# All Questions

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### 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 ...
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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 ...
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### 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 ...
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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 ...
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### Widely accepted mathematical results that were later shown wrong?

I wonder if there are 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 ...
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Is there an integral fusion category of global dimension $210$, such that the simple objects have dimensions $\{1,5,5,5,6,7,7\}$ and the following fusion matrices? $\small{\begin{smallmatrix} 1 & ... 2answers 3k views ### Does the curvature determine the metric? Hello, I ask myself, whether the curvature determines the metric. Concretely: Given a compact Riemannian manifold$M$, are there two metrics$g_1$and$g_2$, which are not everywhere flat, such that ... 0answers 222 views ### Lifting a quadratic system to a non vanishing vector field on$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$. It is constructed in ... 39answers 29k views ### Most interesting mathematics mistake? Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincare's 3d sphere charaterization or the search to prove ... 2answers 2k views ### Similarities between Post's Problem and Cohen's Forcing Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated ... 2answers 944 views ### Generalization of a theorem of Øystein Ore in group theory Theorem (Øystein Ore, 1938): A finite group$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 ... 2answers 3k views ### Distinct numbers in multiplication table Consider multiplication table for numbers$1,2,\cdots, n$. 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 interested ... 3answers 465 views ### Jordan-Hölder theorem for subfactors? All the subfactors$(N\subset M)$are irreducible and finite index inclusions of II$_1$factors. First recall that in this paper, D. Bisch characterizes the Jones projections$e_K$of the ... 21answers 26k views ### 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 ... 35answers 34k 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 ... 15answers 21k 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 ... 10answers 13k 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 ... 2answers 14k views ### Is the analysis as taught in universities in fact the analysis of definable numbers? Ten years ago when I studied in the university I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows: All numbers are divided into two classes: those ... 11answers 4k 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 ... 5answers 3k views ### Is the Riemann Hypothesis equivalent to a$\Pi_1$sentence? 1) Can the Riemann Hypothesis (RH) be expressed as a$\Pi_1$sentence? More formally, 2) Is there a$\Pi_1$sentence which is provably equivalent to RH in PA? (This is mentioned in P. ... 4answers 3k 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 ... 3answers 3k views ### Number of elements in the set$\{1,\cdots,n\}\cdot\{1,\cdots,n\}$Let$A_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for$|A_n|$? Will it be$o(n^2)$? 2answers 2k views ### Euler characteristic of a manifold and self-intersection This is probably quite easy, but how do you show that the Euler characteristic of a manifold M (defined for example as the alternating sum of the dimensions of integral cohomology groups) is equal to ... 2answers 2k views ### Dimension of infinite product of vector spaces This question is motivated by the question link text, which compares the infinite direct sum and the infinite direct product of a ring. It is well-known that an infinite dimensional vector space is ... 7answers 4k 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 386 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 ... 1answer 337 views ### coloring in lattice This is a mathematical question raised from engineering and physics: Is there some established mathematical approach in filling a physical lattice with some colored basis (black and white here)? For ... 0answers 315 views ### Reference for Wang Tile I am working on projects in solving ground state of generalized ising models. One recent work involves tiling with basic tiles that filled the whole lattice. For example, we could obtain results: ... 3answers 639 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 $$\Phi(n)=n-\sum_i\frac n{p_i}\+\sum_{i \lt j}\frac ... 76answers 39k views ### Best online mathematics videos? I know of two good mathematics videos available online, namely: Sphere inside out (part I and part II) Moebius transformation revealed Do you know of any other good math videos? Share. 46answers 32k views ### Ways to prove the fundamental theorem of algebra This seems to be a favorite question everywhere, including Princeton quals. How many ways are there? Please give a new way in each answer, and if possible give reference. I start by giving two: ... 8answers 11k views ### If f is infinitely differentiable then f coincides with a polynomial Let f be an infinitely differentiable function on [0,1] and suppose that for each x \in [0,1] there is an integer n \in \mathbb{N} such that f^{(n)}(x)=0. Then does f coincide on [0,1] ... 42answers 17k views ### Examples of eventual counterexamples Define an "eventual counterexample" to be P(a) = T for a < n P(n) = F n is sufficiently large for P(n) = T\ \ \forall n \in \mathbb{N} to be a 'reasonable' conjecture to make. where ... 9answers 19k views ### solving f(f(x))=g(x) This question is of course inspired by the question How to solve f(f(x))=cosx and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form f(f(x))=g(x) on a ... 7answers 8k views ### Does \mathrm{Aut}(\mathrm{Aut}(…\mathrm{Aut}(G)…)) stabilize? Purely for fun, I was playing around with iteratively applying \DeclareMathOperator{\Aut}{Aut}\Aut to a group G; that is, studying groups of the form$$ {\Aut}^n(G):= \Aut(\Aut(...\Aut(G)...)) $$... 2answers 6k views ### The amplituhedron minus the physics Is it possible to appreciate the geometric/polytopal properties of the amplituhedron without delving into the physics that gave rise to it? All the descriptions I've so far encountered assume ... 4answers 8k views ### 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 ... 37answers 14k views ### Major mathematical advances past age fifty [closed] From A Mathematician’s Apology, G. H. Hardy, 1940: "I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever ... 7answers 6k views ### Arguments against large cardinals I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally ... 10answers 9k views ### Algorithm for finding the volume of a convex polytope It's easy to find the area of a convex polygon by division into triangles, but what is the optimal way of finding the volume of higher-dimensional convex bodies? I tried a few methods for dividing ... 7answers 3k views ### Bijection between irreducible representations and conjugacy classes of finite groups Is there some natural bijection between irreducible representations and conjugacy classes of finite groups (as in case of S_n)? 6answers 2k views ### Is the non-triviality of the algebraic dual of an infinite-dimensional vector space equivalent to the axiom of choice? If V is given to be a vector space that is not finite-dimensional, it doesn't seem to be possible to exhibit an explicit non-zero linear functional on V without further information about V. The ... 2answers 2k views ### Are non-PL manifolds CW-complexes? Can every topological (not necessarily smooth or PL) manifold be given the structure of a CW complex? I'm pretty sure that the answer is yes. However, I have not managed to find a reference for ... 5answers 1k views ### Are the 'semi' trivial zeros of \zeta(s) \pm \zeta(1-s) all on the critical line? The proof that \Gamma(z)\pm \Gamma(1-z) only has zeros for z \in \mathbb{R} or z= \frac12 +i \mathbb{R} has been given here: Are all zeros of \Gamma(s) \pm \Gamma(1-s) on a line with real ... 4answers 25k 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 they are Hermitian and positive-definite? I am investigating ... 5answers 3k views ### totally disconnected and zero-dimensional spaces When do the notions of totally disconnected space and zero-dimensional space coincide? From what I gather, there are at least three common notions of topological dimension: covering dimension, small ... 3answers 621 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 ... 3answers 1k views ### Is there a purely group-theoretic reformulation of an equivalence of subgroups? There is an equivalence relation between inclusion of finite groups coming from the world of subfactors: Definition:$(H_{1} \subset G_{1}) \sim(H_{2} \subset G_{2})$if$(R^{G_{1}} \subset ...
I want to find some condition to construct a continuous finitely additive measure on the natural numbers, i.e. $f:P(\mathbb{N})\rightarrow [0,1]$ such that $f(\{n\})=0$, and $f$ is an additive ...
Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...