Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
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The Gauss–Kuzmin–Wirsing (GKW) operator acts on suitable function spaces on [0,1] by (L f)(x) = Σ_k=1^∞ 1/(x+k)^2 f (1/x+k). This is the transfer operator of the Gauss map T(x) = \1/x\, which generates the continued fraction expansion.
C_22b = b_o is the largest constant for which one has an inequality L ≥ b_o C for all knots that admit an alternating diagram, where L is the ropelength of a knot (or link) with crossing number) C.
C_22a is the largest constant for which one has an inequality L≥ C_22aC^3/4 for all knots, where L is the ropelength of a knot (or link) with crossing number) C.
The constant C_42 is limsup_n→ inftyR_n, where R_n=minmax_1≤ k≤ n lvert Σ_1≤ i≤ nz_i^krvert, where the minimum is taken over all z_1,…,z_n∈ ℂ with max_i lvert z_irvert=1.
Given a function f ∈ L^1(ℝ), set A(f) := inf r > 0: f(x) ≥ 0 hbox for all |x| ≥ r . Let C be the largest constant for which one has A(f) A(hat f) ≥ C for all even f with f(0), hat f(0) < 0. Establish upper and lower bounds for C that are as strong as possible.
Let D=\z∈ℂ:lvert zrvert<1\ and let F be the class of holomorphic functions f:D→ℂ normalized by f'(0)=1. <a href="#BS2023-def-F">[BS2023-def-F]</a> For finF, let B_f denote the radius of the largest univalent disk in f(D).
For a finite non-empty set A of integers, C_3e is the least constant such that |A - A| ≤ |A + A|^C_3e for every such A; equivalently, C_3e = sup_A loglvert A-Arvert / loglvert A+Arvert. This is Problem 6.43 of [GGSWT2025].
Let 1 ≤ p,q,r ≤ ∞ with 1/r + 1 = 1/p + 1/q. Let C(p,q,r) denote the supremum of the quantity Q(f, g) := ‖f * g‖_r/‖f‖_p ‖g‖_q over all non-zero test functions f,g. What is C(p,q,r)?
Zaremba’s conjecture concerns denominators of rational numbers b/d∈(0,1) whose finite continued fraction expansions have all partial quotients bounded by an absolute constant.
