Questions tagged [open-problems]

If it turns out that a problem is equivalent to a known open problem, then the open-problem tag is added. After that, the question essentially becomes, "What is known about this problem? What are some possible ways to approach this problem? What are some ways that people have tried to attack it before, and with what results?"

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6
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0answers
575 views

Tarski Monster group with prime 5

Does the Tarski Monster group with prime 5 exist? I know that for 2 and 3, the group does not exist, but what about 5?
5
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0answers
548 views

Do the banded operators check the invariant subspace problem?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...
3
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1answer
203 views

Equiprojective polyhedra

Seeing Garabed Gulbenkian's question (which was inspired by Joel Hamkins' question), reminds me of an analogous problem which I believe remains open, and which some might find intriguing. Define an ...
16
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2answers
833 views

Tiling the square with rectangles of small diagonals

For a given integer $k\ge3$, tile the unit square with $k$ rectangles so that the longest of the rectangles' diagonals be as short as possible. Call such a tiling optimal. The solutions are obvious in ...
18
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2answers
2k views

Integer-distance sets

Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$. Say that $S$ is an integer-distance set if every pair of points in $S$ is separated by an integer Euclidean distance. ...
9
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1answer
554 views

Small quadrilaterals containing a given convex region

It is easy to prove that (*) Every convex planar set of area 1 is contained in a quadrilateral of area 2. It is also easy to see that statement (*) remains true if the constant 2 is replaced with a ...
45
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0answers
2k views

Set-theoretic reformulation of the invariant subspace problem

The invariant subspace problem (ISP) for Hilbert spaces asks whether every bounded linear operator $A$ on $l^2$ (with complex scalars) must have a closed invariant subspace other than $\{0\}$ and $l^2$...
8
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1answer
814 views

A question about the Axiom of Choice

Let AC denote the Axiom of Choice. Let PP denote the so-called "Partition Principle" which states that "If S is a non-empty set and T is a non-empty set of pairwise disjoint subsets of S, then S can ...
25
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2answers
7k views

Open Problems in Algebraic Topology and Homotopy Theory

Some time ago (I see it was initially written before 1999?) Mark Hovey assembled a list of open problems in algebraic topology. The list can be found here. Some of the problems I know about have been ...
11
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1answer
570 views

A conjecture by Euler about $8n+3$

Euler's conjecture: For any positive integer $n$, $8n+3$ can be represented as a sum $$8n+3=(2k-1)^2+2p,$$ where $k$ is a positive integer, and $p$ is a prime. I want to know whether there has been ...
2
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0answers
370 views

Counting factors: is this approach in the literature on multiperfect numbers?

Does the following approach (or something near it) exist in the number theory literature? I will provide some motivation for $\omega(p^n - 1)$ as $n \rightarrow \infty$ and for this question. First, ...
5
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1answer
816 views

List of open problems of formal languages [closed]

As we know, there are some open problems of formal languages. I am wondering if there is a somehow complete list of open problem of formal languages. If there isn't such a list, can we make it one as ...
2
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2answers
257 views

Does this modified Hasse principle hold for curves?

Let $C$ be a curve over $\mathbb Q$ with a point $P$ on $Pic^1$. For each $\mathbb Q$-rational point $Q$, $Q-P$ is a point on the Jacobian $J$. We can use the map $H^0(\mathbb Q, J) \to H^1(\mathbb Q,...
11
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1answer
387 views

Needle probing for a convex body

Suppose there is an unknown closed convex body $K$ of volume vol$(K) = V$ inside the unit cube $[-\frac{1}{2}, \frac{1}{2}]^d$ in $\mathbb{R}^d$. You are permitted to probe with a (one-dimensional) ...
12
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2answers
1k views

Distinctive property of the primes 17 and 19?

Consider the question whether it is true that a prime number $p$ divides $1^1+2^2+3^3+....+(p-1)^{p-1}$ if and only if $p \in \{17,19\}$. For the obvious heuristic reasons, for large $n$ one would ...
1
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2answers
424 views

A conjecture on closed discrete subset

I am struggling with this old problem, which is also posted here: Let $X$ satisfy countable chain condition(abbreviated as CCC) and $X$ has a regular $G_\delta$-diagonal. Then the cardinality of $X$...
37
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5answers
2k views

Surfaces filled densely by a geodesic

Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no single geodesic $\gamma$ that fills $S$ densely? Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points ...
32
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12answers
14k views

Open problems in PDEs, dynamical systems, mathematical physics

(This question might not be appropriate for this site. If so, I apologize in advance. I would have posted to mathstack, but I'm looking for advice from active researchers.) I am an undergrad in math ...
0
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0answers
425 views

Known and unknown about Ramanujan's tau function

What is a good reference for open problems relating to the Ramanujan tau function? I know about Lehmer's conjecture. I know the following reductions of the problem: the smallest counterexample must ...
3
votes
1answer
887 views

Collatz conjecture— finite state machine transducer construction, origination?

wikipedia has an entry on the Collatz conjecture with a section on As an abstract machine that computes in base two. this apparently describes a construction of a FSM transducer computing sequential ...
15
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1answer
724 views

Is it true that every f.g. infinite simple group has exponential growth?

