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 $n$....

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

Determine the maximum number and relative positions of limit cycles for polynomial vector fields of degree $n$, and investigate the topology of real a...

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

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

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....

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

Suppose $A \subset \mathbb{F}_p^2$ is a set meeting every line in at most 2 points. Is it true that all except $o(p)$ points of $A$ lie on a cubic cur...

Fix $k$. Let $A \subset \mathbb{R}^2$ be a set of $n$ points with no more than $k$ on any line. Suppose at least $\delta n^2$ pairs $(x, y) \in A \tim...

Fix integers $k, \ell$. Given $n \geq n_0(k, \ell)$ points in $\mathbb{R}^2$, is there either a line containing $k$ of them, or $\ell$ of them that ar...

Suppose $A \subset \mathbb{R}^2$ is a set of size $n$ with $cn^2$ collinear 4-tuples. Does it contain 5 points on a line?...

What is the largest subset of the grid $[N]^2$ with no three points on a line? In particular, for $N$ sufficiently large, is it impossible to have a s...

Let $\Gamma$ be a smooth codimension 2 surface in $\mathbb{R}^n$. Must $\Gamma$ intersect some 2-dimensional plane in 5 points, if $n$ is sufficiently...

What is the largest subset of $[N]^d$ with no 5 points on a 2-plane?...

Let $X \subset \mathbb{R}^2$ be a set of $n$ points. Does there exist a line $\ell$ through at least two points of $X$ such that the numbers of points...

Let $A$ be a set of $n$ points in the plane. Can one select $A' \subset A$ of size $n/2$ such that any axis-parallel rectangle containing 1000 points ...

Given $n$ points in the unit disc, must there be a triangle of area at most $n^{-2+o(1)}$ determined by them?...

Suppose $A$ is an open subset of $[0, 1]^2$ with measure $\alpha$. Are there four points in $A$ determining an axis-parallel rectangle with area $\geq...

How many rotated (about the origin) copies of the "pyjama set" $\{(x, y) \in \mathbb{R}^2 : \operatorname{dist}(x, \mathbb{Z}) \leq \varepsilon\}$ are...

Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings?...

What is the densest packing of spheres in dimensions 4 through 23? More generally, what is the optimal sphere packing density in dimension $n$?...

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?...

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

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?...

If a compact set in $\mathbb{R}^d$ has Hausdorff dimension greater than $d/2$, must it determine a set of distances with positive Lebesgue measure?...

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

What is the minimum number of pieces needed to perform a Banach-Tarski decomposition of the ball?...

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?...

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

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

What is the relationship between curvature and Euler characteristic for even-dimensional Riemannian manifolds?...

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?...