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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 ...

94
votes
1answer
27k views

What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis?

In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function $T(s)$, known as the Todd function. My question is, what is the ...
3
votes
1answer
337 views

Updated background on Hilbert 16th problem?

What is the current situation of the second part of the Hilbert 16th problem? What are the most updated news on this problem?
2
votes
1answer
53 views

Existence of a Lipschitz map from a positive measure set to a ball

Question. If $A\subset \mathbb{R}^n$ is any set of positive Lebesgue $n$-measure, does there exists a Lipschitz map $f:A\to\mathbb{R}^n$ such that $f(A)$ is a ball with the same measure? In dimension ...
-2
votes
2answers
309 views

What are some of the unsolved mathematical problems that we generally agree were posed before the beginning of the XX.-th century?

Because mathematics has been extremely well-developed in XX.-th and continues to do so in XXI.-th century and because there is an enormous number of open problems and conjectures and hypotheses posed ...
3
votes
1answer
147 views

A non-rational variety with a full exceptional collection?

Does there exist a smooth non-rational projective variety whose bounded derived category of coherent sheaves admits a full exceptional collection? I could not find any examples in the literature (for ...
18
votes
2answers
1k views

Is there a known Turing machine which halts if and only if the Collatz conjecture has a counterexample?

Some of the simplest and most interesting unproved conjectures in mathematics are Goldbach's conjecture, the Riemann hypothesis, and the Collatz conjecture. Goldbach's conjecture asserts that every ...
7
votes
1answer
322 views

What are some open problems in moduli spaces and moduli stacks?

I would like to know what are the open big and interesting problems related to moduli spaces and moduli stacks ? Thanks in advance for your help.
1
vote
0answers
61 views

Weak Leopoldt Conjecture for the Split Prime $\mathbb{Z}_p$-extension

In his 1973 Annals paper, Iwasawa proved that the weak Leopoldt Conjecture holds for the cyclotomic $\mathbb{Z}_p$-extension of any number field. If $K$ is an imaginary quadratic field and $F/K$ is ...
7
votes
0answers
186 views

A question related to Conways 99 graph problem

I have observed that the number of triangles $\frac{vk}{6}$ of a strongly regular graph with parameters $(v,k,1,2)$ is given by the coefficient $2(k-1)$ in the molien series of the "4-D extraspecial ...
1
vote
0answers
135 views

Algebraic version of unilateral shift

It was confirmed that Wold-type decomposition can be extended from von Neumann algebras to Baer*-rings (see this paper). By algebraic tools the notion of unilateral shifts is successfully transmitted ...
7
votes
4answers
288 views

Upper bound of the expectation of sum of the absolute value pairs

We have two arrays $A,B$ of length $n$. All values are i.i.d drawn from same distribution on $[0,1]$. If we sort $A,B$ in non-decreasing order and let $A_{(i)},B_{(i)}$ denote the i-th value in the ...
1
vote
0answers
129 views

Status of an open problem in isometric aspect of Banach space theory

The following open problem is taken from the book Open Problems in the Geometry and Analysis of Banach Spaces, page $40.$ Problem $84:$ Assume that $X$ is an infinite-dimensional separable Banach ...
9
votes
1answer
496 views

Status of the three-body problem

I find many numerical results on the three-body problem, but what is rigorously proved? Especially I would be interested in the parameter domains for which we have rigorous lower bounds on the ...
7
votes
0answers
238 views

Status of two Banach space theory open problems posted by Pełczyński

In the book 'Open Problems in the Geometry and Analysis of Banach Spaces', I am interested in the following two problems. Problem $1$: Let $X$ be a separable infinite-dimensional Banach space that ...
40
votes
13answers
7k views

PhD dissertations that solve an established open problem

I search for a big list of open problems which have been solved in a PhD thesis by the Author of the thesis (or with collaboration of her/his supervisor). In my question I search for every possible ...
2
votes
1answer
157 views

Lattices that top is the top of its join-irreducibles, such that a random element is almost surely greater then any given join-irreducible

Let $(L,\leq,0,1)$ be a lattice, and let's denote by $JI(L)$ the set of its join-irreducibles (i.e. elements that are not the lowest grater bound of two other elements). We suppose that $\sup JI(L)=...
2
votes
1answer
131 views

An upper bound for a vector with given norm 1 and norm 2

Suppose $X = (x_1, \ldots , x_n)$ is given and we know that $x_i$'s are nonnegative, $\sum_{i=1}^n x_i = n$ and $\sum_{i=1}^n x_i^2 = m $. Just by this information, is it possible to find a vector ...
6
votes
0answers
155 views

Is a stronger version of the Erdős-Turan conjecture on arithmetic progessions reasonable? (And related questions.)

