Category
Problem Set
Status
Kakeya Conjecture
Must a Kakeya set in $\mathbb{R}^n$ have Hausdorff and Minkowski dimension $n$?...
Illumination Problem
Can every convex body in $\mathbb{R}^n$ be illuminated by $2^n$ light sources?...
MLC Conjecture
Is the Mandelbrot set locally connected?...
Weinstein Conjecture
Does every regular compact contact-type level set carry a periodic orbit?...
Birkhoff Conjecture
If a billiard table is strictly convex and integrable, must its boundary be an ellipse?...
Abundance Conjecture
If the canonical bundle of a variety is nef, must it be semiample?...
Vaught Conjecture
Is the number of countable models of a complete first-order theory finite, $\aleph_0$, or $2^{\aleph_0}$?...
Cherlin-Zilber Conjecture
Is every simple group with $\aleph_0$-stable theory an algebraic group over an algebraically closed field?...
Yang-Mills Existence and Mass Gap
Does Yang-Mills theory exist mathematically and exhibit a mass gap in 4D?...
Partition Principle Implies Axiom of Choice
Does the partition principle (PP) imply the axiom of choice (AC)?...
Woodin's GCH below Strongly Compact Cardinals
Does the generalized continuum hypothesis below a strongly compact cardinal imply it everywhere?...
GCH and Diamond Principle
Does the generalized continuum hypothesis entail the diamond principle $\diamondsuit(E_{\text{cf}(\lambda)}^{\lambda^+})$ for every singular cardinal ...
GCH and Suslin Trees
Does the generalized continuum hypothesis imply the existence of an $\aleph_2$-Suslin tree?...
Ultimate Core Model
Does there exist an ultimate core model containing all large cardinals?...
Woodin's Ω-Conjecture
If there is a proper class of Woodin cardinals, does Ω-logic satisfy an analogue of Gödel's completeness theorem?...
Strongly Compact vs Supercompact Cardinals
Does the consistency of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?...
Jónsson Algebra on ℵ_ω
Does there exist a Jónsson algebra on $\aleph_\omega$?...
Open Coloring Axiom and Continuum Hypothesis
Is the open coloring axiom (OCA) consistent with $2^{\aleph_0} > \aleph_2$?...
Reinhardt Cardinals without Choice
Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?...
Sudoku: Unique Solution Puzzles
How many Sudoku puzzles have exactly one solution?...
Sudoku: Minimal Puzzles Count
How many Sudoku puzzles with exactly one solution are minimal (removing any clue creates multiple solutions)?...
Maximum Givens in Minimal Sudoku
What is the maximum number of givens for a minimal Sudoku puzzle?...
Tic-Tac-Toe Winning Dimension
Given the width of a tic-tac-toe board, what is the smallest dimension guaranteeing X has a winning strategy?...
Perfect Chess
What is the outcome of a perfectly played game of chess?...
Perfect Komi in Go
What is the perfect value of komi (compensation points) in Go?...
Cap Set Problem
What is the largest possible cap set in $n$-dimensional affine space over the three-element field?...
Octal Games Periodicity
Are the nim-sequences of all finite octal games eventually periodic?...
Grundy's Game Periodicity
Is the nim-sequence of Grundy's game eventually periodic?...
Rendezvous Problem
What is the optimal strategy for two agents to meet on a network without communication?...
Ibragimov-Iosifescu Conjecture for φ-mixing
Does the Ibragimov-Iosifescu conjecture hold for φ-mixing sequences?...
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, ...
Tammes Problem
For n > 14 points (except n=24), what is the maximum minimum distance between points on a unit sphere?...
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?...
Hopf Conjectures
What are the relationships between curvature and Euler characteristic for higher-dimensional Riemannian manifolds?...
Yau's Conjecture on First Eigenvalue
Is the first eigenvalue of the Laplace-Beltrami operator on an embedded minimal hypersurface of $S^{n+1}$ equal to $n$?...
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?...
Orchard-Planting Problem
What is the maximum number of 3-point lines attainable by a configuration of $n$ points in the plane?...
Unit Distance Problem
How many pairs of points at unit distance can be determined by $n$ points in the Euclidean plane?...
Bellman's Lost-in-a-Forest Problem
What is the shortest path that guarantees reaching the boundary of a given shape, starting from an unknown point with unknown orientation?...
Borromean Rings Question
Can three unknotted space curves (not all circles) be arranged as Borromean rings?...
Danzer's Problem
Do Danzer sets of bounded density or bounded separation exist?...
Brouwer's Conjecture on Graph Laplacians
Can the sum of eigenvalues of the Laplacian matrix of a graph be bounded by the number of edges?...
Eternal Domination vs Domination Number
Does there exist a graph where the dominating number equals the eternal dominating number and both are less than the clique covering number?...
Graham's Pebbling Conjecture
Is the pebbling number of the Cartesian product of two graphs at least the product of their pebbling numbers?...
Meyniel's Conjecture on Cop Number
Is the cop number of a connected n-vertex graph $O(\sqrt{n})$?...