Unsolved Problems

Does every bounded linear operator on a separable Hilbert space over the complex numbers have a non-trivial invariant subspace?...

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_...

If a function on $\mathbb{R}^n$ has zero integral over every congruent copy of a given domain, must the function be identically zero?...

Is one-dimensional dynamics generally hyperbolic?...

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

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

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

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

Let $K \subset \mathbb{R}^N$ be a balanced compact set with normalized Gaussian measure $\gamma_\infty(K) \geq 0.99$. Does $10K$ contain a compact con...

Let $A \subset \mathbb{R}$ be a set of positive measure. Does $A$ contain an affine copy of $\{1, \frac{1}{2}, \frac{1}{4}, \dots\}$?...

Is every set $\Lambda \subset \mathbb{Z}$ either a Sidon set, or a set of analyticity?...

Let $\mathcal{F}$ be all integrable functions $f : [0, 1] \to \mathbb{R}_{\geq 0}$ with $\int f = 1$. For $1 < p \leq \infty$, estimate $c_p := \inf_{...

Let $A$ be a set of $n$ integers. Is there some $\theta$ such that $\sum_{a \in A} \cos(a\theta) \leq -c\sqrt{n}$?...

Let $A \subset \mathbb{Z}$ be a set of size $n$. For how many $\theta \in \mathbb{R}/\mathbb{Z}$ must we have $\sum_{a \in A} \cos(a\theta) = 0$?...

Describe the rough structure of sets $A \subset \mathbb{Z}$ with $|A| = n$ and $\|\hat{1}_A\|_1 \leq K \log n$....

Let $A \subset \mathbb{R}$ be a set of positive measure. Does $A$ contain an affine copy of $\{1, \frac{1}{2}, \frac{1}{4}, \dots\}$?...

Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?...

If $z_1, \ldots, z_n$ are complex numbers that are linearly independent over the rationals, then the transcendence degree of $\mathbb{Q}(z_1, \ldots, ...

If all zeros of a polynomial lie in the closed unit disk, does each zero have at least one critical point within unit distance from it?...

Does the central limit theorem hold for all φ-mixing sequences?...

Is there a constant $c > 1$ such that all non-cyclotomic polynomials have Mahler measure at least $c$?...

Is a measurable set in $\mathbb{R}^d$ spectral if and only if it tiles by translation?...

If a billiard table is strictly convex and integrable, is it necessarily an ellipse?...

Is there a minimum positive Mahler measure for non-cyclotomic polynomials?...

For which domains do non-zero functions exist with zero integrals over all congruent copies?...

Does the Ibragimov-Iosifescu conjecture hold for φ-mixing sequences?...

For conformal maps f into the unit disk, when is $\int |f'(z)|^p dA < \infty$ for $p > 0$?...

Is a measurable set spectral if and only if it tiles $\mathbb{R}^d$ by translation?...

Does every bounded operator on an infinite-dimensional complex Banach space have a nontrivial closed invariant subspace?...

Is there a constant c > 1 such that all non-cyclotomic polynomials have Mahler measure ≥ c?...

For any polynomial f of degree d≥2 and complex z, does there exist a critical point c with $|f(z)-f(c)| \leq |f'(z)||z-c|$?...

Characterize domains where nonzero functions have vanishing integrals over every congruent copy....

If all roots of a polynomial lie in the unit disk, is each root within distance 1 from some critical point?...

What is the exact value of Bloch's constant (the largest radius for which every holomorphic function contains a univalent disk)?...

Is the Mandelbrot set locally connected?...

Does every regular compact contact-type level set of a Hamiltonian carry a periodic orbit?...