Mathematics Problem Archive

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 $C$...

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...

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...

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 ...

Does the three-dimensional cube admit a triangulation into tetrahedra all of whose dihedral angles are acute?...

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...

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...

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...

Is there a near-linear-time algorithm to construct the straight skeleton of a simple polygon? Determine the optimal complexity, including for polygons...

Given motorcycles moving simultaneously along fixed rays and crashing upon reaching another track, determine the motorcycle graph in near-linear time....

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 ...

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 ...

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...

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

Given a polygonal room with perfectly reflecting sides and a point light source, characterize when every point of the room is illuminated. In particul...

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

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?...

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...

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 ...

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...

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...