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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Establish, check and extend rigorous computer-assisted proofs that smooth solutions of the 3D incompressible Euler equations (and related models) develop singularities in finite time.
Does every bounded linear operator on a separable infinite-dimensional complex Hilbert space have a non-trivial closed invariant subspace? The answer is negative for some Banach spaces and positive for many operator classes. The Hilbert space case is open.
Show that every Kakeya (Besicovitch) set in R^n has Hausdorff and Minkowski dimension n. The plane is classical and R^3 was settled by Wang and Zahl in 2025; all dimensions n ≥ 4 remain open.
Lower the known upper bound Λ ≤ 0.2 for the de Bruijn–Newman constant. The Riemann Hypothesis is equivalent to Λ = 0, and Λ ≥ 0 is known.
In harmonic analysis, for λ > 0 let T^λ denote the Bochner–Riesz operator on ℝ^3, initially defined for Schwartz functions f ∈ S(ℝ^3) by T^λ f(x) := ∫_ℝ^3 (1-lvert ξ rvert^2)_+^λ widehatf(ξ)e^ix· ξ dξ, where widehatf denotes the Fourier transform of f and (t)_+ := max\t,0\.
For any dimension n, let C(n) denote the quantity C(n) := π^n/2/Γ(n/2+ 1) inf_f (r/2)^n f(0)/hat f(0) where f ranges over integrable continuous functions f := ℝ^n → ℝ, not identically zero, with hat f(ξ) ≥ 0 for all ξ and f(x) ≤ 0 for all |x| ≥ r for some r>0.
C_1a is the largest constant for which one has max_-1/2 ≤ t ≤ 1/2 ∫_ℝ f(t-x) f(x) dx ≥ C_1a (∫_-1/4^1/4 f(x) dx)^2 for all non-negative f : ℝ → ℝ.
Let C be the smallest constant such that min_0 ≤ t ≤ 1 ∫_ℝ f(x) f(x+t) dx ≤ C ‖f‖_L^1(ℝ)^2 for f ∈ L^1(ℝ). What is C?
Let C be the best constant for which one has max_-1/2 ≤ t ≤ 1/2|∫_ℝ f(t-x) f(x) dx| ≥ C (∫_-1/4^1/4 f(x) dx)^2 for all f : [-1/4,1/4] → ℝ (note f can take negative values). What is C?
In harmonic analysis, the Beurling–Ahlfors transform B (also called the Ahlfors–Beurling operator) is the singular integral operator on L^p(ℂ), 1<p<∞, defined by Bf(z) = -1/π p.v.∫_ℂ f(w)/(z-w)^2 dm(w) = -1/π lim_ε→ 0^+∫_lvert w-zrvert>varepsilonf(w)/(z-w)^2 dm(w), where dm is Lebesgue measure on…
Let D=\z∈ℂ:lvert zrvert<1\. Following standard notation, let F be the class of holomorphic functions f:D→ℂ normalized by lvert f'(0)rvert=1 (equivalently, after rotation, f'(0)=1).
Degree at most d functions f:lbrace ± 1rbrace^n→ℝ have Fourier–Walsh expansion f(x)=Σ_S⊆ [n], |S|≤ d widehat f(S) x^S, x^S:=Π_i∈ Sx_i, [n]:=lbrace 1,…,nrbrace. For d∈ℕ set p_d:=2d/d+1.
Let D^d := z=(z_1,…,z_d)∈ℂ^d: lvert z_1rvert,…,lvert z_drvert<1 be the unit polydisc, and let the Schur class S_d be the set of analytic functions f:D^dtoD.
For any 1 ≤ p < ∞ and n ≥ 2, let C(p,n) be the smallest constant such that for any complex polynomial f of degree n with zeroes z_1,…,z_n satisfying 1/n Σ_i=1^n |z_i|^p ≤ 1, and every zero f(ζ)=0 of f, there exists a critical point f'(ξ) = 0 of f with |ξ - ζ| ≤ C(p,n). What is C(p,n)?
Let Ω⊂ℂ be simply connected with at least two boundary points in the extended complex plane, and let φ:ΩtoD be a conformal map. Brennan's conjecture states that ∫_Ωlvert φ'(z)rvert^p dx dy < ∞ qquadwhenever 4/3<p<4.
C_16 = L is the smallest constant for which the sharp Brezis–Gallouet inequality ‖u‖_L^∞(T^2)^2 ≤ 1/4π ‖∇ u‖_L^2(T^2)^2 Bigl[lnδ(u) + lnbigl(1+lnδ(u)bigr) + LBigr] holds for all zero-mean functions u ∈ H^2(T^2) with sufficiently large frequency ratio δ(u) := ‖Δ u‖_L^2(T^2)^2/‖∇ u‖_L^2(T^2)^2.
