Unsolved Problems

No packing of congruent spheres in three dimensions has density greater than $\frac{\pi}{\sqrt{18}} \approx 0.74048$....

What is the densest packing of congruent spheres in $n$ dimensions for $n \geq 4$?...

A Kakeya set (containing a unit line segment in every direction) in $\mathbb{R}^n$ must have Hausdorff dimension $n$....

For the Newtonian $n$-body problem with positive masses, are there only finitely many central configurations (relative equilibria) for each $n$?...

Is the $C^r$ closing lemma true for dynamical systems?...

Does every simple closed curve in the plane contain all four vertices of some square?...

Determine the structure of centralizers of generic diffeomorphisms....

Strengthen mathematical theory for isometric and rigid embedding relevant to protein folding....

Develop mathematics for creating optimal symmetric structures through nanoscale self-assembly....

Establish appropriate distance metrics on genome space incorporating biological utility....

Among all centrally symmetric convex bodies in $\mathbb{R}^n$, does the cube (or cross-polytope) minimize the product of the body's volume and the vol...

Can every convex body in $n$-dimensional space be illuminated by at most $2^n$ point light sources?...

What is the minimum area of a region in the plane in which a unit line segment can be continuously rotated through 360 degrees?...

Can a solid cube be completely covered by finitely many smaller homothetic cubes with ratio less than 1, such that the interiors are disjoint?...

Does every simple closed curve in the plane contain four points that form the vertices of a square?...

Can the unit ball in $\mathbb{R}^n$ be illuminated by fewer than $2^n$ directions?...

What is the classification of complete minimal hypersurfaces in spheres of all dimensions?...

Does every convex, closed, twice-differentiable surface in $\mathbb{R}^3$ have at least two umbilical points?...

Does the isoperimetric inequality hold for Cartan-Hadamard manifolds?...

Does the Euler characteristic of a compact affine manifold vanish?...

What minimal hypersurfaces in spheres have constant mean curvature?...

Does a hemisphere have minimum area among shortcut-free surfaces with a given boundary length?...

Is every Osserman manifold either flat or locally isometric to a rank-one symmetric space?...

Is the first eigenvalue of the Laplace-Beltrami operator on a minimal hypersurface in $S^{n+1}$ equal to $n$?...

Can every $n$-dimensional convex body be covered by at most $2^n$ smaller homothetic copies?...

What is the minimum number of points in the plane needed to guarantee a convex $n$-gon?...

What is the largest minimum area of a triangle determined by $n$ points in a unit square?...

Does every centrally symmetric $d$-dimensional polytope have at least $3^d$ faces?...

What is the maximum number of unit distances determined by $n$ points in the plane?...

Does a convex body in $\mathbb{R}^n$ with one interior lattice point at its center of mass have volume at most $(n+1)^n/n!$?...

Can every bounded set in $\mathbb{R}^n$ be partitioned into $n+1$ sets of smaller diameter?...

What is the maximum number of non-overlapping unit spheres that can touch a central unit sphere in $n$ dimensions?...

Is the sphere the worst-packing convex solid?...

Is there a dense set of points in the plane with all pairwise distances rational?...

Can every convex body in $\mathbb{R}^n$ be illuminated by $2^n$ light sources?...

What is the kissing number (maximum number of non-overlapping unit spheres that can touch a central unit sphere) in dimensions other than 1, 2, 3, 4, ...

Does every convex, closed, twice-differentiable surface in 3D Euclidean space have at least two umbilical points?...

Does the isoperimetric inequality extend to Cartan-Hadamard manifolds (complete simply-connected manifolds of nonpositive curvature)?...

Does the Euler characteristic of a compact affine manifold vanish?...

Can every $n$-dimensional convex body be covered by at most $2^n$ smaller positively homothetic copies?...

What is the minimum number $g(n)$ of points in general position in the plane guaranteeing a convex $n$-gon?...

What configuration of $n$ points in the unit square maximizes the area of the smallest triangle they determine?...

Does every centrally symmetric $d$-dimensional polytope have at least $3^d$ faces?...

How many pairs of points at unit distance can be determined by $n$ points in the Euclidean plane?...

Do Danzer sets of bounded density or bounded separation exist?...