All Questions

Filter by
Sorted by
Tagged with
149 votes
38 answers
38k views

Computer algebra errors

In the course of doing mathematics, I make extensive use of computer-based calculations. There's one CAS that I use mostly, even though I occasionally come across out-and-out wrong answers. After ...
148 votes
7 answers
21k views

Homotopy groups of Lie groups

Several times I've heard the claim that any Lie group $G$ has trivial second fundamental group $\pi_2(G)$, but I have never actually come across a proof of this fact. Is there a nice argument, ...
Matt Noonan's user avatar
  • 3,984
128 votes
10 answers
19k views

Are there any very hard unknots?

Some years ago I took a long piece of string, tied it into a loop, and tried to twist it up into a tangle that I would find hard to untangle. No matter what I did, I could never cause the later me any ...
gowers's user avatar
  • 28.7k
116 votes
8 answers
32k views

Zagier's one-sentence proof of a theorem of Fermat

Zagier has a very short proof (MR1041893, JSTOR) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares. The proof defines an involution of the set $S= \lbrace (x,y,z) \...
Keivan Karai's user avatar
  • 6,064
111 votes
2 answers
15k views

Does every non-empty set admit a group structure (in ZF)?

It is easy to see that in ZFC, any non-empty set $S$ admits a group structure: for finite $S$ identify $S$ with a cyclic group, and for infinite $S$, the set of finite subsets of $S$ with the binary ...
Konrad Swanepoel's user avatar
54 votes
6 answers
7k views

What is the smallest unsolved Diophantine equation?

If $P=\sum_{\bf i} a_{\bf i}x^{\bf i}\in {\mathbb Z}[x_1,\dots,x_d]$, let $|P|=\sum_{\bf i}|a_{\bf i}|x^{\bf i}$ and $h(P)=|P|(2,\dots,2)$, so that there is only a finite number of $P$ with $h(P)\leq ...
Zidane's user avatar
  • 917
42 votes
8 answers
10k views

The finite subgroups of SU(n)

This question is inspired by the recent question "The finite subgroups of SL(2,C)". While reading the answers there I remembered reading once that identifying the finite subgroups of SU(3) is still an ...
Q.Q.J.'s user avatar
  • 2,083
42 votes
3 answers
19k views

Which integers can be expressed as a sum of three cubes in infinitely many ways?

For fixed $n \in \mathbb{N}$ consider integer solutions to $$x^3+y^3+z^3=n \qquad (1) $$ If $n$ is a cube or twice a cube, identities exist. Elkies suggests no other polynomial identities are known. ...
joro's user avatar
  • 24.2k
36 votes
1 answer
9k views

Infinite tensor products

Let $A$ be a commutative ring and $M_i, i \in I$ be a infinite family of $A$-modules. Define their tensor product $\bigotimes_{i \in I} M_i$ to be a representing object of the functor of multilinear ...
Martin Brandenburg's user avatar
34 votes
5 answers
11k views

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?

What mathematical treatment is there on the renormalization group flow in a space of Lagrangians?
user4's user avatar
  • 911
27 votes
7 answers
7k views

Asymptotic density of k-almost primes

Let $\pi_k(x)=|\{n\le x:n=p_1p_2\cdots p_k\}|$ be the counting function for the k-almost primes, generalizing $\pi(x)=\pi_1(x)$. A result of Landau is $$\pi_k(x)\sim\frac{x(\log\log x)^{k-1}}{(k-1)!\...
Charles's user avatar
  • 8,974
24 votes
3 answers
12k views

Fourier transform of the unit sphere

The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula $$ \int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu ...
Francois Ziegler's user avatar
23 votes
1 answer
5k views

On equation $f(z+1)-f(z)=f'(z)$

Original Problem If $f$ is an entire function such that $$ f(z+1)-f(z)=f'(z) $$ for all $z$. Is there a non-trivial solution? ($f(z)=az+b$ is trivial) And here is something uncertainty If we use ...
Lwins's user avatar
  • 1,531
16 votes
1 answer
2k views

The cyclic subfactors theory: a quantum arithmetic?

Context: First recall some results: Actions of finite groups on the hyperfinite type $II_{1}$ factor $R$ (Jones 1980). A Galois correspondence for depth 2 irreducible subfactors (Izumi-Longo-Popa ...
13 votes
2 answers
2k views

Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus

I was exploring some raising and lowering operators related to an infinitesimal generator for fractional integro-derivatives and found an Appell sequence of polynomials, i.e., an infinite sequence of ...
Tom Copeland's user avatar
  • 9,937
13 votes
3 answers
1k views

Bound the error in estimating a relative totient function

Let $n=p_1^{e_1}\cdots p_k^{e_k}$ be an integer with $k$ prime factors. We know that the number of integers less than $n$ and coprime to it is $$\Phi(n)=n-\sum_i\frac n{p_i}+\sum_{i \lt j}\frac n{...
Aaron Meyerowitz's user avatar
12 votes
0 answers
2k views

Would the following conjectures imply $\lim\inf_{n\to\infty}p_{n+k}-p_{n}=O(k\log k)$?