In ℝ^d (d≥ 1), let M_d denote the centered Hardy–Littlewood maximal operator associated to cubes, defined by M_d f(x) := sup_r>0 1/lvert Q(x,r)rvert∫_Q(x,r) lvert f(y)rvert dy, where Q(x,r) is a closed ℓ_∞ ball of radius r and center x in ℝ^d, that is, a closed cube centered at x, with sides…
For n ≥ 4, let Ω(n) be the set of pairs (α,β) ∈ ℝ_+^2 such that, whenever P is a degree n polynomial whose roots z_1,…,z_n sum to zero, and ξ_1,…,ξ_n-1 are the critical points (roots of P'), that |ξ_1|^4 + … + |ξ_n-1|^4 ≤ α (|z_1|^4 + … + |z_n|^4) + β (|z_1|^2 + … + |z_n|^2)^2. What is Ω(n)?
For any transcendental entire function f(z)=Σ_n≥ 0 a_n z^n, define . M(r,f):=max_|z|=r|f(z)|, μ(r,f):=max_n≥ 0|a_n| r^n. Following [Er1961], define β(f):=liminf_r→∞μ(r,f)/M(r,f). We define C_51 = B to be the supremum of β(f) over all transcendental entire functions f.
The Falconer distance problem threshold C_34 = s_Δ(ℝ^2) in the plane is defined as s_Δ(ℝ^2) : :=\ infBigl s∈[0,2] : ∀ compact E⊂ℝ^2,\ dim_H(E)>s Longrightarrow lvertΔ(E)rvert>0 Bigr.
Let 1 ≤ q ≤ ∞, and let j and m be non-negative integers such that j < m. Furthermore, let 1 ≤ r ≤ ∞, p ≥ 1 be real and θ ∈ [0, 1] such that the following relations hold: 1/p = j + θ ( 1/r - m ) + 1 - θ/q, j/m ≤ θ < 1.
Let C denote the best constant for which | x: sup_h>0 1/2h ∫_x-h^x+h f(y) dy ≥ λ | ≤ C/λ ∫_ℝ f(x) dx for absolutely integrable non-negative f : ℝ → ℝ. What is C?
For 1 ≤ p ≤ 2, let C(p) be the best constant such that ‖ hat f ‖_L^p'(ℝ) ≤ C(p) ‖ f ‖_L^p(ℝ) holds for all test functions f : ℝ → ℝ. Here p' := p/p-1 is the dual exponent of p. What is C(p)?
Let C be the best constant for which one has ‖f f‖_L^2(ℝ)^2 ≤ C ‖ff‖_L^1(ℝ) ‖f * f‖_L^∞(ℝ) for non-negative f : ℝ → ℝ. What is C?
Let D:=\z∈ℂ:lvert zrvert<1\. The Bergman space A^2(D) consists of analytic functions f on D with lVert frVert_2 := (1/π∫_D lvert f(z)rvert^2 dA(z))^1/2 < ∞, where dA(z) denotes the Lebesgue area measure. For c∈(0,1), write A(c,1) := z∈ℂ: c<lvert zrvert<1.
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 L_f denote the radius of the largest disk contained in f(D).
For positive-semidefinite d × d matrices A_1, …, A_n and any unitarily invariant norm |||·||| (including the operator norm and Schatten p-norms) and m ≤ n, define C(n,m,d) := inf frac 1/n^m Σ_j_1, j_2, …, j_m = 1^n |||A_j_1A_j_2… A_j_m||| (n-m)!/n! Σ_substackj_1, j_2, …, j_m = 1 \ all distinct^n…
For integers m,n≥ 1, let B_ℝ,m(n) be the smallest constant such that every m-linear form T:(ℓ_∞^n)^m → ℝ satisfies the (multilinear) Bohnenblust–Hille inequality (Σ_j_1,…,j_m=1^n bigl|T(e_j_1,…,e_j_m)bigr|^2m/m+1)^m+1/2m ≤ B_ℝ,m(n) ‖T‖, where ‖T‖:=sup_‖x^(1)‖_∞,…,‖x^(m)‖_∞ ≤…
C_46 is the infimal exponent p such that one has the global bound ‖widehatf dσ‖_L^p(ℝ^3) lesssim_p ‖f‖_L^∞(S^2) qquadfor all f∈ L^∞(S^2).
