Mathematics Problem Archive

Consider two nested convex plane domains, and let $T_1,T_2$ be their billiard ball maps on the oriented lines intersecting both domains. If $T_1\circ ...

Let $f(x,y)$ be a polynomial for which $0$ is a nonsingular value, let $\gamma$ be an oval component of $f(x,y)=0$, and assume that $\gamma_\varepsilo...

Assume the origami hyperbolic-paraboloid pattern has invisible straight folds along a chosen diagonal of each elementary trapezoid. What is the shape ...

Find conceptual proofs and possible generalizations of the new projective-geometry configuration theorems described in the source. In particular, prov...

For a fixed combinatorial dissection of a square into $n$ triangles, the ordered triangle areas satisfy a polynomial relation depending only on the co...

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

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....

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

Given a sufficiently dense sample of points on a surface (technically, an $\epsilon$-sample), reconstruct a surface homeomorphic to the original....

An optimization problem that naturally arises in the study of ``swarm robotics'' is to wake up a set of ``asleep'' robots, starting with only one ``aw...

Consider minimal immersions $S^n\to S^N(1)$ with total area at most a fixed constant $A$, identifying immersions which differ by a motion of the ambie...

Let $B$ be a set of $n+2$ hyperplanes in general position in complex projective space $\mathbb{P}^n(\mathbb{C})$. Prove that there is no nondegenerate...

Let $\Delta$ be a ball in $\mathbb{C}^n$, $n\geq2$, and let $F$ be a complete Hermitian manifold. Does every holomorphic map from $\Delta\setminus\{0\...

There almost surely exists a horofunction h such that lim n→∞ − 1 n h(xn) = A, where A = lim n→∞ 1 n d(x0, xn). A theorem of Karlsson states that ∀ϵ >...

Determine all single sided (resp. double sided) sequences that are pattern Sturmian....

Prove that the pictured standard triple soap bubble is the least-perimeter way to enclose and separate three given volumes in $\mathbb{R}^3$....

Is every Laplacian algebra of polynomials maximal?...

Does every curve bounding a surface of positive curvature in 3-space have (at least) four points where the torsion vanishes?...

Does connectedness of the shadows imply that f(M) is convex?...

Are there any complete surfaces of negative curvature in Euclidean 3-space whose principal curvatures are bounded away from zero?...

Does every convex polyhedron have Rupert's property?...