Unsolved Problems

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....

Does every simple closed curve in the plane contain all four vertices of some square?...

Find all integer solutions to $n! + 1 = m^2$....

Every bridgeless graph has a cycle double cover: a collection of cycles that covers each edge exactly twice....

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....

Develop a mathematical framework that axiomatizes physics, particularly mechanics, thermodynamics, and probability theory....

Prove that the general equation of the seventh degree cannot be solved using functions of only two variables....

Is one-dimensional dynamics generally hyperbolic?...

Determine the structure of centralizers of generic diffeomorphisms....

Create a mathematically consistent, predictive model of brain function that goes beyond biological inspiration....

Develop high-dimensional mathematics to model and predict behavior in large-scale distributed networks....

Develop methods that capture persistence in stochastic environments, addressing Mumford's call for new mathematics....

Extend classical fluid dynamics to handle complex substances like foams, suspensions, gels, and liquid crystals....

Apply quantum and statistical field theory methods to model and potentially control pathogen evolution....

Determine whether algebraic geometry can systematically replace linear algebra in optimization....

Develop the mathematics required to control the quantum world for computation....

Create scalable mathematics for differential games, replacing traditional PDE approaches....

Apply Shannon's information theory to biological evolution....

Develop asymptotics for systems with massive degrees of freedom....

Identify governing principles for biological systems, analogous to physical laws....

Use mathematical duality and geometry as foundations for developing novel computational algorithms....

Find lower bounds for sensing complexity as data collection grows, addressing entropy maximization....

Apply Perelman's proof of the Poincaré conjecture to materials fabrication across scales....

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....

Extend understanding of symmetries and action principles in biology to include robustness, modularity, evolvability, and variability....

Connect the Langlands program to fundamental physics symmetries....

Explore homotopy theory's role in Langlands programs....

If $\alpha$ is algebraic and irrational, and $\beta$ is algebraic and irrational, is $\alpha^\beta$ transcendental?...

Generalize the reciprocity law of number theory to arbitrary number fields....

Extend the theory of quadratic forms with algebraic numerical coefficients....

Is the ring of invariants of a linear algebraic group acting on a polynomial ring always finitely generated?...

Rigorously justify Schubert's enumerative geometry....

Can every non-negative rational function be expressed as a sum of squares of rational functions?...

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

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 ...

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...

Suppose that $A \subset [0, 1]$ is open and has measure greater than $1/3$. Is there necessarily a solution to $xy = z$ with $x, y, z \in A$?...

Which finite groups have the smallest largest product-free sets?...

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 ...

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...

Fix an integer $d$. What is the largest sum-free subset of $[N]^d$?...

Is $r_5(N) \ll N(\log N)^{-c}$? Is $r_4(\mathbb{F}_5^n) \ll N^{1-c}$ where $N = 5^n$?...

Let $S \subset \mathbb{N}$ be random. Under what conditions is Roth's theorem for progressions of length 3 true with common differences in $S$?...

Let $G$ be an abelian group of size $N$, and suppose that $A \subset G$ has density $\alpha$. Are there at least $\alpha^{15}N^{10}$ tuples $(x_1, \ld...

Suppose that $A \subset \mathbb{Z}/N\mathbb{Z}$ has density $\alpha$ and is Fourier uniform (all Fourier coefficients of $1_A - \alpha$ are $o(N)$). D...

Does there exist a Lipschitz function $f : \mathbb{N} \to \mathbb{Z}$ whose graph $\Gamma = \{(n, f(n)) : n \in \mathbb{Z}\} \subset \mathbb{Z}^2$ is ...

What is the largest subset of $[N]$ with no solution to $x + 3y = 2z + 2w$ in distinct integers $x, y, z, w$?...

Suppose that $A \subset \mathbb{F}_3^n$ is a set of density $\alpha$. Under what conditions on $\alpha$ is $A$ guaranteed to contain a 3-term progress...