Let d ≥ 2 and D ≥ 1. For p ∈ 4,∞, let C^p(d,D) be the maximum of the ratio frac‖u‖_L^p(S^d)‖u‖_L^2(S^d) where u ranges over (real) spherical harmonics of degree D on the d-dimensional sphere S^d, which we normalize to have unit measure.
For each n ≥ 2, let C(n) be the smallest constant such that for any complex polynomial f of degree n ≥ 2 with zeros z_1, …, z_n in the unit disk and critical points w_1, …, w_n-1, and for any nonnegative weights l_1, …, l_n ≥ 0 satisfying Σ_k=1^n l_k = 1, we have min_1 ≤ j ≤ n-1 | Σ_k=1^n l_k z_k -…
Let f:ℂ→ℂ be a polynomial of degree n≥ 2 whose zeroes all lie in the closed unit disk D(0,1)=\z:lvert zrvert≤ 1\. Sendov's conjecture states that if λ_0 is one of these zeroes, then f' has at least one zero in D(λ_0,1). every zero λ_0 of f has a critical point in D(λ_0,1).
For n ≥ 2, let C(n) be the least constant such that for any polynomial f of degree n, and any z ∈ ℂ with f'(z) ≠ 0, there exists a critical point f'(ξ)=0 such that |f(z)-f(ξ)/z-ξ| ≤ C(n) |f'(z)|. Establish upper and lower bounds for C(n) that are as strong as possible.
C_10c is the least constant K for which one has disc(A) ≤ K√(n) qquadfor all n and all A∈[-1,1]^n× n. Here the discrepancy disc(A) is defined as disc(A) := min_x∈± 1^n ‖Ax‖_∞.
The complex Grothendieck constant (often denoted K_G^ℂ) is the smallest number C_10b such that, for every m,n≥ 1 and every complex matrix A=(a_ij)∈ℂ^m× n, max_substacku_1,…,u_m∈ S^∞\ v_1,…,v_n∈ S^∞ |Σ_i=1^mΣ_j=1^n a_ij⟨ u_i, v_j⟩| ≤ C_10b\ max_substack|s_1|=⋯=|s_m|=1\ |t_1|=⋯=|t_n|=1…
Let Q_n = -1,1^n be the Hamming cube (two vertices are adjacent if they differ in exactly one coordinate). For a set A ⊂ Q_n define the function h_A:Q_n→ 0,1,...,n by - h_A(x)=0 if x∉ A; - if x∈ A, then h_A(x) is the number of neighbors of x that lie in the complement A^c.
C_2 is the Crouzeix constant (sometimes denoted Q). It is the smallest constant C such that for every n ≥ 1, every complex matrix A ∈ ℂ^n × n, and every complex polynomial p one has ‖p(A)‖ ≤ C max_z ∈ W(A) |p(z)|, where ‖·‖ is the operator norm induced by the Euclidean norm (i.e.
C_11a is the smallest constant such that, for every n≥ 1 and every function f:-1,1^n → ℝ Ebigl|f(x)-Ef(x)bigr| ≤ C_11aE|∇ f|(x), where x=(x_1,…,x_n) is uniform on -1,1^n and |∇ f|(x)=Bigl(Σ_j=1^n |D_j f(x)|^2Bigr)^1/2, D_j f(x)=f(x)-f(x^(j))/2, with x^(j)=(x_1,...,x_j-1,-x_j,x_j+1,...,x_n).
C_10 is the real Grothendieck constant K_G^ℝ. It is the smallest constant C such that for every m,n ≥ 1 and every real matrix A=(a_ij) ∈ ℝ^m× n one has max_substacku_1,…,u_m, v_1,…,v_n ∈ S^∞ Σ_i=1^m Σ_j=1^n a_ij ⟨ u_i, v_j⟩ ≤ C max_ε_1,…,ε_m, δ_1,…,δ_n = ± 1 Σ_i=1^m Σ_j=1^n a_ij ε_i δ_j.
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).
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)?
The Banach–Mazur rotation problem asks whether every separable Banach space whose group of linear isometric equivalences acts transitively on the unit sphere is linearly isometric to a Hilbert space.
Lower bound for c(p) for 1 < p ≤ ∞, improving the known value √(4/7) at p = 2 or the known value 0.64 at p = ∞.
Let A ⊂ R be a set of positive measure. Does A contain an affine copy of 1, 1/2, 1/4, . . . ?
Ahlfors and Grunsky also conjectured in [AG37] that this upper bound is the precise value of the Bloch constant.
Brennan's conjecture, part 1: B(-2) = 1.
The MLC conjecture, stating that the mandelbrot set is locally connected.
The Flint Hills series summing csc(n)^2 / n^3 from n=1 to ∞ converges. (Note that we 0-index the series below.)
