Mathematics Problem Archive

Are there smooth convex curves, other than ellipses, simultaneously admitting one-parameter families of $p$- and $q$-periodic billiard trajectories fo...

For billiards inside an oval in the Lorentz plane with metric $ds^2=dx^2-dy^2$, is there an analogue of Birkhoff's theorem, with separate existence st...

For a smooth strictly convex hypersurface $M\subset\mathbb{R}^{2n}$, find lower bounds for the number of $p$-periodic outer-billiard orbits for values...

For the ellipsoid $x^2/a+y^2/b+z^2/c=1$ in Minkowski space with metric $dx^2+dy^2-dz^2$, find conditions on $a,b,c>0$ ensuring the existence of an $(n...

Let $\gamma$ be a smooth arc agreeing to all orders with the $x$-axis at endpoints $(0,0)$ and $(1,0)$, and iterate $T(\gamma)=\gamma+\gamma'$. Unless...

Given an oriented oval $\gamma$, let $\gamma_1$ be the locus of endpoints of its oriented unit tangent segments, and iterate this construction. If eve...

Extend the known results on projectively self-dual polygons and curves to projectively self-dual polyhedra and to projectively self-dual polygons in m...

(Mitra) Let XG be a finite 2-complex with fundamental group G. Let XH be a cover corresponding to the f.p. subgroup H. Let I(x) denote the injectivity...

(Epstein) Let $G$ be a word-hyperbolic group and $\partial G$ its boundary. Is there an algorithm to compute $\check H^i(\partial G)\cong H^{i+1}(G,\m...

(Kim Ruane) Let $G$ be a Coxeter group, for example a right-angled Coxeter group, acting properly discontinuously by isometries on a CAT(0) space $X$....

(Swarup) If G is a Fuchsian group, define area(G) to be the area of the convex core of H2/A. Since area(A) = 2π(rk(A)−1) for Fuchsian groups which are...

Characterize groups of the form Fm ∗Z Fn which are limit groups....

For $x\in[F_n,F_n]$, define its genus as the least $g$ such that $x$ is a product of $g$ commutators. Let $f(g)$ be a uniform bound, independent of $x...

Is there a proper closed subgroup of Homeo+(S1) that acts transitively on (unordered) 4-tuples? Or k-tuples (k ≠ 3)? Relation to earthquakes?...

(Misha Kapovich) Suppose $G$ is a finitely generated Kleinian group in $\operatorname{Isom}(\mathbb{H}^n)$. Is $$\delta(G)\leq\operatorname{vcd}(G),$$...

(Swarup) Starting with a finitely presented group $G$, take maximal graph-of-groups decompositions alternately over finite and over two-ended edge gro...

(Shalen) If a finitely presented group G acts nontrivially (i.e. without global fixed points) on an R-tree, does it act nontrivially on a simplicial t...

(Mohan Ramachandran) For a finitely presented group $G$, consider: (A) some finite-index subgroup of $G$ admits a nontrivial action on a simplicial tr...

Suppose $H=F/N$ is finitely presented and contains $C^n$ for every $n$. Write $G_n=H*_{C^n}H=F*F/N_n$. If $C$ is nontrivial and finite, is $$\lim_{n\t...

(Kapovich) (a) For closed orientable surfaces $\Sigma_g$ and $\Sigma_h$, is there a faithful representation $\pi_1(\Sigma_g)\to\operatorname{MCG}(\Sig...

Assuming that the fixed subgroup Fix(α) is cyclic, find a bound on the length of a generator of Fix(α) in terms of the complexity of α....

Which α preserve an order (invariant under right translations) on Fn? If α has periodic elements it cannot preserve an order. Are there other obstruct...

Can the mapping class group (or a finite index subgroup of it) of a closed surface be embedded into Out(Fn)? Into the mapping class group of a punctur...

(Grigorchuk) Aut(Fn) acts on the space ∂3Fn of triples of distinct ends of Fn. Denote by Yn the compact space (Cantor set) the quotient space of ∂3Fn ...

(Seymour Bachmut) Is SL2(K) finitely generated for K = Z[X, X−1] or K = F[X, X−1, Y, Y −1] for a field F?...

Can the odometer acting on the rooted binary tree be embedded in a nonabelian free pro-$2$ group?...

Determine the asymptotic length of a shortest nontrivial group law for the $n$-fold iterated wreath product of $C_2$....

Suppose $G$ and $H$ are Cayley, or vertex-transitive, expanders on the same number of vertices and can be matched with asymptotically vanishing edge d...

If $G$ is a finite vertex-transitive expander, must every almost automorphism of $G$ be close to an automorphism?...

For a locally convergent sequence $(G_n)$ of bounded-degree integer-labeled graphs, does the normalized rank modulo $p$ of the adjacency matrix conver...

Let $(G_n)$ be a locally convergent graph sequence and let $\mu_n$ be the probability distribution of the roots of the chromatic polynomial of $G_n$. ...

Let $G$ be a $d$-regular Cayley graph with spectral measure $\mu$. Is $\mu$ a weak limit of spectral measures of finite $d$-regular graphs?...

Let $\Gamma$ have property (T), and let $(G_n)$ be a sofic approximation of a Cayley graph of $\Gamma$. Can $(G_n)$ be changed by an asymptotically va...

Can free spanning forests be used to prove basic properties of the first $L^2$ Betti number, such as multiplicativity upon passage to a finite-index s...

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...

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?...

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{...

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 $1/n$?...

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

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 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...