$$
When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD,
$$
\# Ш(E_p) = 2^{2k-2}.
$$
Together with the fact that…
R.P.
- 4,745
78
votes
21 answers
Is rigour just a ritual that most mathematicians wish to get rid of if they could?
"No". That was my answer till this afternoon! "Mathematics without proofs isn't really mathematics at all" probably was my longer answer. Yet, I am a mathematics educator who was one of the panelists of a discussion on "proof" this afternoon,…
Amir Asghari
- 2,277
- 3
- 40
- 58
78
votes
1 answer
The topology of Arithmetic Progressions of primes
The primary motivation for this question is the following: I would like to extract some topological statistics which capture how arithmetic progressions of prime numbers "fit together" in a manner that will be made precise below.
Setup
Consider a…
Vidit Nanda
- 15,397
78
votes
12 answers
What practical applications does set theory have?
I am a non-mathematician. I'm reading up on set theory. It's fascinating, but I wonder if it's found any 'real-world' applications yet. For instance, in high school when we were learning the properties of i, a lot of the kids wondered what it was…
user2929
- 799
77
votes
3 answers
5/8 bound in group theory
The odds of two random elements of a group commuting is the number of conjugacy classes of the group
$$ \frac{ \{ (g,h): ghg^{-1}h^{-1} = 1 \} }{ |G|^2} = \frac{c(G)}{|G|}$$
If this number exceeds 5/8, the group is Abelian (I forget which groups…
john mangual
- 22,599
77
votes
9 answers
Can we unify addition and multiplication into one binary operation? To what extent can we find universal binary operations?
The question is the extent to which we can unify addition
and multiplication, realizing them as terms in a single
underlying binary operation. I have a number of questions.
Is there a binary operation $n\star m$ on the integers
$\mathbb{Z}$ such…
Joel David Hamkins
- 224,022
77
votes
11 answers
Applications of mathematics
All of us have probably been exposed to questions such as: "What are the applications of group theory...".
This is not the subject of this MO question.
Here is a little newspaper article that I found inspiring:
Madam, – In response to Marc…
André Henriques
- 42,480
77
votes
15 answers
Each mathematician has only a few tricks
The question "Every mathematician has only a few tricks" originally had approximately the title of my question here, but originally admitted an interpretation asking for a small collection of tricks used by all mathematicians. That question now has…
Jon Bannon
- 6,987
- 6
- 68
- 110
77
votes
5 answers
Inaccessible cardinals and Andrew Wiles's proof
In a recent issue of New Scientist (16 Aug 2010), I was surprised to read that a part of Wiles' proof of Taniyama-Shimura conjecture relies on inaccessible cardinals.
Here's the article
Richard Elwes, To infinity and beyond: The struggle to save…
Cosmonut
- 1,041
77
votes
3 answers
Gromov's list of 7 constructions in differential topology
At the 2010 Clay Research Conference, Gromov explained that we know of only 7 different methods for constructing smooth manifolds. Working from memory, and hence not necessarily respecting the order he used:
Algebraic geometry (affine and…
Mohammed Abouzaid
- 1,614
77
votes
28 answers
Good papers/books/essays about the thought process behind mathematical research
Papers in mathematics are generally written as if the major insights suddenly appeared, unbidden, in a notebook on the researcher's desk and then were fleshed out into the final paper.
While this is great for finding out about results, it's terrible…
DoubleJay
- 2,353
- 3
- 33
- 45
77
votes
2 answers
Complex structure on $S^6$ gets published in Journ. Math. Phys
A paper by Gabor Etesi was published that purports to solve a major outstanding problem:
Complex structure on the six dimensional sphere from a spontaneous symmetry breaking
Journ. Math. Phys. 56, 043508-1-043508-21 (2015) journal version, current…
Misha Verbitsky
- 9,105
77
votes
9 answers
Irreducibility of polynomials in two variables
Let $k$ be a field. I am interested in sufficient criteria for $f \in k[x,y]$ to be irreducible. An example is Theorem A of this paper (Brindza and Pintér, On the irreducibility of some polynomials in two variables, Acta Arith. 1997).
Does anyone…
Hailong Dao
- 30,261
76
votes
7 answers
Example of a manifold which is not a homogeneous space of any Lie group
Every manifold that I ever met in a differential geometry class was a homogeneous space: spheres, tori, Grassmannians, flag manifolds, Stiefel manifolds, etc. What is an example of a connected smooth manifold which is not a homogeneous space of any…
MTS
- 8,419
76
votes
34 answers
Dimension leaps
Many mathematical areas have a notion of "dimension", either rigorously or naively, and different dimensions can exhibit wildly different behaviour. Often, the behaviour is similar for "nearby" dimensions, with occasional "dimension leaps" marking…
Andrew Stacey
- 26,373