Category
Problem Set
Status
Smale's 7th Problem: Distribution of Points on the 2-Sphere
What is the optimal arrangement of $n$ points on the 2-sphere to minimize energy for various potential functions?...
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....
Frankl's Union-Closed Sets Conjecture
For every finite union-closed family of sets (other than the empty family), there exists an element that belongs to at least half of the sets....
Inscribed Square Problem (Toeplitz Conjecture)
Does every simple closed curve in the plane contain all four vertices of some square?...
Brocard's Problem
Find all integer solutions to $n! + 1 = m^2$....
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....
The Erdős-Straus Conjecture
For every integer $n \geq 2$, the equation $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ has a solution in positive integers x, y, z....
Hilbert's 6th Problem: Axiomatization of Physics
Develop a mathematical framework that axiomatizes physics, particularly mechanics, thermodynamics, and probability theory....
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 Mathematics of the Brain
Create a mathematically consistent, predictive model of brain function that goes beyond biological inspiration....
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....
Biological Quantum Field Theory
Apply quantum and statistical field theory methods to model and potentially control pathogen evolution....
Beyond Convex Optimization
Determine whether algebraic geometry can systematically replace linear algebra in optimization....
Mathematics of Quantum Computing
Develop the mathematics required to control the quantum world for computation....
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....
Fundamental Laws of Biology
Identify governing principles for biological systems, analogous to physical laws....
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....
Symmetries and Action Principles for Biology
Extend understanding of symmetries and action principles in biology to include robustness, modularity, evolvability, and variability....
Geometric Langlands and Quantum Physics
Connect the Langlands program to fundamental physics symmetries....
Arithmetic Langlands, Topology, and Geometry
Explore homotopy theory's role in Langlands programs....
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 9th Problem: Reciprocity Laws
Generalize the reciprocity law of number theory to arbitrary number fields....
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....
Hilbert's 17th Problem: Expression of Definite Forms
Can every non-negative rational function be expressed as a sum of squares of rational functions?...
Hilbert's 18th Problem: Polyhedra and Space-Filling
Are there only finitely many essentially different space-filling convex polyhedra? Is there a polyhedron which tiles space but not in a lattice arrang...
Large Sum-Free Sets
Let $A$ be a set of $n$ positive integers. Does $A$ contain a sum-free set of size at least $n/3 + \Omega(n)$, where $\Omega(n) \to \infty$ as $n \to ...
Restricted Sumset Problem
Let $A \subset \mathbb{Z}$ be a set of $n$ integers. Is there a subset $S \subset A$ of size $(\log n)^{100}$ such that $S \hat{+} S$ is disjoint from...
Product-Free Sets in [0,1]
Suppose that $A \subset [0, 1]$ is open and has measure greater than $1/3$. Is there necessarily a solution to $xy = z$ with $x, y, z \in A$?...
Product-Free Sets in Alternating Groups
What is the largest product-free set in the alternating group $A_n$?...
Product-Free Sets in Finite Groups
Which finite groups have the smallest largest product-free sets?...
Ulam's Sequence
Define Ulam's sequence $1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, \ldots$ where $u_1 = 1, u_2 = 2$, and $u_{n+1}$ is the smallest number uniquely ...
Almost Sum-Free Sets
Suppose that $A \subset [N]$ has no more than $\varepsilon N^2$ solutions to $x + y = z$. Can one remove $\varepsilon' N$ elements to leave a sum-free...
Sum-Free Subsets of [N]^d
Fix an integer $d$. What is the largest sum-free subset of $[N]^d$?...
Progressions in Subsets of Z/NZ
Is $r_5(N) \ll N(\log N)^{-c}$? Is $r_4(\mathbb{F}_5^n) \ll N^{1-c}$ where $N = 5^n$?...