Is it true that every finitely generated infinite simple group has exponential (word-)growth? Remark: As Mark Sapir has pointed out, the question whether every finitely generated group of ...
10
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4answers
2k views

problems from the scottish book

Which of the problems from the Scottish Book (pdf of English version) by Stefan Banach are still open? I know that one of the problems was solved by Per Enflo for which he got a live goose from ...
7
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1answer
390 views

Is there any o-minimal expansion of the real field with functions of growth higher than exponential?

Let $\bar{\mathbb{R}}$ be the structure of the real field, that is $(\mathbb{R},0,1,+,-,*,<)$ . We say that a function $f$ is of growth higher than exponential if for all $N\in \mathbb{N}$ there $f(...
1
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1answer
938 views

An open problem on general topology

There is an open problem in this paper: Classes defined by stars and neighbourhood assignments by van Mill and others. Problem 4.8. Is a regular (Tychonoff) star compact space metrizable if it has a $...
16
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3answers
2k views

open problems in Seiberg-Witten Theory on 4-Manifolds

What are some of the open problems in Seiberg-Witten Theory on 4-Manifolds.I tried googling but couldn't any. I tried googling it, but couldn't find any resources.The places where I can a survey or ...
7
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0answers
430 views

How many values a polynomial map misses?

Let $F$ be a field. For a uni-variate polynomial $f(x)$ over $F$,let $M_f(F)$ denote the number of values that $f$ misses, that is, the cardinality of the subset $F - f(F)$ in $F$. Assume that $f$ is ...
17
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1answer
769 views

Randomly switching street lights, in a square city

This is a combinatorics-probability question, best stated however in "recreational" terms. Imagine a $N\times N$ city, meaning that we have $N$ horizontal streets, and $N$ vertical streets. At each ...
12
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7answers
2k views

Open problems in the theory of compact quantum groups

What are the important open problems in the theory of compact quantum groups? Or conjectures? Here is an example from An De Rijdt's Ph.D. thesis: Is every compact quantum group with the fusion rules ...
12
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4answers
2k views

What are the major open problems in design theory nowaday?

I gather that the question whether the Bruck-Chowla-Ryser condition was sufficient used to top the list, but now that that's settled - what is considered the most interesting open question?
20
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2answers
1k views

Center of mass from the abstract point of view, or could the ancient Greeks invent modern analysis?

This is a very open-ended question, which may or may not have a perfect answer, and for which I have a few ideas but nothing like a clear picture. However, I guess it won't hurt to ask to see if ...
27
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4answers
3k views

Open problems in Birational Geometry, after BCHM

Rencently a breakthrough was made in the context of the Minimal Model Program by the work of Birkar-Cascini-Hacon-McKernan. They proved that the canonical ring of a smooth or mildly singular ...
10
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1answer
1k views

Transcendentality of all irrationals in the Cantor set

Hi, I am a student researcher trying to prove that all irrationals within the Cantor set are transcendental. This is grounded, intuitively, in Cantor set members' being non-normal; since algebraic ...
54
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1answer
2k views

Intersecting Family of Triangulations

Let $\cal T_n$ be the family of all triangulations on an $n$-gon using $(n-3)$ non-intersecting diagonals. The number of triangulations in $\cal T_n$ is $C_{n-2}$ the $(n-2)$th Catalan number. Let $\...
47
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1answer
2k views

A function whose fixed points are the primes

If $a(n) = (\text{largest proper divisor of } n)$, let $f:\mathbb{N} \setminus \{ 0,1\} \to \mathbb{N}$ be defined by $f(n) = n+a(n)-1$. For instance, $f(100)=100+50-1=149$. Clearly the fixed points ...
15
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2answers
1k views

Why Donaldson's Four-Six Conjecture?

