Category
Problem Set
Status
Collatz Conjecture
Starting with any positive integer $n$, repeatedly apply the function: if $n$ is even, divide by 2; if $n$ is odd, multiply by 3 and add 1. Does this ...
Twin Prime Conjecture
Are there infinitely many twin primes? Twin primes are pairs of primes that differ by 2, such as (3, 5), (5, 7), (11, 13), (17, 19), (29, 31)....
Goldbach's Conjecture
Every even integer greater than 2 can be expressed as the sum of two primes....
Hadwiger Conjecture
Every graph with chromatic number $k$ has a $K_k$ minor (where $K_k$ is the complete graph on $k$ vertices)....
Erdős-Faber-Lovász Conjecture
If a graph is the union of $n$ cliques of size $n$, no two of which share more than one vertex, then the chromatic number is $n$....
Kepler Conjecture
No packing of congruent spheres in three dimensions has density greater than $\frac{\pi}{\sqrt{18}} \approx 0.74048$....
Sphere Packing in Higher Dimensions
What is the densest packing of congruent spheres in $n$ dimensions for $n \geq 4$?...
Inverse Galois Problem
Is every finite group the Galois group of some Galois extension of the rational numbers $\mathbb{Q}$?...
Kaplansky's Conjectures
A set of conjectures about group rings: (1) Zero divisor conjecture: If $G$ is a torsion-free group and $K$ is a field, then $K[G]$ has no zero diviso...
Invariant Subspace Problem
Does every bounded linear operator on a separable Hilbert space over the complex numbers have a non-trivial invariant subspace?...
Schanuel's Conjecture
Given $n$ complex numbers $z_1, \ldots, z_n$ that are linearly independent over the rationals, the transcendence degree of $\mathbb{Q}(z_1, \ldots, z_...
Are there infinitely many Mersenne primes?
Are there infinitely many prime numbers of the form $M_p = 2^p - 1$ where $p$ is prime?...
The Kakeya Conjecture
A Kakeya set (containing a unit line segment in every direction) in $\mathbb{R}^n$ must have Hausdorff dimension $n$....
The Volume Conjecture
For a hyperbolic knot $K$, the limit of normalized colored Jones polynomials equals the hyperbolic volume of the knot complement....
The Triangulation Conjecture
Every topological manifold can be triangulated....
The Abundance Conjecture
For a minimal model $X$ of non-negative Kodaira dimension, the canonical divisor $K_X$ is semi-ample....
The Regularity Problem for Euler Equations
Do solutions to the 3D Euler equations for incompressible fluid flow remain smooth for all time, given smooth initial data?...
Singular Cardinals Hypothesis
If $\kappa$ is a singular strong limit cardinal, then $2^\kappa = \kappa^+$....
Whitehead Problem
Is every abelian group $A$ such that $\text{Ext}^1(A, \mathbb{Z}) = 0$ a free abelian group?...
The Unique Games Conjecture
For certain constraint satisfaction problems (unique games), it is NP-hard to approximate the maximum fraction of satisfiable constraints beyond a cer...
Landau's Fourth Problem: Primes of the Form n² + 1
Are there infinitely many primes of the form $n^2 + 1$?...
Smale's 4th Problem: Integer Zeros of Polynomials
Find efficient algorithms for deciding whether a polynomial with integer coefficients has an integer root....
Smale's 5th Problem: Height Bounds for Diophantine Curves
Find effective uniform bounds for the heights of rational points on algebraic curves....
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 9th Problem: Linear Programming in Polynomial Time
Find a strongly polynomial algorithm for linear programming....
Smale's 10th Problem: The Pugh Closing Lemma
Is the $C^r$ closing lemma true for dynamical systems?...
The Jacobian Conjecture
If $F: \mathbb{C}^n \to \mathbb{C}^n$ is a polynomial map with constant non-zero Jacobian determinant, then $F$ is invertible....
Inscribed Square Problem (Toeplitz Conjecture)
Does every simple closed curve in the plane contain all four vertices of some square?...
Catalan's Conjecture (Mihăilescu's Theorem)
The only solution to $x^p - y^q = 1$ in natural numbers x, y > 0 and p, q > 1 is $3^2 - 2^3 = 1$....
The Cycle Double Cover Conjecture
Every bridgeless graph has a cycle double cover: a collection of cycles that covers each edge exactly twice....
Hilbert's 13th Problem: Seventh Degree Equations
Prove that the general equation of the seventh degree cannot be solved using functions of only two variables....
Smale's 11th Problem: One-Dimensional Dynamics
Is one-dimensional dynamics generally hyperbolic?...
Smale's 12th Problem: Centralizers of Diffeomorphisms
Determine the structure of centralizers of generic diffeomorphisms....
The Dynamics of Networks
Develop high-dimensional mathematics to model and predict behavior in large-scale distributed networks....
Capture and Harness Stochasticity in Nature
Develop methods that capture persistence in stochastic environments, addressing Mumford's call for new mathematics....
21st Century Fluids
Extend classical fluid dynamics to handle complex substances like foams, suspensions, gels, and liquid crystals....
Beyond Convex Optimization
Determine whether algebraic geometry can systematically replace linear algebra in optimization....
Game Theory at Scale
Create scalable mathematics for differential games, replacing traditional PDE approaches....
Information Theory for Virus Evolution
Apply Shannon's information theory to biological evolution....
Computation at Scale
Develop asymptotics for systems with massive degrees of freedom....
Computational Duality
Use mathematical duality and geometry as foundations for developing novel computational algorithms....
Occam's Razor in Many Dimensions
Find lower bounds for sensing complexity as data collection grows, addressing entropy maximization....
Physical Consequences of Perelman's Proof
Apply Perelman's proof of the Poincaré conjecture to materials fabrication across scales....
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....
Hilbert's 7th Problem: Transcendence of Certain Numbers
If $\alpha$ is algebraic and irrational, and $\beta$ is algebraic and irrational, is $\alpha^\beta$ transcendental?...
Hilbert's 11th Problem: Quadratic Forms over Algebraic Number Fields
Extend the theory of quadratic forms with algebraic numerical coefficients....
Hilbert's 14th Problem: Finite Generation of Rings
Is the ring of invariants of a linear algebraic group acting on a polynomial ring always finitely generated?...
Hilbert's 15th Problem: Schubert's Enumerative Calculus
Rigorously justify Schubert's enumerative geometry....