Sphere Packing in Higher Dimensions
What is the densest packing of congruent spheres in $n$ dimensions for $n \geq 4$?...
The Kakeya Conjecture
A Kakeya set (containing a unit line segment in every direction) in $\mathbb{R}^n$ must have Hausdorff dimension $n$....
Smale's 6th Problem: Finiteness of Central Configurations
For the Newtonian $n$-body problem with positive masses, are there only finitely many central configurations (relative equilibria) for each $n$?...
Smale's 10th Problem: The Pugh Closing Lemma
Is the $C^r$ closing lemma true for dynamical systems?...
Inscribed Square Problem (Toeplitz Conjecture)
Does every simple closed curve in the plane contain all four vertices of some square?...
Smale's 12th Problem: Centralizers of Diffeomorphisms
Determine the structure of centralizers of generic diffeomorphisms....
Algorithmic Origami and Biology
Strengthen mathematical theory for isometric and rigid embedding relevant to protein folding....
Optimal Nanostructures
Develop mathematics for creating optimal symmetric structures through nanoscale self-assembly....
The Geometry of Genome Space
Establish appropriate distance metrics on genome space incorporating biological utility....
Mahler's Conjecture
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...
The Illumination Conjecture
Can every convex body in $n$-dimensional space be illuminated by at most $2^n$ point light sources?...
Kakeya Needle Problem
What is the minimum area of a region in the plane in which a unit line segment can be continuously rotated through 360 degrees?...
The Knaster Problem
Can a solid cube be completely covered by finitely many smaller homothetic cubes with ratio less than 1, such that the interiors are disjoint?...
The Inscribed Square Problem
Does every simple closed curve in the plane contain four points that form the vertices of a square?...
The Shephard's Problem
Can the unit ball in $\mathbb{R}^n$ be illuminated by fewer than $2^n$ directions?...
The Spherical Bernstein Problem
What is the classification of complete minimal hypersurfaces in spheres of all dimensions?...
The Carathéodory Conjecture
Does every convex, closed, twice-differentiable surface in $\mathbb{R}^3$ have at least two umbilical points?...
The Cartan-Hadamard Conjecture
Does the isoperimetric inequality hold for Cartan-Hadamard manifolds?...
Chern's Affine Conjecture
Does the Euler characteristic of a compact affine manifold vanish?...
Chern's Conjecture for Hypersurfaces in Spheres
What minimal hypersurfaces in spheres have constant mean curvature?...
The Filling Area Conjecture
Does a hemisphere have minimum area among shortcut-free surfaces with a given boundary length?...
The Osserman Conjecture
Is every Osserman manifold either flat or locally isometric to a rank-one symmetric space?...
Yau's Conjecture on First Eigenvalues
Is the first eigenvalue of the Laplace-Beltrami operator on a minimal hypersurface in $S^{n+1}$ equal to $n$?...
The Hadwiger Covering Conjecture
Can every $n$-dimensional convex body be covered by at most $2^n$ smaller homothetic copies?...
The Happy Ending Problem
What is the minimum number of points in the plane needed to guarantee a convex $n$-gon?...
The Heilbronn Triangle Problem
What is the largest minimum area of a triangle determined by $n$ points in a unit square?...
Kalai's $3^d$ Conjecture
Does every centrally symmetric $d$-dimensional polytope have at least $3^d$ faces?...
The Unit Distance Problem
What is the maximum number of unit distances determined by $n$ points in the plane?...
Ehrhart's Volume Conjecture
Does a convex body in $\mathbb{R}^n$ with one interior lattice point at its center of mass have volume at most $(n+1)^n/n!$?...
Borsuk's Conjecture
Can every bounded set in $\mathbb{R}^n$ be partitioned into $n+1$ sets of smaller diameter?...
The Kissing Number Problem
What is the maximum number of non-overlapping unit spheres that can touch a central unit sphere in $n$ dimensions?...
Ulam's Packing Conjecture
Is the sphere the worst-packing convex solid?...
Erdős-Ulam Problem
Is there a dense set of points in the plane with all pairwise distances rational?...
Illumination Problem
Can every convex body in $\mathbb{R}^n$ be illuminated by $2^n$ light sources?...
Kissing Number Problem
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, ...
Carathéodory Conjecture
Does every convex, closed, twice-differentiable surface in 3D Euclidean space have at least two umbilical points?...
Cartan-Hadamard Conjecture
Does the isoperimetric inequality extend to Cartan-Hadamard manifolds (complete simply-connected manifolds of nonpositive curvature)?...
Chern's Conjecture (Affine Geometry)
Does the Euler characteristic of a compact affine manifold vanish?...
Hadwiger Conjecture (Covering)
Can every $n$-dimensional convex body be covered by at most $2^n$ smaller positively homothetic copies?...
Happy Ending Problem
What is the minimum number $g(n)$ of points in general position in the plane guaranteeing a convex $n$-gon?...
Heilbronn Triangle Problem
What configuration of $n$ points in the unit square maximizes the area of the smallest triangle they determine?...
Kalai's 3^d Conjecture
Does every centrally symmetric $d$-dimensional polytope have at least $3^d$ faces?...
Unit Distance Problem
How many pairs of points at unit distance can be determined by $n$ points in the Euclidean plane?...
Danzer's Problem
Do Danzer sets of bounded density or bounded separation exist?...
Geometry of Continued Fractions — Further open questions
v1.3 research notesStudy geometric properties of Markov spectrum....
Chromatic number of the plane
v1.3 research notesDetermine the least number of colors needed to color the Euclidean plane so that points at unit distance receive different colors....
Covering points by congruent rectangles
v1.3 research notesGiven a finite planar point set and a prescribed rectangle, approximate efficiently the minimum number of congruent copies of the rectangle needed to ...
Odd rep-tiling by a 14-omino
v1.3 research notesCan the $3\times6$ rectangle with a $2\times2$ corner removed tile a rectangle using an odd number of congruent copies?...
Comparing sums of square roots
v1.3 research notesCan sums of square roots of integers be compared in polynomial time on a Turing machine? Equivalently, obtain effective polynomial bit bounds for a no...
Triangulations with many distinct areas
v1.3 research notesFind the largest function $t(n)$ such that every convex $n$-gon has a triangulation containing at least $t(n)$ distinct triangle areas; also determine...