Simon Donaldson apparently made the following conjecture: Two closed symplectic 4-manifolds $(X_1,\omega_1)$ and $(X_2,\omega_2)$ are diffeomorphic if and only if $(X_1\times S^2,\omega_1\oplus\omega)$...
68
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16answers
4k views

Important open problems that have already been reduced to a finite but infeasible amount of computation

Most open problems, when formalized, naturally involve quantification over infinite sets, thereby obviating the possibility, even in principle, of "just putting it on a computer." Some questions (e.g....
2
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2answers
526 views

If all real conjugacy classes are strongly real, then all real irreps are “strongly real”(symmetric), true?

Question Is true that if all real conjugacy classes of a finite group are strongly real, then all its real irreducible representations (irreps) are "strongly real" (symmetric)? And vice versa? ...
5
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1answer
411 views

Open problems about CMC hypersurfaces with symmetries?

Recently, Andrews and Li announced a complete classification of CMC ($H=const.$) tori in $S^3$, confirming a conjecture of Pinkall and Sterling. Their main result is that any such torus is ...
8
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1answer
697 views

approximate uncertainty principle for finite abelian groups

Edit: we cannot find such an example. It would imply a negative solution to the KS${}_2$ conjecture which has now been proven by Marcus, Spielman, and Srivastava in this paper. In fact, their ...
12
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1answer
611 views

A maximal element, where Schur gives a minimal element

Let me recall a result due to I. Schur, which I learnt from F. Goldberg's answer to my MO question Hadamard-like inequalites for positive definite symmetric matrices. If $H$ is a subgroup of $\frak ...
11
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1answer
499 views

Why would dim primitive irrep divide size of some conjugacy class ?

From Isaacs et.al. 2005 Conjecture C. Let χ be a primitive irreducible character of an arbitrary finite group G. Then χ(1) divides | clG(g)| for some element g ∈ G. Here, of course, we ...
14
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3answers
608 views

Kaplansky's 6th conjecture: dim(Irrep) | dim(algebra) - for semi-simple Hopf algebras

Let $H$ be a semisimple Hopf algebra. One of the Kaplansky's conjectures states that the dimension of any irreducible $H$-module divides the dimension of $H$. In which cases the conjecture is known ...
11
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1answer
585 views

Are Galois groups of Q with restricted ramification supposed to be finitely generated?

Fix a finite set $S$ of places of $\mathbb Q$. Let $G_{\mathbb Q,S}$ be the Galois group of the maximal extension of $\mathbb Q$ unramified outside S$. I believe that it is an open question whether ...
20
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0answers
523 views

The oriented homeomorphism problem for Haken 3-manifolds

Haken famously described an algorithm to solve the homeomorphism problem for the 3-manifolds that bear his name (fleshed out by many others, including Hemion and Matveev who fixed some gaps). But it'...
8
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0answers
263 views

Is $Q_n(x)=\sigma_{n+1}(x)/\sigma_n(x)$ logarithmically convex on $\mathbf{R}$?

In 1975 J. van de Lune considered the monotony properties of the canonical Riemann Upper and Lower sums for $\int_0^1 t^xdt$, with $x>0$. Writing $\sigma_n(x) := 1^x+2^x+\cdots+n^x$ these sums are ...
6
votes
2answers
494 views

Measures of full Hausdorff dimension for self-affine sets

Consider the iterated function system $T_{1}(x)=(\beta x,\tau y)$, $T_{2}(x,y)=(\beta x+(1-\beta),\tau y+ (1-\tau))$ for $\beta\in(1/2,1)$ and $\tau\in (0,1/2)$ with self affine set $\Lambda_{\beta,\...
9
votes
3answers
1k views

Kronecker's Jugendtraum for real quadratic fields?

Kronecker's Jugendtraum (or Hilbert's 12'th problem) is to find abelian extensions of arbitrary number fields by adjoining `special' values of transcendental functions. The Kronecker-Weber theorem was ...
11
votes
1answer
1k views

The Class Number One Problem for Real Quadratic Fields

An approach to the Gauß class number one problem for imaginary quadratic fields is to determine the integral points on the modular curve $Y_{nonsplit}(n)$ for a suitable $n$. Here follows a quick ...
3
votes
2answers
490 views

S-matrix conjecture: status?

Is the $S$-matrix conjecture still open? I mean the one listed as Problem 7 in this survey.
14
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1answer
656 views

Minimizing the excursion of a sum of unit vectors

I have $n$ unit-length vectors $v_i$ in $\mathbb{R}^3$, whose sum is zero: $$ v_1 + v_2 + \cdots + v_n = 0 \; .$$ Now I form the closed polygon $P$ in space by placing them head to tail. So the ...

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