Problems
Everything in the archive: the problem as it was posed, what has been attempted against it, and who checked each attempt. The mark down the left of the list says who has looked — a person, a machine, or nobody yet. Human reviews and machine checks are counted separately and are never added together.
31 problems
For a measurable set Ω⊂ R^3, let E(Ω)=P(Ω)+frac12iint_Ω×Ωdx dy/|x-y|, where P is De Giorgi perimeter, and set V_*=5frac2-2^2/32^2/3-1≈3.51. The conjecture asks for the complete fixed-volume minimization picture. Chodosh and Gianocca prove…
Denjoy's 1932 theorem says a C^1+bv circle diffeomorphism with irrational rotation number has no wandering interval. Whether it is sharp in regularity: for every concave modulus of continuity ω weaker than Lipschitz, there is a C^1+ω…
A Banach space is primary if in every decomposition into two complemented subspaces one summand is isomorphic to the whole. Lechner, Motakis, Müller and Schlumprecht identified the primariness of L_p(L_1) as a prominent remaining open…
The realisation problem asks which unital Banach algebras arise as the Calkin algebra B(X)/K(X) of some Banach space. Recorded in Tarbard's thesis and studied by Horváth and Kania. The paper exhibits a unital Banach algebra that cannot be…
For f(z) = ∏_i=1^n (z - z_i) with all |z_i| ≤ 1, let ρ(f) be the radius of the largest disc contained in z : |f(z)| < 1. Is ρ(f) ≫ 1/n? The worst case is now known to be Θ(1/n), with the explicit bound ρ(f) ≥ (log 2)/n.
How large can a measurable A ⊆ [0,R]^2 be while avoiding the vertices of upward-oriented axis-aligned right triangles of area 1/2? At most O_c(R^2/(log R)^c), with a matching-shaped lower bound construction.
After L^2 normalization, stable phase retrieval holds over the L^2-spans of independent real-valued centered random variables exactly when all but possibly one coordinate satisfies a uniform two-sided L^1 bound. This confirms the…
The dissipative barrier method suppresses spectral pollution when a differential operator is truncated, but can it hide genuine spectral points? Known as the graveyard problem, the question stayed open in dimension two and above for more…
Davis, Figiel, Johnson and Pełczyński showed their interpolation space admits a Schauder basis when the range space has a shrinking one. Can the DFJP space always be chosen with a basis whenever the range space has a basis? The paper…
The Elton–Odell theorem gives, in every infinite-dimensional normed space, a unit-sphere sequence with mutual distances at least 1+ε. Over C, identifying vectors differing by a unimodular scalar gives a toroidal distance. Does every…
Let p be a complex polynomial of degree n≥2 whose zeros all lie in the closed unit disk. For every zero a of p, there is a critical point ζ satisfying |ζ-a|<1, except when |a|=1 and p is a nonzero scalar multiple of z^n-a^n.