Mathematics Problem Archive

What is the largest area of a shape that can be maneuvered through an L-shaped corridor of unit width?...

What is the optimal arrangement of $n$ points on the 2-sphere to minimize energy for various potential functions?...

Are there only finitely many essentially different space-filling convex polyhedra? Is there a polyhedron which tiles space but not in a lattice arrang...

What is the shortest path that guarantees escape from a forest of known shape and size, starting from an unknown location?...

What are necessary and sufficient conditions for an integral curve defined by two periodic functions to be closed?...

For n > 14 points (except n=24), what is the maximum minimum distance between points on a unit sphere?...

What is the maximum number of 3-point lines attainable by a configuration of $n$ points in the plane?...

What is the shortest path that guarantees reaching the boundary of a given shape, starting from an unknown point with unknown orientation?...

Can three unknotted space curves (not all circles) be arranged as Borromean rings?...

Conjecture Let $k$ be a field of characteristic zero. A collection $f_1,\ldots,f_n$ of polynomials in variables $x_1,\ldots,x_n$ defines an automorphi...

Conjecture Let $X$ be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subva...

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 ...

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...

Does every polygonal outer billiard in the hyperbolic plane have periodic orbits, possibly lying on the circle at infinity?...

Describe the polygonal outer billiard tables in the hyperbolic plane for which every orbit is periodic....

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...

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...

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...

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

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...

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

Does there exist a totally skew embedded $3$-disc in $\mathbb{R}^7$?...

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

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...

Find an integer cosine rule for integer triangles in integer trigonometry....

{\bf(IKEA problem.)} Classify all $n$-tuples of LLS-sequences for the angles that form integer $n$-gons....

Classify all combinatorial possible types of faces....

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

Which $n$-gons are realizable as faces of an $m$-dimensional continued fraction? Here are two essentially geometrically different subcases: ; {\bf Fac...

Describe all finite two-dimensional sails (and the corresponding continued fractions)....

{\bf (Multidimensional IKEA problem.)} Describe the collections of the sails of the cones for all polytopes of a given combinatorial type....

{\bf (V. Arnold.)} Does there exist an algorithm to decide whether a given type of fundamental domain is realizable by a periodic continued fraction?...

{\bf (V. Arnold.)} Torus decompositions of integer noncongruent Klein sails are distinct....

{\bf (V. Arnold.)} Describe all torus decompositions that are realized by periodic two-dimensional continued fractions....

{\bf (V. Arnold.)} Classify continued fractions that correspond to the same cubic extension of the field of rational numbers....

Prove the existence of a cone for a single non-periodic combinatorial structure ($n\ge 3$)....

Find frequencies on $n$-dimensional continued fractions with the highest relative frequencies....

For every positive integer constant $C$ there exist only finitely many pairwise integer non-congruent faces with frequencies exceeding $C$....

Is that true that sum of all relative frequencies for all possible faces is finite for higher dimensions $(n\ge 3)$?...

In case of positive answer to the above question find the generalization of the Gauss map and compare the corresponding frequencies of faces with the ...

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...

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...