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_...

Is one-dimensional dynamics generally hyperbolic?...

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

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

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?...

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?...

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)?...