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### Examples of common false beliefs in mathematics

The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while ...
4k views

### Finding a 1-form adapted to a smooth flow

Let $M$ be a smooth compact manifold, and let $X$ be a smooth vector field of $M$ that is nowhere vanishing, thus one can think of the pair $(M,X)$ as a smooth flow with no fixed points. Let us say ...
26k views

### Important formulas in combinatorics

Motivation: The poster for the conference celebrating Noga Alon's 60th birthday, fifteen formulas describing some of Alon's work are presented. (See this post, for the poster, and cash prizes offered ...
70k views

### Not especially famous, long-open problems which anyone can understand

Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please. Motivation: I plan to use this list ...
2k views

### Limit cycles as closed geodesics (in negatively or positively curved space)

EDIT: Here is a related post which concern quadratic vector fields rather than Van der Pol equation. In this linked post we see that the convexity of limit cycle play a crucial role. On ...
28k views

### Is there an introduction to probability theory from a structuralist/categorical perspective?

The title really is the question, but allow me to explain. I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory (in the sense of ...
7k views

### When is $A$ isomorphic to $A^3$?

This is totally elementary, but I have no idea how to solve it: let $A$ be an abelian group such that $A$ is isomorphic to $A^3$. is then $A$ isomorphic to $A^2$? probably no, but how construct a ...
2k views

### Generalization of a theorem of Øystein Ore in group theory

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive. Proof: see below. Let $(H \subset G)$ be an inclusion of finite groups and ...
88k views

### Widely accepted mathematical results that were later shown to be wrong?

Are there any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time later, possibly ...
9k views

### What do epimorphisms of (commutative) rings look like?

(Background: In any category, an epimorphism is a morphism $f:X\to Y$ which is "surjective" in the following sense: for any two morphisms $g,h:Y\to Z$, if $g\circ f=h\circ f$, then $g=h$. Roughly, "...
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### Thinking and Explaining

How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words differ, ...
29k views

### Which journals publish expository work?

I wonder if anyone else has noticed that the market for expository papers in mathematics is very narrow (more so than it used to be, perhaps). Are there any journals which publish expository work, ...
10k views

### Compelling evidence that two basepoints are better than one

This question is inspired by an answer of Tim Porter. Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his ...
6k views

### Does the formal power series solution to $f(f(x))= \sin( x)$ converge?

I have spent some time using gp-pari. There is, of course, a formal power series solution to $f(f(x)) = \sin x.$ It is displayed below, identified by the symbol $g$ because I am not entirely sure ...
3k views

### Why is there a connection between enumerative geometry and nonlinear waves?

Recently I encountered in a class the fact that there is a generating function of Gromov--Witten invariants that satisfies the Korteweg--de Vries hierarchy. Let me state the fact more precisely. ...
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### Similarities between Post's Problem and Cohen's Forcing

Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated 4/...
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### Are the Sierpiński cardinal $\acute{\mathfrak n}$ and its measure modification $\acute{\mathfrak m}$ equal to some known small uncountable cardinals?

This question was motivated by an answer to this question of Dominic van der Zypen. It relates to the following classical theorem of Sierpiński. Theorem (Sierpiński, 1921). For any countable partition ...
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### Ways to prove the fundamental theorem of algebra

This seems to be a favorite question everywhere, including Princeton quals. How many ways are there? Please give a new way in each answer, and if possible give reference. I start by giving two: ...
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### Does $\mathrm{Aut}(\mathrm{Aut}(…\mathrm{Aut}(G)…))$ stabilize?

Purely for fun, I was playing around with iteratively applying $\DeclareMathOperator{\Aut}{Aut}\Aut$ to a group $G$; that is, studying groups of the form $${\Aut}^n(G):= \Aut(\Aut(...\Aut(G)...))$$ ...
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### How small can a sum of a few roots of unity be?