Define the size, possibly $\infty$, of a set $S\subseteq \mathbb{N}$ as $|S|=\sum\limits_{n\in S} \frac{1}{n}$. Then the Erdős-Turan conjecture states that if $|S|=\infty$, S must contain arbitrarily ...
1
vote
0answers
126 views

A “conjectured” concentration inequality for operators, probably related with random matrix theory

I am working on some open problem. And I have reduced the original problem to the "conjecture" (actually I am not familiar with random matrix theory or other fields that may have such a result) as ...
2
votes
2answers
277 views

A possible dynamical approach to the “Invariant Subspace Problem”

In the literature, is there any paper or research investigating the invariant subspace problem with consideration of differential operators acting on an appropriate Sobolev space?In particular is ...
2
votes
0answers
68 views

Open problems concerning Araujo's biseparating maps

Araujo stated the following four open questions at the end of his paper, page $518$ and $519.$ Question $1:$ Assume that there exists a biseparating map $T:A^n(\Omega:E)\to A^m(\Omega',F)$ which is ...
11
votes
1answer
505 views

What is known about the relationship between Fermat's last theorem and Peano Arithmetic?

As far as I know, whether Fermat's Last Theorem is provable in Peano Arithmetic is an open problem. What is known about this problem? In particular, what is known about the arithmetic systems $PA + \...
2
votes
0answers
185 views

Open problems in the theory of manifolds relating to construction [closed]

A while ago I stumbled across a paper of Thurston: Some Simple Examples of Symplectic Manifolds, where Thurston constructs closed symplectic manifolds with no Kaehler structure. My question is: What ...
10
votes
4answers
1k views

What is the smallest sphere whose surface includes 100 integer points?

Let $S(r)$ be the surface of the origin-centered sphere in $\mathbb{R}^3$. A point is an integer point if all its coordinates are integers. What is the smallest radius $r_n$ such that $S(r_n)$ ...
7
votes
0answers
320 views

On a question of Coste & Roy from 1979

On page 44 of their 1979 paper, Topologies for real algebraic geometry, Coste & Roy define a structure sheaf on the real Zariski spectrum of a commutative ring (which can be regarded as the real ...
0
votes
0answers
228 views

Why are questions about taxicab numbers so difficult?

The generalized taxi cab number $\mathrm{Taxicab}(k, j, n)$ is the smallest positive integer such that it can be expressed as the sum of $j$ $k$-th powers in $n$ different ways. We have the famous ...
3
votes
0answers
239 views

If $a^3+b^3+c^3=N$, then $x^3+y^3+z^3+t^3 = N$ in infinitely many ways?

It is well-known that, $$a^3+b^3+c^3 = N\tag1$$ for $N=1,\,2$ is solvable in the integers in infinitely many ways . However, it is an open question (but is conjectured) that if for general $N$ it has ...
3
votes
1answer
187 views

Asymptotic form of pdf of Escape Time of arithmetic fBm

I am trying to apply the Girsanov formula and Doobs optional sampling theorem to obtain an asymptotic form of first passage density of an fbm process with drift, but the answer i am getting seems ...
1
vote
1answer
185 views

Portability of Thompson theorem about solvability to Moufang loops

Say we have a finite Moufang Loop $Q$, $|Q|<\infty$. There is a theorem proved by Thompson that states: Group $G$, $|G|<\infty$ is solvable $\iff$ $\forall a, b \in G \langle a, b\rangle$ is ...
11
votes
3answers
784 views

Undecidable easy arithmetical statement

Is there a basic arithmetic statement which is known to be undecidable ? By basic arithmetic statement I do mean an easy statement in the spirit of the Collatz conjecture . By the way is there some ...
6
votes
1answer
126 views

On statistical bases in Banach spaces

Let $K$ be a subset of the positive integers $\mathbb{N}$. For each $n\in \mathbb{N}$, $K_{n}$ denotes the set $\{k\in K: k\leq n\}$ and $|K_{n}|$ denotes the number of the elements in $K_{n}$. The ...
3
votes
1answer
178 views

Can one find a Jordan curve which has exactly one inscribed rectangle?

In On the number of inscribed squares of a simple closed curve in the plane it is shown that Theorem: For every positive integer $n$ there is a simple closed curve in the plane (which can be ...
10
votes
0answers
146 views

Equilaterally triangulated surfaces with prescribed boundary

There is a problem in Richard Kenyon's list which I would like to post here, because although I have thought about it from time to time, I have not been able to make the slightest progress on it: ...
1
vote
0answers
83 views

What is an umbilic point of a convex polyhedron?

