Mathematics Problem Archive

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

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....

For every positive integer constant $C$ there exist only finitely many pairwise integer non-congruent faces with frequencies exceeding $C$....

Is that true that sum of all relative frequencies for all possible faces is finite for higher dimensions $(n\ge 3)$?...

In case of positive answer to the above question find the generalization of the Gauss map and compare the corresponding frequencies of faces with the ...

Find useful ways of verifying polyconvexity and quasiconvexity for stored-energy functions arising in anisotropic nonlinear elasticity....

Prove or disprove that, under reasonable growth conditions on $W$, an energy-minimizing deformation satisfies $\det Dy^*(x)\ge\varepsilon>0$....

Devise general methods for proving the existence of local but nonglobal minimizers and other weak equilibria in nonlinear elastostatics....

Develop a qualitative dynamics for dynamic theories of elasticity....

Establish the status of elasticity theory with respect to atomistic models....

For free-energy functions $\psi(A,\theta)$ of elastic crystals, determine boundary conditions under which the minimum is attained and conditions under...

Under which additional assumptions does this principle become a rigorous theorem?...

Describe the symplectic invariants of stable rank-one singularities described by V. V. Kalashnikov. For such singularities, one of the action variable...

[N. T. Zung] Study the topology and geometry of these singular fibers and their small neighbourhoods....

[N. T. Zung] Give a clear description of these special singular fibers....

Complete the above table: {\rm (1)} construct new examples of natural Hamiltonian systems on closed two-dimensional surfaces admitting polynomial inte...

Construct stationary axially symmetric 4-dimensional Einstein metrics admitting Killing tensors of higher order....

[Gilkey ] In the Riemannian case, is every $1$-homogeneous manifold locally homogeneous?...

{\it Construct a natural Hamiltonian system on the 2-sphere with a nonconstant potential which is superintegrable by integrals of degree $\ge 3$ and a...

Prove the algebraic version of Birkhoff conjecture for outer billiards in a non-Euclidean surface of constant curvature....

Prove Conjecture [source label: Descon] it the case that the foliation admits (i) a rational, (ii) an algebraic first integral....

Is it possible to choose the domain so that the dynamics of the corresponding billiard map are locally (near the 2-periodic orbit) conjugated to the d...

Are there plane billiards, other than ellipses, that possess rational caustics with two different values of the rotation numbers? Same question for ou...