Mathematics Problem Archive

Are there smooth convex curves, other than ellipses, simultaneously admitting one-parameter families of $p$- and $q$-periodic billiard trajectories fo...

For billiards inside an oval in the Lorentz plane with metric $ds^2=dx^2-dy^2$, is there an analogue of Birkhoff's theorem, with separate existence st...

For a smooth strictly convex hypersurface $M\subset\mathbb{R}^{2n}$, find lower bounds for the number of $p$-periodic outer-billiard orbits for values...

For the ellipsoid $x^2/a+y^2/b+z^2/c=1$ in Minkowski space with metric $dx^2+dy^2-dz^2$, find conditions on $a,b,c>0$ ensuring the existence of an $(n...

Let $\gamma$ be a smooth arc agreeing to all orders with the $x$-axis at endpoints $(0,0)$ and $(1,0)$, and iterate $T(\gamma)=\gamma+\gamma'$. Unless...

Given an oriented oval $\gamma$, let $\gamma_1$ be the locus of endpoints of its oriented unit tangent segments, and iterate this construction. If eve...

Extend the known results on projectively self-dual polygons and curves to projectively self-dual polyhedra and to projectively self-dual polygons in m...

Let $m,p\ge 2$, put $d=m+p-1$, and let $F(x)=(1,x,\ldots,x^d)$ be the rational normal curve. For $j=1,\ldots,mp$, let $X_j$ be the $p$-plane spanned b...

Classify all empty simplices of dimension $n$ up to lattice congruence....

Find a natural generalization of the Farey tessellation to higher-dimensional hyperbolic geometry....

{\bf (Jacobi's last theorem.)} Let $K$ be a totally real cubic number field. Consider arbitrary elements $y$ and $z$ of $K$ such that $0<y,z<1$ (here ...

Generalize continued fractions to describe 3-bridge knots....

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

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...

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 ...

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...

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 ...

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...