Mathematics Problem Archive

Which theorems about boundary behavior of Riemann maps have analogues for geometric coding trees?...

Find a collection of representative examples of transcendental entire maps whose dynamics may serve as models for general phenomena....

Describe the dynamics of Newton's method when applied to broad natural classes of transcendental entire functions....

Under the hypotheses of Eremenko–Lyubich Question 2, can even a single orbit converge to $z_0$?...

Call a polynomial bad if its Newton map has an attracting cycle that is not a root. Prove that every bad degree-$d$ polynomial $f_1$ belongs to a one-...

For $P_c(z)=z^2+c$ with connected Julia set, characterize the parameters $c$ for which $J(P_c)$ is locally connected....

Find a general property of a lifting family that guarantees convergence of the real Thurston algorithm for every periodic or preperiodic kneading sequ...

Find a general property of a lifting family that guarantees convergence of the real Thurston algorithm for arbitrary kneading sequences....

In the rank-zero case, if the limiting map on an invariant stable component is constant with value $x_0\in\partial U$, prove that one eigenvalue at $x...

Can the subsequential limit map on an invariant stable component of a polynomial diffeomorphism of $\mathbb C^2$ be a retraction onto a Herman ring or...

In the rank-two case for a polynomial diffeomorphism of $\mathbb C^2$, can an invariant stable component be a product of two Herman rings, or a produc...

For every degree $d$, prove that expanding (hyperbolic, Axiom A) maps are dense in the spaces $\operatorname{Rat}_d$ of rational maps and $\operatorna...

Develop an analogue of the Haken decomposition for geometrically finite rational maps. In particular, if the Julia set is disconnected, can the map be...

What is the combinatorial complexity of the Voronoi diagram of a set of lines (or line segments) in three dimensions?...

What is the complexity of the union of ``fat'' objects in $\mathbb{R}^3$?...

What is the maximum number of $k$-sets? (Equivalently, what is the maximum complexity of a $k$-level in an arrangement of hyperplanes?)...

Is there a deterministic, linear-time polygon triangulation algorithm significantly simpler than that of Chazelle?...

Is there an $O(n)$-space data structure that supports $O(\log n)$-time point-location queries in a three-dimensional subdivision of $n$ faces?...

Can the number of simple polygonalizations of a set of $n$ points in the plane be computed in polynomial time? A simple polygonalization is a simple p...

Given a visibility graph $G$ and a Hamiltonian circuit $C$, determine in polynomial time whether there is a simple polygon whose vertex visibility gra...

What is the complexity of the vertical decomposition of $n$ surfaces in $\mathbb{R}^d$, $d \ge 5$?...

Can a minimum-link path among polygonal obstacles be found in subquadratic time?...

Can an $n$-vertex polygonal curve be simplified in time nearly linear in $n$?...

How efficiently can one compute a polyhedral surface that is an $\epsilon$-approximation of a given triangulated surface in $\mathbb{R}^3$?...

Is the flip graph connected for general-position points in $\mathbb{R}^3$? Given a set of $n$ points in $\mathbb{R}^3$, the flip graph has a node for ...

Can every convex polytope in $\mathbb{R}^3$ be partitioned into tetrahedra such that the dual graph has a Hamiltonian path?...

Is it possible to trap all the light from one point source by a finite collection of two-sided disjoint segment mirrors? A light ray is trapped if it ...

How many intersections among an arrangement of pseudosegments in the plane must be added as vertices to allow the pseudosegment arrangment to be exten...

How many polyominoes on $n$ squares are there? A polyomino is a connected interior-disjoint union of axis-aligned unit squares joined edge-to-edge, in...

Is it true that every two sets of $n$ planar points in general position with the same number points on their convex hulls have compatible triangulatio...

For a planar point set $S$, is the number of pointed pseudotriangulations always at least the number of triangulations? A pseudotriangle is a planar p...

Given two sets of numbers, each of size $n$, how quickly can the set of all pairwise sums be sorted? In symbols, given two sets $X$ and $Y$, our goal ...

Consider a polyhedron with simply connected facets (no holes on a facet) and without boundary (every edge is incident to exactly two facets). Can the ...

Can every closed polyhedron be cut along its surface and unfolded into one piece in the plane without overlap? Such an unfolding is called a general u...

Does every simple graph with maximum vertex degree $\Delta \leq 6$ have a 3D orthogonal point-drawing with no more than two bends per edge? A 3D ortho...

What is the complexity of finding a tour of maximum Euclidean length for a planar point set?...

What is the complexity of finding a shortest tour in a solid planar grid graph? A planar grid graph is a graph whose vertices are any set of points on...

What is the complexity of the pallet loading problem? Given two pairs of numbers, $(A,B)$ and $(a,b)$, and a number $n$, decide whether $n$ small rect...

What is the optimal partition of a square into convex pieces such that the circularity of the pieces is optimized? The circularity of a polygon is the...

Given an arbitrary polygon, transform it by a finite sequence of ``vertex-centroid'' moves to a regular polygon. A vertex-centroid move is a translati...

Given a set of $n$ unit-radius balls in $\mathbb{R}^3$, what is the number of lines that are tangent to four of the balls in the set, and miss all the...

Let $C$ be a convex piece of paper; its boundary may be a smooth curve, or a polygon. A perimeter halving folding is a folding of $C$ obtained by iden...

Is there any genus-zero orthogonal polyhedron $P$ built by gluing together cubes face-to-face that cannot be edge-unfolded, where all cube edges on th...

Label the faces of a unit cube with numbers $1$--$6$ as in a die. Place the cube to sit on an integer lattice grid, with one corner at the origin and ...

Let $M$ be a closed polyhedral surface homeomorphic to $S^2$ which is entirely composed of equal regular pentagons. If $M$ is immersed in 3-space, is ...

Partition a given polygon $P$ into $n$ mutually congruent pieces so that the area of $P$ not covered by the union of the pieces is as small as possibl...

Let us say that two rectangles in the place are independent if both their $x$- and $y$-axis projections are disjoint. A set of rectangles is then inde...

Does every convex polyhedron $P$ have a zipper unfolding? A zipper unfolding cuts open $P$ via a single path, necessarily a Hamiltonian path (to span ...

Let $M$ be a compact surface, $I=(-1,1)$, and $f:M\times I\to\mathbb{R}^3$ a differentiable map such that each $f_t$ is an immersion and its induced m...

Is every bubble cluster in $\mathbb{R}^3$ made of spherical pieces meeting according to Plateau's rules necessarily stable, in the sense of having non...