Unsolved Problems

(Montgomery--Yang problem). Does there exist a pseudo-free, smooth, $S^{1}$ action on $S^{5}$ with more than three multiple orbits?...

Is there a closed aspherical 5-manifold that is not triangulable?...

Let $M_{1}$ and $M_{2}$ be smooth manifolds of dimension $n$. Suppose $M_{1}$ admits an $S$-map into $\R^{p}$. If $M_{2}$ is homeomorphic to $M_{1}$, ...

The Andrews--Curtis Conjecture [AC65] for the trivial group: a presentation of the trivial group can be changed to the trivial presentation by Andrews...

(Whitehead's Asphericity Question). Is every subcomplex of an aspherical 2-complex aspherical?...

(Zeeman Conjecture). If $K$ is a finite contractible 2-complex, then $K\times I$ collapses to a point [Zee64, Conjecture (1)]....

Let $M$ be a finite-volume hyperbolic $n$-manifold. - Does it always have a finite cover with $b_{1}>0$? - Does it always have a finite cover with f...

Does there exist a 1-cusped finite-volume hyperbolic $n$-manifold for any $n\geq 5$?...

Suppose $M$ is a manifold with a complete Riemannian metric with nonnegative Ricci curvature. Is the fundamental group of $M$ finitely generated?...

Let $R$ be $\Z$ or a field. Let $A$ and $B$ be differential graded algebras so that either: \begin{itemize} - As a graded algebra, $A$ (respectively ...

Let $(W,\omega,V_{i},\phi_{i})$ be two Weinstein structures on a fixed symplectic manifold $(W,\omega)$ (or equivalently consider two Weinstein handle...

- In higher dimensions, find the `non-analytic' cohomology module $\HH^{*}_{?}$ analogous to the analytic lattice cohomology $\HH^{*}_{\mathrm{an}}$, ...

Let $A,B$ be positive semidefinite, by Jensen's inequality, it is easy to see $[tr(A^s+B^s)]^{\frac{1}{s}}\leq [tr(A^r+B^r)]^{\frac{1}{r}}$, whenever ...

Problem Given a Matrix $A$, the $k$-th elementary symmetric function of $A$, namely $S_k(A)$, is defined as the sum of all $k$-by- $k$ principal mino...

Conjecture There exists a finite lattice which is not the congruence lattice of a finite algebra....

Conjecture In the category of continuous funcoids (defined similarly to the category of topological spaces) the following is a direct categorical prod...

Question I've been working on this for a long time and I'm getting nowhere. Could you help me or at least tell me where to look for help. Suppose D is...

Conjecture For $c \geq m \geq 1$, let $P(c,m)$ be the statement that given any exact $c$-coloring of the edges of a complete countably infinite graph ...

Question What is the Waring rank of the determinant of a $d \times d$ generic matrix? For simplicity say we work over the complex numbers. The $d \ti...

Conjecture Any $C^r$ structurally stable diffeomorphism is hyperbolic....

Problem Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?...

Question Give a necessary and sufficient criterion for the sequence $(a_n)$ so that the power series $\sum_{n=0}^{\infty} a_n x^n$ is bounded for all ...

Conjecture Let $D$ be the open unit disk in the complex plane and let $U_1,\dots,U_n$ be open sets such that $\bigcup_{j=1}^nU_j=D\setminus\{0\}$. Sup...

Conjecture For all $\alpha=(\alpha_1,\alpha_2,\ldots)\in l_2(\cal{C})$ the following inequality holds $$\sum_{n\geq 1}|\alpha_n|^2\geq \frac{6}{\pi^2}...

For $k\in \mathbb{N}$ let $T_k$ denote the minimal number $t\in \mathbb{N}$ such that there is a rainbow $AP(k)$ in every equinumerous $t$-coloring of...

Problem Do 4-colorings of $\mathbb{Z}_{p}$, for $p$ a large prime, always contain a rainbow $AP(4)$ if each of the color classes is of size of either ...

Question Is it true that every permutation of positive integers must contain monotone 4-term arithmetic progressions?...

A latin square is even if the product of the signs of all of the row and column permutations is 1 and is odd otherwise. Conjecture For every positive...

Question Is the set of prime numbers 2-accessible?...

Question Is the set of Fibonacci numbers 3-accessible?...

Conjecture An integer partition is wide if and only if it is Latin....

Given integers $k,n\ge2$, let $d(k,n)$ be the smallest integer $d\ge2$ such that the symmetric group $\frak S$ on the set of all words of length $n$ o...

Let $E$ be a non-empty finite set. Given a partition $\bf h$ of $E$, the stabilizer of $\bf h$, denoted $S(\bf h)$, is the group formed by all permuta...

Begin with the generating function for unrestricted partitions: (1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)... Now change some of the plus signs to ...

Conjecture Define a $2 \times n$ array of positive integers where the first row consists of some distinct positive integers arranged in increasing ord...

Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an $n \times n$ grid. The first player (if...

Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an $n \times n$ grid. The first player (if...

Conjecture Every surreal number has a unique sign expansion, i.e. function $f: o\rightarrow \{-, +\}$, where $o$ is some ordinal. This $o$ is the leng...

Let $S_n$ denote the set of all permutations of $[n]=\set{1,2,\ldots,n}$. Let $i(\pi)$ and $d(\pi)$ denote respectively the number of increasing and t...

Conjecture The largest measure of a Lebesgue measurable subset of the unit sphere of $\mathbb{R}^n$ containing no pair of orthogonal vectors is attain...

Question What is the saturation number of cycles of length $2\ell$ in the $d$-dimensional hypercube?...

Problem Determine the smallest percolating set for the $4$-neighbour bootstrap process in the hypercube....

Problem Bound the extremal number of $C_{10}$ in the hypercube....

Conjecture Does there exist a nontrivial perfect 2-error-correcting code over any finite alphabet, other than the ternary Golay code?...

A $(v, k, t)$ covering design, or covering, is a family of $k$-subsets, called blocks, chosen from a $v$-set, such that each $t$-subset is contained i...

Conjecture If ${\mathbb F}$ is a finite field with at least 4 elements and $A$ is an invertible $n \times n$ matrix with entries in ${\mathbb F}$, the...

Conjecture For every prime $p$, there is a constant $c(p)$ (possibly $c(p)=p$ ) so that the union (as multisets) of any $c(p)$ bases of the vector spa...

Conjecture If $A$ is an invertible $n \times n$ matrix, then there is an $n \times n$ submatrix $B$ of $[A A]$ so that $perm(B)$ is nonzero....

Conjecture If $B_1,B_2,\ldots B_p$ are invertible $n \times n$ matrices with entries in ${\mathbb Z}_p$ for a prime $p$, then there is a $n \times (p-...

Let $M$ be a matroid of rank $r$, and for $0 \le i \le r$ let $w_i$ be the number of closed sets of rank $i$. Conjecture $w_0,w_1,\ldots,w_r$ is unim...