An umbilic point of a smooth ($C^2$) convex surface in Euclidean 3-space is a point where the principal curvatures are equal. Is there some good way to generalize this notion to convex polyhedra? See ...
11
votes
2answers
411 views

Intrinsic vs Extrinsic geometry of convex surfaces

By Alexandrov's isometric embedding theorem, any locally convex metric prescribed on the sphere admits a realization as a convex surface in Euclidean 3-space, which, by Pogorelov's rigidity result, is ...
60
votes
0answers
1k views

Converse to Euclid's fifth postulate

There is a fascinating open problem in Riemannian Geometry which I would like to advertise here because I do not think that it is as well-known as it deserves to be. Euclid's famous fifth postulate, ...
8
votes
0answers
175 views

Plank invariant measures on convex bodies

Let $K\subset R^2$ be a convex body, i.e., a compact convex set with interior points. A plank $P$ is the region between a pair of parallel lines in $R^2$. Let us say that $P$ intersects $K$ properly ...
32
votes
0answers
898 views

Converse of the Archimedean property of the sphere

In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of ...
16
votes
1answer
329 views

Characterisation of bell-shaped functions

This is an open problem that I learned from Thomas Simon. I will completely understand if the question is judged as non-research level (and it is indeed not related to my research), but I believe a ...
1
vote
0answers
187 views

Does this idea give an algorithm for constructing Hadamard matrices?

Fedor Petrov's answer of my preceding question shows that my question reduces to the famous Hadamard conjecture about Hadamard matrices of order $4k$. So I decided to study this conjecture and I got ...
13
votes
1answer
453 views

Does the image of the exponential map generate the group?

Let $G$ be a connected Fréchet-Lie group and let $\mathfrak g$ be its Lie algebra. Does the image $\exp(\mathfrak g) \subset G$ of the exponential map generate $G$?
9
votes
4answers
4k views

Is there always one integer between these two rational numbers?

It appears that for each integer $k\geq2$, there is always one integer $c$ that satisfies the inequalities below. Can this be proved? $$\frac{3^k-2^k}{2^k-1}<c\leq \frac{3^k-1}{2^k}.$$ Note that ...
12
votes
1answer
530 views

Who conjectured the Cartan determinant conjecture

The Cartan determinant conjecture states that every finite dimensional algebra of finite global dimension has the property that the determinant of its Cartan matrix is equal to one. Who stated this ...
1
vote
0answers
53 views

Modular which is metrizing but does not satisfy the $\Delta_2$ condition

Let $\Phi$ be a nice Young function (N-function) and $(\Omega,\mathcal{F},P)$ a probability space such that either $P$ is diffuse on a set of non-zero probability or $P$ is purely atomic and there are ...
2
votes
0answers
44 views

Splitting of ordinals of oscillation ranks of a Baire $1$ function

Denny and Tang proved that Theorem $2.3$ Let $(f_n)$ be a sequence in $\mathfrak{B}_1(K)$ converging pointwise to a function $f.$ Suppose $\sup\{ \beta(f_n):n\in\mathbb{N} \} \leq \beta_0$ and $\...
7
votes
0answers
464 views

Limit cycles as closed geodesics(2)

Hilbert 16th problem asks for a uniform upper bound $H(n)$ for the number of limit cycles of a polynomial vector field of degree $n$ on the plane. Here is an updated proof of the ...
4
votes
0answers
342 views

Limit cycles of quadratic systems and closed geodesics(Finitness of $H(2)$)

This question is inspired by this answer to the question Finding a 1-form adapted to a smooth flow. Assume that $V$ is a polynomial vector field of degree $2$ as follows:$$\begin{cases} x'=P(...
11
votes
2answers
568 views

Current state of the Komlos conjecture on vector balancing

Komlos Conjecture: the exists an absolute constant $K>0$ such that for all $d$ and any collection of vectors $v_1,\ldots, v_n\in \mathbb{R}^d$ with $\left\lVert v_i\right\rVert _2=1$ we can find ...
34
votes
6answers
7k views

Open problems in mathematical physics

What are good, still unsolved problems in mathematical physics that are in vogue? I always get the same answers: reference to Millennium Problems by the Clay Institute, or "there's still a lot to do ...
13
votes
3answers
400 views

Random N-body problem

Suppose there are $N$ unit-mass particles whose initial positions are uniformly distributed in a unit-radius disk. Each particle is assigned a randomly oriented initial velocity vector $v_i$ of length ...