Mathematics Problem Archive

Let $C$ be an analytic invariant curve of a rational function $f$, suppose $f:C\to C$ is not a homeomorphism, and assume $C$ contains a repelling fixe...

Let $D\subset\mathbb R^n$ have regular boundary for the Dirichlet problem. Prove or disprove that the following are equivalent: (PI) there is a nonzer...

A spherical quadrilateral is a disk with four marked boundary vertices, curvature-one metric, geodesic sides, and interior angles $\pi\alpha_j>0$. Cla...

For a local inverse germ $\phi_z$ of a meromorphic function $f$ at a noncritical value $w=f(z)$, Gross's theorem gives analytic continuation along alm...

Let $F$ be entire in two variables and let a holomorphic germ $\phi$ satisfy $F(z,\phi(z))=0$ near a nonsingular point. Must $\phi$ admit analytic con...

Let subharmonic functions $u_k$ on the unit square converge uniformly to $u(x,y)=x$. If $D_k=\{u_k<0\}$ and $D_k^*$ is the component containing $-1/2$...

Let $u$ be subharmonic on $1<|z|<2$, and let pairwise disjoint open sets $D_k$ be unions of components of $\{u<0\}$. Suppose that for every $r\in(1,2)...

Let positive masses $a_k$ be placed at a discrete set $x_k\in\mathbb R^n$ and suppose $\sum_k a_k/|x_k|^{n-1}<\infty$, so that $$F(x)=\sum_k\frac{a_k(...

Let $f$ be holomorphic in the unit disk with exactly one simple zero $z_0$ and two $1$-points $z_1,z_2$, counted with multiplicity. Determine the larg...

Let $f$ be holomorphic in the unit disk, with $f(0)=0$, $f'(0)\ne0$, no other zeros, and exactly two solutions $z_1,z_2$ of $f(z)=1$, counted with mul...

Solve the two-one-point extremal problem for real holomorphic $f$: determine the minimum of $\max(|z_1|,|z_2|)$ and maximum of $|f'(0)|$ when $f(0)=0$...

Let $f$ be real and holomorphic in the unit disk, with one simple zero at $0$ and two simple $1$-points at $\pm ia$. Determine the minimum possible va...

For $\phi(n)=\sup_{\deg p=n}\int_{|z|<1}|p'|/(1+|p|^2)\,dm$, let $\alpha=\limsup\log\phi(n)/\log n$. For a regular compact set $E$, define $\beta_E$ f...

Obtain better rigorous estimates for the Littlewood exponents $\alpha$, $\beta$, and for $\sup_c P_c$, where $P_c$ is the pressure for the hyperbolic ...

Are the iterates $p_c^n$ of hyperbolic quadratic polynomials extremal, or nearly extremal, for the Littlewood exponent $\alpha$ governing mean spheric...

For which parameters $c$ is the pressure $P_c$ of the hyperbolic quadratic polynomial $p_c(z)=z^2+c$, for the potential $|(p_c^n)'|^{-1}$, close to or...

For the rectangular lattice $\Lambda=\{an+ibm:n,m\in\mathbb Z\}$ with $a^2+b^2=1$ and $a\in(0,1)$, let $f_a:\mathbb D\to\mathbb C\setminus\Lambda$ be ...

Let $f=(f_0,\ldots,f_n)$ be a linearly nondegenerate holomorphic curve, let $T(r,f)$ have finite lower order $\lambda$, and let $N_1(r)$ be the averag...

Let $f$ be holomorphic in the unit disk with a simple zero at $0$, exactly two simple $1$-points at $a$ and $\overline a$, and no other zeros or $1$-p...

Let $f$ be holomorphic in the unit disk with a simple zero at $0$, exactly two simple $1$-points at $b$ and $-b$, and no other zeros or $1$-points. De...

Does every three-dimensional convex polytope have a non-self-intersecting edge unfolding? Does a minimum spanning tree of the dual edge graph, with a ...

Given $n$ points and $m$ hyperplanes in $\mathbb R^d$ whose incidence graph contains no $K_{s,t}$, determine the maximum number of incidences. Of spec...

For an $n$-point planar set, determine the maximum number of halving lines. More generally, determine the maximum number of $k$-sets, subsets obtained...

For $n$ pseudocircles in general position, determine the maximum number of tangent pairs and the maximum number of digon cells. Determine whether the ...

Determine the worst-case complexity of the medial surface and of an offset surface of an $n$-feature polyhedron, and of the Voronoi diagram of $n$ lin...

Determine the exact Erdős–Szekeres number $f(n)$, the least number of planar points in general position forcing a convex $n$-gon. Also determine sharp...

Determine the combinatorial complexity of the visibility complex of $n$ pairwise disjoint unit spheres in three-dimensional space....

Given $n$ planar points, find a subquadratic algorithm for the minimum-area triangle or prove a quadratic lower bound in a suitable computation model....

Given $n$ points in $\mathbb C^2$, determine in quadratic time whether three lie on a complex line, or prove a quadratic lower bound; the known algori...

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...

On a centrally symmetric convex body, must every pair of points at maximum intrinsic surface distance be antipodal? Resolve this even for rectangular ...

Determine the minimum number of $d$-simplices needed to triangulate the $d$-dimensional cube, and its asymptotic growth with $d$....

Does the hyperbolic plane admit a smooth isometric immersion into $\mathbb R^4$? More generally, determine the least Euclidean dimension for such an i...

Are the Hermite constants associated with densest lattice sphere packings always rational? Determine their arithmetic nature in dimensions where the e...

What is the largest possible ratio between the sum of edge lengths of a tetrahedron that can pass through or fit inside another tetrahedron and the su...

Does there exist a nondegenerate triangle whose side lengths, three medians, three altitudes, and area are all rational?...

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...

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...

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...