Mathematics Problem Archive

For finitely many positive charges $a_k$ at points $x_k\in\mathbb R^3$, is the critical set of $u(x)=\sum_{k=1}^n a_k/|x-x_k|$ always finite?...

Let $u_1,u_2,u_3$ be subharmonic in the plane with $u_j(0)=0$, and let $v_1\le v_2\le v_3$ be their pointwise increasing rearrangement. Put $I(r,v)=\i...

Let $f:\mathbb C\to\mathbb P^2$ be linearly nondegenerate and let $\delta(a,f)$ be the Nevanlinna deficiency of a point $a\in\mathbb P^2$. Prove that ...

Let $f:\mathbb C\to\mathbb P^n$ be holomorphic with spherical derivative $\|f'\|(z)=O(|z|^\sigma)$ for some $\sigma>-1$, and let $a_1,\ldots,a_q$ be h...

For $p\ge3$, let $V(D)$ consist of zero-free holomorphic vectors $(f_1,\ldots,f_p)$ on $D$ with $\sum f_j=0$, and use the source's definition of a $C$...

For real matrices $A\in\operatorname{Mat}_{n\times n}$, $B\in\operatorname{Mat}_{n\times p}$, and $C\in\operatorname{Mat}_{m\times n}$ with $n=mp$, de...

A facet of a $d$-polytope is degenerate if it has more than $d$ vertices. Determine the maximum number of degenerate facets of an $n$-vertex $d$-polyt...

Determine the maximum total number of faces of a $d$-dimensional convex polytope with $n$ vertices and $n$ facets. In dimension four, do such fat-latt...

For fixed $d>3$, determine whether every point of an $n$-point set in $\mathbb R^d$ is a convex-hull vertex faster than the best known near-$n^{2\lflo...

Given a sorted list of $n$ real numbers, find for every $1\le k\le n$ the shortest interval containing exactly $k$ entries. Find a subquadratic algori...

Given lines in the plane and two vertices $s,t$ of their arrangement, find a subquadratic algorithm for the shortest $s$-$t$ path along arrangement ed...

Can every convex polytope be triangulated so that every vertex degree, or every edge degree, is bounded by a constant or by a polylogarithmic function...

Determine the least number of colors needed to color the Euclidean plane so that points at unit distance receive different colors....

Given a finite planar point set and a prescribed rectangle, approximate efficiently the minimum number of congruent copies of the rectangle needed to ...

Do there exist seven planar points in general position—no three collinear and no four concyclic—such that every pairwise distance is an integer?...

Can the $3\times6$ rectangle with a $2\times2$ corner removed tile a rectangle using an odd number of congruent copies?...

Can sums of square roots of integers be compared in polynomial time on a Turing machine? Equivalently, obtain effective polynomial bit bounds for a no...

Find the largest function $t(n)$ such that every convex $n$-gon has a triangulation containing at least $t(n)$ distinct triangle areas; also determine...

What is the coarse Ricci curvature of discrete or continuous nilpotent groups? In particular, for the natural random walk generated by $a,b$ on the di...

Suppose the global infimum of coarse Ricci curvature is zero, while its infimum on every finite-radius ball about an origin is positive. Is there a sy...

Define curvature up to $\delta$ by $$T_1(m_x,m_y)\leq(1-\kappa(x,y))d(x,y)+\delta.$$ Which theorems for positive coarse Ricci curvature extend to this...

Define a scalar-curvature candidate by $S(x)=\int\kappa(x,y)\,dm_x(y)$, possibly with a distance-dependent weight. Does this quantity have useful geom...

Under the strengthened transport estimate $$T_1(m_x^{*t},m_{x'}^{*t'})\leq e^{-\kappa\min(t,t')}d(x,x')+C\frac{(\sqrt t-\sqrt{t'})^2}{2d(x,x')},$$ the...

For permutation groups with the transposition random walk, coarse Ricci curvature is positive but gives concentration of the wrong order. Can this dis...