Assume Goldbach's conjecture. Then for every $n\ge 2$ there exists at least one non-negative integer $r\le n-2$ such that both $n+r$ and $n-r$ are primes. Let's write $r_{0}(n):=\inf\{r\le n-2, (n-r,n+...
Sylvain JULIEN's user avatar
11 votes
1 answer
1k views

Necessary conditions for cofibrancy in global projective model structure on simplicial presheaves

Consider the global projective model category of simplicial presheaves on some category (the category of smooth manifolds is particularly interesting to me). In Section 9.1 of Dugger's paper “...
Dmitri Pavlov's user avatar
11 votes
2 answers
1k views

Elliptic operators corresponds to non vanishing vector fields

Added, June 19, 2019: The main motivation of this post is to associate an index to differential operator associated to a dynamical system such that the index has an interesting ...
Ali Taghavi's user avatar
9 votes
1 answer
791 views

Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory

In the background sections below, I establish the relations among characterizations of the action of Virasoro / Kac–Schwarz operators of 2D gravity models presented in terms of Laurent series by ...
Tom Copeland's user avatar
  • 9,937
5 votes
1 answer
872 views

Series solution of the trinomial equation

The roots of trinomial equations $x^p+x-q=0$ ($p\in\mathbb{N}$) can be expressed in terms of the hypergeometric functions. I am wondering if at least one real root, for instance given by the following ...
yarchik's user avatar
  • 482
280 votes
47 answers
109k views

Examples of unexpected mathematical images

I try to generate a lot of examples in my research to get a better feel for what I am doing. Sometimes, I generate a plot, or a figure, that really surprises me, and makes my research take an ...
160 votes
23 answers
29k views

Do you read the masters?

I often hear the advice, "Read the masters" (i.e., read old, classic texts by great mathematicians). But frankly, I have hardly ever followed it. What I am wondering is, is this a ...
121 votes
12 answers
27k views

How to solve $f(f(x)) = \cos(x)$?

I found the following equation on some web page I cannot remember, and found it interesting: $$f(f(x))=\cos(x)$$ Out of curiosity I tried to solve it, but realized that I do not have a clue how to ...
user4503's user avatar
  • 1,551
86 votes
5 answers
116k views

Eigenvalues of matrix sums

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite? I am ...
Jean-Pierre Gunman's user avatar
86 votes
11 answers
30k views

Is there a complex structure on the 6-sphere?

I don't know who first asked this question, but it's a question that I think many differential and complex geometers have tried to answer because it sounds so simple and fundamental. There are even a ...
51 votes
6 answers
12k views

What does Mellin inversion "really mean"?

Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its Mellin transform is ...
Frank Thorne's user avatar
  • 7,199
46 votes
8 answers
5k views

Can a problem be simultaneously polynomial time and undecidable?

The Robertson-Seymour theorem on graph minors leads to some interesting conundrums. The theorem states that any minor-closed class of graphs can be described by a finite number of excluded minors. As ...
Gordon Royle's user avatar
  • 12.3k
44 votes
2 answers
10k views

Does the curvature determine the metric?

I ask myself, whether the curvature determines the metric. Concretely: Given a compact manifold $M$, are there two metrics $g_1$ and $g_2$, which are not everywhere flat, such that they are not ...
Bernhard Boehmler's user avatar
44 votes
2 answers
7k views

What is known about the sum x^{n^2}/n?

It follows from a general theorem of Honda that the formal group with the logarithm $$ x+x^{2^s}/2+x^{3^s}/3+x^{4^s}/4+\cdots $$ has integer coefficients. I became interested in it because its $p$-...
მამუკა ჯიბლაძე's user avatar
35 votes
3 answers
7k views

Distinct numbers in multiplication table

Consider the multiplication table for the numbers $1,2,\dots, n$. How many different numbers are there? That is, how many different numbers of the form $ij$ with $1 \le i, j \le n$ are there? I'm ...
falagar's user avatar
  • 2,761
27 votes
8 answers
5k views

Representability of finite metric spaces

There have been a couple questions recently regarding metric spaces, which got me thinking a bit about representation theorems for finite metric spaces. Suppose $X$ is a set equipped with a metric $d$...
Matt Noonan's user avatar
  • 3,984
26 votes
2 answers
2k views

Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map $D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the differential operator corresponding ...
Ali Taghavi's user avatar
23 votes
2 answers
2k views

Uniqueness of compactification of an end of a manifold

Let $M$ be an $n$-dimensional manifold (smooth or topological). I call $\bar{M}$ a compactification of $M$ if it is an $n$-dimensional compact manifold with boundary $\partial \bar{M}$, an $(n-1)$-...
Igor Khavkine's user avatar
22 votes
3 answers
2k views

Hecke equidistribution

For a prime $p\equiv 1\pmod{4}$, we can write $p=a^2+b^2=N(a+bi)$. Therefore $$ a+bi=p^{1/2}e^{i\varphi} $$ where $\varphi\in [0,2\pi]$. I know that Hecke proved that $\varphi$ is equidistributed. I ...
M.B's user avatar
  • 2,468
19 votes
3 answers
4k views

sum of squares in ring of integers

Lagrange proved that every positive integer is a sum of 4 squares. Are there general results like this for rings of integers of number fields? Is this class field theory? Explicitly, suppose a ...
user avatar
18 votes
2 answers
2k views

Explaining Mukai-Fourier transforms physically

A core concept in mathematics, engineering, and physics is the Fourier Transform (FT) and its many variants (Generalized Fourier Series, Green's Function, Pontryagin duality). The basic algorithm is ...
Tom Copeland's user avatar
  • 9,937
18 votes
0 answers
977 views

"Special" meanders

One of the open problems in combinatorics is enumeration of meanders. Here on MO I only could find them under the heading not-especially-famous-long-open-problems-which-anyone-can-understand. Since my ...
მამუკა ჯიბლაძე's user avatar
18 votes
0 answers
854 views

Is the universality of the surreal number line a weak global choice principle?

I'd like to consider the principle asserting that the surreal number line is universal for all class linear orders, or in other words, that every linear order (including proper-class-sized) linear ...
Joel David Hamkins's user avatar
17 votes
2 answers
3k views

Does ZFC prove the universe is linearly orderable?

It is consistent with ZFC that the universe is well-ordered, e.g. in $V=L$ where global choice holds. I also know that it is consistent that global choice fails (although I have no immediate example ...
Asaf Karagila's user avatar
  • 38.1k
17 votes
3 answers
2k views

How many tacks fit in the plane?

Call a tack the one point union of three open intervals. Can you fit an uncountable number of them on the plane? Or is only a countable number?
nonlinearality's user avatar
16 votes
2 answers
2k views

Intuitive explanation why "shadow operator" $\frac D{e^D-1}$ connects logarithms with trigonometric functions?

Consider the operator $\frac D{e^D-1}$ which we will call "shadow": $$\frac {D}{e^D-1}f(x)=\frac1{2 \pi }\int_{-\infty }^{+\infty } e^{-iwx}\frac{-iw}{e^{-i w}-1}\int_{-\infty }^{+\infty } e^...
Anixx's user avatar
  • 9,306
2 votes
0 answers
419 views

Lifting a quadratic system to a non-vanishing vector field on $S^{3}$ or $T^{1} S^{2}$

Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non-vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...
Ali Taghavi's user avatar
290 votes
125 answers
90k views

What are some examples of colorful language in serious mathematics papers?

The popular MO question "Famous mathematical quotes" has turned up many examples of witty, insightful, and humorous writing by mathematicians. Yet, with a few exceptions such as Weyl's "angel of ...
249 votes
37 answers
169k views

Best algebraic geometry textbook? (other than Hartshorne)

I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, webpage, etc. One suggestion ...
192 votes
94 answers
105k views

Famous mathematical quotes [closed]

Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of? Standard community wiki rules apply: one ...
167 votes
50 answers
55k views

17 camels trick

The following popular mathematical parable is well known: A father left 17 camels to his three sons and, according to the will, the eldest son should be given a half of all camels, the middle son ...
135 votes
26 answers
28k views

What are some famous rejections of correct mathematics?

Dick Lipton has a blog post that motivated this question. He recalled the Stark-Heegner Theorem: There are only a finite number of imaginary quadratic fields that have unique factorization. They are $...
112 votes
19 answers
41k views

What is the definition of "canonical"?

I just received a referee report criticizing that I would too often use the word "canonical". I have a certain understanding of what "canonical" should stand for, but the report ...
109 votes
10 answers
23k views

Set theories without "junk" theorems?

Clearly I first need to formally define what I mean by "junk" theorem. In the usual construction of natural numbers in set theory, a side-effect of that construction is that we get such theorems as $...
Jacques Carette's user avatar

15 30 50 per page