Problems

For a transcendental entire function, how fast can |f(z)| be forced to grow along a path to infinity, and how short can such a path be in terms of the maximum modulus M(r, f)?

Localizing Bernstein theory to prove lower bounds for the Lebesgue constants of Lagrange interpolation, with application to a problem of Erdős and Turán and to a conjectured bound from the interpolation literature.

Two free ergodic measure-preserving flows whose L^1 full groups are isomorphic as abstract groups are conjugate up to a scalar time change. This proves the flow analogue of Belinskaya's theorem, answering a question posed by François Le…

Heil, Ramanathan and Topiwala conjectured in 1996 that any finite set of time-frequency shifts of a nonzero square-integrable function is linearly independent. This refutes it: there is a Schwartz function admitting 12 linearly dependent…

Douglas and Yang attach to each nonzero vector x of a quasinilpotent operator T a local resolvent-growth exponent k_x, giving the power set Λ(T) = k_x : x ≠ 0. Ji and Zhang asked whether 1 always belongs to Λ(T). It does, for every…

The arithmetic Kakeya conjecture asserts the infimum of sum-difference exponents is 1, which would imply the Kakeya conjecture in all dimensions. In the bounded-slope-count regime, Tao establishes that the exponents converge to 2 at a rate…

Fuglede's conjecture asks whether a set tiles exactly when it is spectral. The paper proves it for an infinite sequence of square-free order cyclic groups: the tile-to-spectral direction for all square-free cyclic groups, and the…

Iterates of a firmly nonexpansive operator converge weakly but not strongly, by Genel and Lindenstrauss. Whether their Cesaro means converge strongly was open. They need not: an explicit curve gives a counterexample.

Maz'ya and Shaposhnikova introduced a non-classical maximal operator M^diamond, the maximal convolution with the vector-valued signum kernel truncated to centered balls. One of Maz'ya's 75 open problems in analysis asks whether it can be…

Whether the real Kalton-Peck space Z_2 is isomorphic to its hyperplanes. It is not: no hyperplane of Z_2 is isomorphic to Z_2, proved through a rank parity theorem for symplectic spaces applied to the Kalton-Swanson symplectic structure.

The Howland-Kato conjecture that every nonzero positive commutator i[f(P),g(Q)] must arise from functions in appropriate Kato classes is false: i[arctan(P),arctan(Q)] is nonzero and nonnegative.

A convex body is in Faber-Krahn position if it minimizes the first Dirichlet eigenvalue within its volume-preserving linear orbit. The paper proves this position is unique up to orthogonal transformations, answering a question of…

Among all nonconstant monic polynomials f whose roots lie in [-1, 1], determine inf_f |x ∈ R : |f(x)| < 1|.

For p ≥ 2, does Carbery's proposed many-function almost-orthogonality inequality hold with the pairwise overlap coefficients raised to the power 2 - and if not, what is the largest possible exponent?

For a continuous bounded-variation path with signature g, logarithmic signature l and increment v, the modified Lyons–Sidorova conjecture predicts the structure of g when R(l)=∞. The paper proves it: g=1 when v=0, and otherwise a prefix α…

Aldroubi, Cabrelli, Krishtal and Molter conjectured that for a bounded normal operator T and any vector g, the normalized orbit T^k g / |T^k g| : k ≥ 0 is never a frame. It can be: an explicit construction produces a normalized orbit that…

Energy measures of any two nonconstant harmonic functions on the standard Sierpinski gasket are mutually absolutely continuous. Strichartz and Tse reported numerical evidence that the Radon-Nikodym densities are L^p-integrable for 1 < p <…

For a positive projection P on a Dedekind complete Banach lattice whose largest central operator below P is α id, Wickstead conjectured α must be 0 or 1/n for some natural n, and proved the finite-dimensional case. The paper proves the…

Can one hear the shape of a drum, in the Steklov setting and in the plane? No: there exist pairs of noncongruent bounded plane domains with identical Steklov spectra including multiplicities, simply connected, strictly convex, with…

For s ∈ (1/4,1) and any degree, the only W^s,1/s-minimizers among maps S^1 → S^1 are Blaschke products. This resolves Open Problems 23 and 24 of Brezis and Mironescu's book on mappings to the circle, and Brezis's Favorite Open Problem 5.4…