Let $(X,\rho)$ be a finite metric space. Its fundamental polytope $R_{X,\rho}$ is the convex hull of the vectors $e_{x,y}=(\delta_x-\delta_y)/\rho(x,y...

Are there geodesic flows or Birkhoff billiards with fractal caustics? More specifically, for every $1\leq s<2$, is there a caustic of a convex billiar...

For a Hamiltonian system, call the good set the maximal invariant subset on which the invariant Liouville measure is almost periodic; call the bad set...

Are there quasiperiodic global minimals for metrics on the torus that are close to a flat three-dimensional torus? Here a geodesic is a global minimal...

Is there an analogue of Pesin theory for suitably defined smooth maps of the objects that arise naturally in algebraic dynamical systems—compact sets ...

Let $X=\{0,1\}^{\mathbb{Z}}$ and $Y=\{y\in\{0,1,2\}^{\mathbb{Z}}:y_i\neq y_{i+1}\}$. Both are mixing shifts of finite type with entropy $\log2$, but t...

Let $T:[0,1)\to[0,1)$ be the doubling map $x\mapsto2x\pmod1$, and let $\mu$ be an ergodic measure for $T$ with $0<h(\mu)<1$. Call $f:\mathbb{R}\to\mat...

Let $A=\{0,1,\ldots,d-1\}$ and let $\sigma$ be the shift on $A^{\mathbb{Z}}$. Define $(v_n(x))_i=\#\{0\leq j\leq n:x_j=i\}$ and $(w_n(x))_i=\#\{0\leq ...

With $\mathcal{F}^+$ and $\mathcal{F}^-$ defined from the forward and backward symbol-count tail fields on the full shift $A^{\mathbb{Z}}$, if $\mathc...

Let $(M,T)$ be a smooth map on a manifold with a good symbolic cover: a mixing shift of finite type factors onto $(M,T)$ and is injective on a set of ...

Characterize mixing shifts of finite type up to topological orbit equivalence....

Let $S$ and $T$ be subshifts. If $S$ is a mixing shift of finite type and $T$ is topologically orbit equivalent to $S$, must $T$ also be a mixing shif...

Consider billiard tables formed by two concentric semicircles joined by two line segments. Consider trajectories with a fixed circle, concentric with ...

If a nonnegative integer matrix $A$ has a unique, simple, nonzero eigenvalue $n$, prove that $A$ is strong shift equivalent over $\mathbb Z_+$ to the ...

Classify shifts of finite type up to topological conjugacy; in particular, give a decision procedure determining whether two nonnegative integer matri...

Given a mixing shift of finite type $S_A$, determine the range of the dimension representation $\operatorname{Aut}(S_A)\to\operatorname{Aut}(G_A)$....

Let $A,B$ be irreducible integer matrices of the same spectral radius. Suppose $\operatorname{tr}(A^n)>0$ implies $\operatorname{tr}(B^n)>0$ for every...

Prove that two mixing Markov shifts admit a magic-word isomorphism exactly when they have the same beta function, the same ratio group $\Delta$, and t...

Characterize the functions that occur as stochastic zeta functions of mixing Markov shifts....

Suppose a $\mathbb Z^2$ action $\alpha$ has $\alpha^{\boldsymbol n}$ an SFT for some $\boldsymbol n$. Can $\alpha$ have infinitely many expansive comp...

If $S$ is an expansive automorphism of an irreducible shift of finite type, must $S$ itself be a shift of finite type?...

If $S$ and $T$ are mixing shifts of finite type, prove that $S^i$ and $T^j$ can commute for all sufficiently large integers $i,j$....

Determine all shift-invariant Borel probability measures and all subsystems of the symbolic system $X$ constructed in Section 14 of the source from th...

For $d>1$, must every $\mathbb Z^d$ sofic shift be a factor of a $\mathbb Z^d$ shift of finite type having the same entropy?...

For $d>1$, if a continuous factor map sends a $\mathbb Z^d$ SFT $X$ onto a sofic shift $Y$, must $X$ contain a sofic subshift $W$ with $h(W)=h(Y)$ and...

Characterize the stable limit sets of one-dimensional cellular automata....