Mathematics Problem Archive

(S. Ivanov) Is there a f.p. slender group which is not polycyclic-by-finite?...

Let $L$ be a flag triangulation of $S^{2k-1}$, let $f_i$ be the number of $i$-simplices of $L$, and define $$\chi=1-\sum_{i=0}^{2k-1}(-1)^i\frac{f_i}{...

Do there exist groups G with balanced presentation (same number of generators and relations), with H1(G) = 0 and with unsolvable word problem?...

Is there a sequence of (perfect, of course) groups with balanced presentations among which one cannot recognize trivial groups?...

Let $G$ be a one-relator group whose relator $W$ is a cyclically reduced word in the generators, and let $P$ be the submonoid of $G$ generated by all ...

Let a finitely presented pro-$p$ group have $r+2$ generators and $r$ relators. Must it have an open subgroup that maps continuously onto a nonabelian ...

Let $G$ be a closed transitive subgroup of the automorphism group of a rooted tree. Must every level stabilizer of $G$ contain an element acting witho...

Can the sequence of metric balls in an infinite Cayley graph form a family of expanders?...

Let $\Gamma$ be finitely generated and let $\{H_n:n\geq1\}$ be a property-$\tau$ family of finite-index normal subgroups. Does the chain $\Gamma_n=\bi...

Let $\Gamma$ be finitely presented with a chain of finite-index normal subgroups having trivial intersection and property $\tau$. Must $\Gamma$ contai...

For a graph sequence $(G_n)$ define $e((G_n))=\liminf |E(G_n)|/|V(G_n)|$, and define its combinatorial cost as the infimum of $e((H_n))$ over graph se...

Compactness implies that for every $\varepsilon>0$ there is $K>0$ such that every finite graph can be approximated within error $\varepsilon$ by a fin...

Which probability measures can occur as eigenvalue distributions of finite $d$-regular graphs? Find natural restrictions....

Do uniformly random $d$-regular graphs converge, in local-global convergence, to the weak closure of independent identically distributed processes?...

Is the i.i.d. action of the free group $F_2$ a local-global limit of finite actions of $F_2$?...

Can every ergodic unimodular random network that is almost surely an infinite tree be obtained as the limit of an expander family?...

Let $G$ be an infinite vertex-transitive graph, let $A$ be a finite vertex set, let $b$ be a vertex, and let $\partial A$ be the set of vertices at di...

Define the first $L^2$ Betti number of a vertex-transitive graph $G$ from the expected degree of a free spanning forest. Do $G$ and its square $G^2$ h...

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...

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 $C(k)$?...

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...

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

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 $\operatorname{Aut}(F)$?...

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\...

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

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/(...

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 $...

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

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

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

Let $V$ be a complex algebraic variety with coordinate ring $R$. If the module of derivations of $R$ is free over $R$, must $V$ be smooth?...

Find an integer cosine rule for integer triangles in integer trigonometry....

{\bf(IKEA problem.)} Classify all $n$-tuples of LLS-sequences for the angles that form integer $n$-gons....

Classify all combinatorial possible types of faces....

Which $n$-gons are realizable as faces of an $m$-dimensional continued fraction? Here are two essentially geometrically different subcases: ; {\bf Fac...

Describe all finite two-dimensional sails (and the corresponding continued fractions)....

{\bf (Multidimensional IKEA problem.)} Describe the collections of the sails of the cones for all polytopes of a given combinatorial type....

{\bf (V. Arnold.)} Does there exist an algorithm to decide whether a given type of fundamental domain is realizable by a periodic continued fraction?...

{\bf (V. Arnold.)} Torus decompositions of integer noncongruent Klein sails are distinct....

{\bf (V. Arnold.)} Describe all torus decompositions that are realized by periodic two-dimensional continued fractions....

{\bf (V. Arnold.)} Classify continued fractions that correspond to the same cubic extension of the field of rational numbers....

Prove the existence of a cone for a single non-periodic combinatorial structure ($n\ge 3$)....

Find frequencies on $n$-dimensional continued fractions with the highest relative frequencies....