Mathematics Problem Archive

Determine the optimal complexity of computing the volume of the union of axis-aligned boxes in fixed dimension at least three. In particular, is there...

Given a planar point set $P$, sample uniformly from the simple polygons with vertex set $P$ in polynomial time, or determine the complexity of countin...

Develop exact polynomial-time algorithms for the constructive forms of Aleksandrov's, Cauchy's, Minkowski's, Steinitz's, and Koebe's polytope-realizat...

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...

For every positive integer $k$, let $R_k$ be a $1/k$ by $1/(k+1)$ rectangle. Can the entire collection $(R_k)_{k\ge1}$ be packed without overlap into ...

For Ollivier's coarse Ricci curvature, smooth uniformly strictly log-concave measures on $\mathbb{R}^N$ have positive curvature. What can be said for ...

The space $\mathbb{R}^N$ equipped with an $L^p$ norm has zero coarse Ricci curvature. Does this observation yield useful results for Finsler manifolds...

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...

For a continuous-time Markov semigroup $(m_x^t)$ define $$\kappa(x,y)=\liminf_{t\to0^+}\frac1t\frac{d(x,y)-T_1(m_x^t,m_y^t)}{d(x,y)}.$$ Under a natura...

Positive coarse Ricci curvature gives a spectral-gap bound for reversible random walks and on finite spaces. What spectral-radius, operator-norm, or P...

For the $\varepsilon$-step random walk on an $N$-dimensional Riemannian manifold, the coarse-curvature argument gives the lower spectral-gap estimate ...

Can estimates based on a uniform lower bound for coarse Ricci curvature be extended to spaces where curvature has only a controlled number of negative...

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...

Can the bounded-local-variance hypothesis used for concentration under positive coarse Ricci curvature be relaxed while retaining estimates governed b...

Can concentration consequences of positive coarse Ricci curvature be formulated as transportation or other functional inequalities? In a coarse settin...

What is the relationship, if any, between Ollivier coarse Ricci curvature and the Sturm–Lott–Villani displacement-convexity notion, including its $CD(...

Is there an analogue, for positive coarse Ricci curvature, of the Bishop–Gromov theorem or the isoperimetric form of the Gromov–Lévy theorem? Identify...

Does positive coarse Ricci curvature imply a useful exponential entropy-decay statement analogous to that obtained from logarithmic Sobolev inequaliti...

Let the metric of a Markov space evolve by $$\frac{d}{dt}d(x,y)=-\kappa(x,y)d(x,y),$$ where $\kappa$ is computed from the current metric, with either ...

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...

Replace the $T_1$ distance in the coarse-Ricci definition by $L^\infty$ transport, requiring a coupling that moves every point by at most $d(x,y)$. Do...

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...

Do spaces with positive sectional curvature in the sense of Alexandrov have positive coarse Ricci curvature for a natural choice of Markov kernels? Ca...

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...

What is the measure of the set of initial conditions of the Newtonian $N$-body problem that lead to global solutions? The complementary set of singula...

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...

Does a free gas coupled to a convex rigid container by conservation of momentum converge weakly to equilibrium, with the container—which moves only by...

Does there exist a Hamiltonian system with a smooth invariant torus on which the induced dynamics is mixing? No such mixing can occur on a two-dimensi...

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...

For $\mathbb{Z}^d$-actions by automorphisms of a connected group, mixing actions are mixing of all orders. Can this result be proved using simpler ide...

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 ...

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 positive square matrices $A,B$ are shift equivalent over $\mathbb Q_+$, prove that they are strong shift equivalent over $\mathbb Q_+$....

Let $S\subset\mathbb R$ be a unital subring. Prove that a tuple $\Lambda=(\lambda_1,\ldots,\lambda_k)$ is the nonzero spectrum of a primitive matrix o...

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

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

Prove that every expansive automorphism of a one-sided full shift is topologically conjugate to a two-sided full shift....

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 each $1\le r\le\infty$, characterize the entropy structures that can occur for $C^r$ diffeomorphisms of compact Riemannian manifolds....

Prove that the jointly periodic points of every surjective one-dimensional cellular automaton are dense....

Prove that for some $N>1$ there is a surjective one-dimensional cellular automaton $f$ with $\nu(f,S_N)<N$....

Consider the planar cubic rigid systems $$\begin{cases}\dot x=-y+x(a+bx+cy+dx^2+exy),\\ \dot y=x+y(a+bx+cy+dx^2+exy).\end{cases}$$ Is $2$ the maximum ...