Mathematics Problem Archive

Determine the measure and Hausdorff dimension of the parameter hairs in the exponential, sine, and cosine families....

Let $\varphi$ be a holomorphic germ fixing $z_0$ with multiplier $e^{2\pi i\alpha}$ for irrational $\alpha$. Can $\varphi^n(z)\to z_0$ uniformly on so...

Let all roots of a degree-$d$ polynomial $f$ lie in the unit disk, let $\alpha$ be a root of multiplicity $m$, and let $A^*_{h}(\alpha)$ be its immedi...

If the quadratic polynomial $P_c(z)=z^2+c$ is nonrecurrent, does there exist a conformal metric $\rho(z)|dz|$ with integrable singularities in which $...

For monic polynomials $z^n+a_{n-1}z^{n-1}+\cdots+a_1z$ with $|a_1|\ge1$, external rays landing at the fixed point $0$ have a rotation number. Does thi...

For a piecewise monotone interval map, iteratively replace its critical values by those of a polynomial with the same ordered critical data and conjug...

For the lift family $x\mapsto k-k|2x-1|^\alpha$, $\alpha>1$, does the real Thurston algorithm converge whenever the initial interval map has a periodi...

Let $f$ be a polynomial diffeomorphism of $\mathbb C^2$ with Jacobian determinant $\delta$, let $U$ be a component of the interior of the bounded-forw...

For a geometrically finite rational map $f$, prove that either its Julia set is the whole sphere and $f$ is ergodic there, or its Julia set has Hausdo...

Prove that every connected component of the Julia set of a geometrically finite rational map is locally connected....

Extend Thurston's finite combinatorial classification from critically finite rational maps to all geometrically finite rational maps: give finite topo...

For $n>1$, do there exist nontrivial critically finite rational maps $f:\mathbb P^n\to\mathbb P^n$ whose postcritical hypersurface $V$ has Kobayashi-h...

Can the Euclidean minimum spanning tree (MST) of $n$ points in $\mathbb{R}^d$ be computed in time close to the lower bound of $\Omega(n \log n)$?...

What is the complexity of computing a minimum-cost Euclidean matching for $2n$ points in the plane? The cost of a matching is the total length of the ...

Can the class of 3SUM hard problems be solved in subquadratic time? These problems can be reduced from the problem of determining whether, given three...

What is the best output-sensitive convex hull algorithm for $n$ points in $\mathbb{R}^d$?...

How many $\pi$-floodlights are always sufficient to illuminate any polygon of $n$ vertices, with at most one floodlight placed at each vertex? An $\al...

Can the interior of every simply connected polyhedron whose surface is meshed by an even number of quadrilaterals be partitioned into a hexahedral mes...

For a point set $P$ in $\mathbb{R}^d$, let $f_d(P)$ be the number of unit-distance point pairs: $$f_d(P) = \left| \{ (u,v) \mid u, v \in P, \, \|u-v\|...

For any (planar) triangle $T$, is there is a $3$-coloring of the (infinite) plane with no monochromatic copy of $T$? We imagine congruent copies of $T...

Is there a data structure maintaining a set of $n$ points in the plane subject to insertions, deletions, and nearest-neighbor queries in $O(\log n)$ t...

Let $\rho(S)$ be the fewest number of reflex vertices in a polygonization of a 2D point set $S$, i.e., the fewest reflexivities of any simple polygon ...

Is the Yao-Yao Graph a $t$-spanner for constant $t$? A geometric graph is a $t$-spanner (or just a spanner) if, for every pair of nodes, the shortest ...

Identify or construct all $k$-equiprojective polyhedra. A polyhedron $P$ is $k$-equiprojective if its orthogonal projection to a plane is a $k$-gon in...

Do there exist rectangles that may be partitioned into a finite number $n$ of rectangular pieces of equal area but with all perimeters different?...

Is there a stable cluster of bubbles in $\mathbb{R}^3$ in which some bubble is topologically a torus?...

For $k\leq n+1$ and prescribed volumes in $\mathbb{R}^n$, prove that the standard $k$-bubble is uniquely area-minimizing. For $n=3$, are standard clus...

Let $A_X(v_1,\ldots,v_k)$ be the area of a minimizing $k$-bubble of volumes $v_i$ in a Riemannian manifold $X$. For $X=\mathbb{R}^n$, is $A_X$ strictl...

Are all regions in every area-minimizing bubble cluster in $\mathbb{R}^n$ connected?...

Determine the shapes of minimizing clusters of immiscible fluids when different interfaces have different surface tensions, even for planar double bub...

Determine double crystals in $\mathbb{R}^3$ minimizing orientation-dependent surface energy for a cubic Wulff shape and for general Wulff shapes. Must...

Let $k(n)$ be the maximum number of regions meeting at one point in an area-minimizing partition in $n$ dimensions. Determine the growth of $k(n)$ and...

Is the known list of boundary singularities of soap films in $\mathbb{R}^3$ complete? Classify what happens for singular or immersed boundaries and wh...

Determine the smallest density greater than one of a minimal cone in $\mathbb{R}^n$, with variants imposing area-minimizing or isolated-singularity hy...

Do least-area partitions of $\mathbb{R}^n$ into regions of unit volume exist? Give the correct definition and determine their regularity....

Is the Weaire--Phelan A15 foam the optimal partition of three-space into equal volumes, and can the existence of a foam in the A15 pattern be proved?...

Prove or disprove successively that the Kelvin cell is the least-area fundamental domain for the BCC torus, that it minimizes among all unit-volume fl...

Prove or disprove that no two-dimensional froth of average bubble area one has shear modulus in any direction exceeding that of the regular hexagonal ...

Find the best constant $C$ in the inequality $A\leq C(\int_B H+L)^2$ for a bubble cluster of area $A$, mean curvature $H$, and boundary length $L$. De...

Extend Gromov's Knothe-based proof of the isoperimetric inequality to multiple regions, and determine whether its vector field can be chosen canonical...

Is the cone over the regular tetrahedron the smallest soap film having the entire tetrahedral frame as boundary? For a frame of two rectangles sharing...

For a regular octahedral wire frame, determine the least-area soap film separating the eight regions, both with ordinary and fractional-density soap f...

If a convex planar curve $\Gamma$ has length less than $2\pi$, does it bound a graph of constant mean curvature one? Determine the volume threshold $V...

Is every embedded constant-mean-curvature surface, or every immersed constant-mean-curvature disk, with boundary a round circle necessarily a spherica...

Find the best constant $C$ such that the principal curvatures of an area-minimizing hypersurface in low dimensions are bounded by $C/r$ at a point who...

For a statistical manifold $(M,g,\nabla,\nabla^*)$, find conditions under which it can be realized as an $n$-dimensional submanifold of an $m$-dimensi...

The smooth positive probability densities on $S^1$ are diffeomorphic to the manifold of smooth equiaffine transformations of $S^1$. Find the correspon...

Given a Riemannian manifold $(M,g)$, can one always introduce a symmetric $(0,3)$-tensor $T$ so that $(M,g,\nabla,\nabla^*)$ is dually flat? If the co...

Determine the fundamental group of a compact affine flat manifold....

Do affine minimal surfaces admit isothermal coordinates with respect to their affine fundamental form, analogously to minimal surfaces in Euclidean th...