Let $n$ be a large natural number, and let $z_1, \ldots, z_{10}$ be (say) ten $n^{th}$ roots of unity: $z_1^n = \ldots = z_{10}^n = 1$. Suppose that the sum $S = z_1+\ldots+z_{10}$ is non-zero. How ...
7k views

### Does the curvature determine the metric?

Hello, I ask myself, whether the curvature determines the metric. Concretely: Given a compact Riemannian manifold $M$, are there two metrics $g_1$ and $g_2$, which are not everywhere flat, such that ...
67k 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 ...
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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 $$\pi_k(x)\sim\frac{x(\log\log x)^{k-1}}{(k-1)!\... 3answers 9k views ### About Goldbach's conjecture let's consider a composite natural number n greater or equal to 4. Goldbach's conjecture is equivalent to the following statement: "there is at least one natural number r such as (n-r) and (n+... 7answers 9k views ### Lower bound for sum of binomial coefficients? Hi! I'm new here. It would be awesome if someone knows a good answer. Is there a good lower bound for the tail of sums of binomial coefficients? I'm particularly interested in the simplest case \... 2answers 4k views ### What is known about the sum x^{n^2}/n? It follows from a general theorem of Honda that the formal group with the logarithm$$ x+x^{2^s}/2+x^{3^s}/3+x^{4^s}/4+\cdots $$has integer coefficients. I became interested in it because its p-... 2answers 1k views ### Elliptic operators corresponds to non vanishing vector fields Added, June 19, 2019: The main motivation of this post is to associate an index to differential operator associated to a dynamical system such that the index has an interesting ... 3answers 1k views ### Hecke equidistribution For a prime p\equiv 1\pmod{4}, we can write p=a^2+b^2=N(a+bi). Therefore$$ a+bi=p^{1/2}e^{i\varphi} $$where \varphi\in [0,2\pi]. I know that Hecke proved that \varphi is equidistributed. I ... 0answers 790 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 ... 0answers 364 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 ... 34answers 70k views ### Why is a topology made up of 'open' sets? [closed] I'm ashamed to admit it, but I don't think I've ever been able to genuinely motivate the definition of a topological space in an undergraduate course. Clearly, the definition distills the essence of ... 69answers 173k views ### Mathematical “urban legends” When I was a young and impressionable graduate student at Princeton, we scared each other with the story of a Final Public Oral, where Jack Milnor was dragged in against his will to sit on a committee,... 67answers 102k views ### Awfully sophisticated proof for simple facts [closed] It is sometimes the case that one can produce proofs of simple facts that are of disproportionate sophistication which, however, do not involve any circularity. For example, (I think) I gave an ... 15answers 47k views ### Why worry about the axiom of choice? As I understand it, it has been proven that the axiom of choice is independent of the other axioms of set theory. Yet I still see people fuss about whether or not theorem X depends on it, and I don't ... 2answers 50k views ### Estimating the size of solutions of a diophantine equation A. Is there natural numbers a,b,c such that \frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} is equal to an odd natural number ? (I do not know any such numbers). B. Suppose that \frac{a}{b+c} + \... 11answers 63k views ### Sum of 'the first k' binomial coefficients for fixed n I am interested in the function$$f(N,k)=\sum_{i=0}^{k} {N \choose i}$$for fixed N and 0 \leq k \leq N . Obviously it equals 1 for k = 0 and 2^{N} for k = N, but are there any other ... 6answers 7k views ### Kahler differentials and Ordinary Differentials What's the relationship between Kahler differentials and ordinary differential forms? 10answers 8k views ### Riemannian surfaces with an explicit distance function? I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds (M,g)) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ... 2answers 4k views ### What interesting/nontrivial results in Algebraic geometry require the existence of universes? Brian Conrad indicated a while ago that many of the results proven in AG using universes can be proven without them by being very careful (link). I'm wondering if there are any results in AG that ... 3answers 16k views ### Which integers can be expressed as a sum of three cubes in infinitely many ways? For fixed n \in \mathbb{N} consider integer solutions to$$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. ...

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