Mathematics Problem Archive

Does every infinite Cayley graph, or every infinite vertex-transitive graph, have a spanning tree without leaves?...

In every infinite Cayley graph, does the density of dead ends tend to zero?...

Are factors of i.i.d. on the $3$-regular tree closed in the weak topology? In particular, is the weak limit of majority functions on $n$-balls a facto...

Does every infinite Cayley graph $G$ admit a $G$-invariant proper coloring with $\chi(G)$ colors?...

Let $X$ be the space of $k$-regular Cayley graphs with the local-convergence topology and let $T\subset X$ be the closed subset of transient graphs. I...

For every $k>1$, does there exist $C(k)<1$ such that the return probability of every transient $k$-regular Cayley graph is at most $C(k)$?...

For a nonamenable group $\Gamma$, does the Bernoulli shift $\{0,1\}^{\Gamma}$ factor onto $\{0,1,2\}^{\Gamma}$?...

Can every $d$-regular graphing without multiple edges be properly edge-colored by $d+1$ colors?...

Let $(G_n)$ and $(H_n)$ converge to the same graph limit, and let $(T_n)$ be a convergent sequence with each $T_n$ a spanning tree of $G_n$. Do there ...

Let $G$ be a bounded-degree expander, or strongly ergodic, graphing that can be properly colored by $C$ colors with arbitrarily small error. Can it be...

Let $G$ be a bounded-degree expander, or strongly ergodic, graphing that weakly contains a finite graph $H$. Does $G$ factor onto $H$?...

Does every higher-rank semisimple real lattice have rank gradient zero?...

If $A$ and $B$ are countably infinite groups, does $A\times B$ have fixed price $1$?...

Does every countable group have fixed price; that is, do all its free probability-measure-preserving actions have the same cost?...

Does every residually finite property-(T) group have rank gradient zero?...

Let $\Gamma$ act on $X$ by probability-measure-preserving maps and let $H\leq\Gamma$ have finite index. Is $\operatorname{cost}(H,X)-1=(\operatorname{...

For a free action of $\Gamma$ on $X$, is $\operatorname{cost}(\Gamma,X)=\operatorname{cost}(\Gamma,X\times X)$ for the diagonal action on $X\times X$?...

Let an amenable group $\Gamma$ act ergodically and essentially faithfully on $X$. Is the groupoid cost of the action $1$? If $\Gamma$ is finitely pres...

Given a graphing of an equivalence relation, is there a subgraphing whose cost is arbitrarily close to the cost of the relation?...

Can a nonabelian free group $F$ have a nontrivial pseudocharacter invariant under $\operatorname{Aut}(F)$?...

Are two independent random subsets of $\mathbb{Z}$ almost surely quasi-isometric as metric spaces? What is the answer for other Cayley graphs, such as...

Let $P$ be a finite $p$-group of order $n$ and let $w$ be any word. Is the probability that $w$ is satisfied in $P$ at least $1/n$?...

Let $M$ be the set of measurable real functions, and define $H_f(x,y)=(x,y+f(x))$ and $V_f(x,y)=(x+f(y),y)$. What group is generated by $\{H_f:f\in M\...

Determine the asymptotic length of the shortest nontrivial word that is a law in every group of order $2^n$....

Given a triple of positive integers $(k,d,n)$, calculate $\operatorname{rk}_k^\circ(kd,n).$...

The $k$-rank of a general form of degree $kd$ in $n$ variables is given by $$ \operatorname{rk}_k^\circ(kd,n)=\begin{cases} \min \left\{s\ge 1 | s\bin...

Given a triple of positive integers $(k,d,n)$, calculate $\operatorname{rk}_k^{\max}(kd,n).$...

For any positive integers $k,d$, the maximal $k$-rank $\operatorname{rk}^{\max}_k(kd,2)$ of binary forms equals $k$. Additionally, in the above notati...

Given $k \geq 3$ and a monomial $m$ of degree $kd$, determine the monomial $k$-rank $\operatorname{rk}_k(m)$....

Given $k \geq 3$ and a monomial $x^a y^b$ of degree $a+b=kd$, it is known that $\operatorname{rk}_k(x^{a}y^{b}) \leq \max(s,t)+1$, where $s$ and $t$ a...

Calculate the degree of $\widetilde{W}_{k,d}$ for perfect pairs $(k,d)$....

[Fr\"oberg's Conjecture, 1985] Let $f_1, \dots, f_r$ be generic forms of degrees $d_1, \dots, d_r$, respectively. Then the Hilbert series of the quoti...

[Fr\"oberg-Iarrobino Conjecture] Given generic linear forms $\ell_1, \ldots, \ell_r$ and a positive integer $d$, let $I$ be the power ideal generated ...

For $\mu \neq (d)$, does a generic $\mu$-power ideal have the same Hilbert function as in [source label: eq:RALF]?...

[] For generic forms $g_1,\ldots,g_r$ of degree $d>1$, the ideal $(g_1^k,\ldots,g_r^k)$ has the same Hilbert series as the one generated by $r$ generi...

It has been conjectured that each complete intersection $R=S/(f_1,\ldots,f_n)$ satisfies the WLP and also the SLP, see . Does the same hold for $R=S/(...

When are the WLP and the SLP true for $T_{n,d,k}$?...

For $R= S/(f_1,\ldots,f_r)$, where $f_1,\ldots,f_r$ are generic forms, does $R$ satisfy the $\mu$-Lefschetz property for all partitions $\mu$?...

Given a scheme of generic fat points $X \subset \mathbb{P}^{n_1-1}\times\ldots\times \mathbb{P}^{n_t-1}$, what is the multi-graded Hilbert function $\...

For the ideal $I$ of $s$ general points in $\mathbb P^{n-1}$, what is the difference between the Hilbert series of the $m$-th symbolic power and the $...

$\mathcal{S}$ is cyclotomic if and only if $k[\mathcal{S}]$ is a complete intersection....

Are $B_{n,m}$ and $B'_{n,m}$ finite for any pair $(n,m)$ with even $n$?...

For any given pair $(n,m)$ with even $n$, $B'_{n,m}=\left(\frac{n}{2}\right)^{m-1}$....

Determine $\lim_{n\to\infty}\frac{B_{n,3}}{n^2}$....

For $n=1$ and given $p$, what are the (lengths of the) possible periods of $\phi$?...

For $n = 1$ and given $p$, find the minimal positive integer $i$ such that $\psi^i$ is the identity map on the space of polynomials of degree at most ...

Let $f$ be a form of odd degree $d$ in $E$. Is it true that $({\rm Ann}(f))_i = (f)_i$, for $i < (n-d)/2$?...

Let $f$ and $g$ be generic quadratic forms in $E$ and let $\ell_1$ and $\ell_2$ be two generic linear forms in $S$. Then the Hilbert series of $E/(f,g...

For a non-hyperelliptic algebraic curve, is its Clifford index determined by the extent to which its canonical embedding has linear syzygies?...

Prove the conjectured local-to-global principle for linear ordinary differential equations known as the Grothendieck–Katz $p$-curvature conjecture....