Operator theory, harmonic analysis and the Riemann hypothesis: progress here is mostly partial results, sharper constants and formalised lemmas, each reviewed or kernel-checked.
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.
The MLC conjecture, stating that the mandelbrot set is locally connected.
How close can the Mahler measure of a ±1 polynomial of degree n get to its L2 norm √(n+1)? The best known ratio was raised above 0.954 in 2025; whether it can approach 1 is open.
How much larger than its entrywise ℓ1 norm can the cheapest decomposition Σ‖x_k‖₁² of a PSD matrix be? The worst ratio C_n is between c√n and O(n).
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; n ≥ 4 is open, with October 2026 claims (OpenAI) for n = 4 and, via local smoothing, all dimensions under review.
Lower the published upper bound Λ ≤ 0.2 for the de Bruijn–Newman constant (unrefereed 2026 claims reach about 0.178 and below). 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?
C_1c is the smallest constant for which one has min_0 ≤ t ≤ 1 ∫_ℝ f(x) f(x+t) dx ≤ C_1c (∫_ℝ f(x) dx)^2 for all non-negative f ∈ L^1(ℝ). Equivalently, in the notation of [K2021], τ = C_1c^-1/2 is the infimum of ∫ f over non-negative f with ∫ f(x) f(x+t) dx ≥ 1 for all t ∈ [0,1].
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…
A distinct covering system is a finite collection of residue classes mathcal C=a_i+n_iℤ:1≤ i≤ k, a_i∈ℤ, 1<n_1<⋯<n_k, bigcup_i=1^k(a_i+n_iℤ)=ℤ. The classes may overlap; distinctness refers to their moduli. Define C_90:=max_mathcal Cmin_1≤ i≤ kn_i =max_mathcal Cn_1.
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)‖_∞ ≤…
For a 2-colouring of the edges of the complete graph K_n, let k_4 be the number of 4-element vertex sets whose six edges all have the same colour.
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.
C_1d is the largest constant for which one has max_t ∈ ℝ |∫_ℝ f(t-x) f(x) dx| ≥ C_1d (∫_-1/4^1/4 f(x) dx)^2 for all real-valued (not necessarily non-negative) f : ℝ → ℝ supported on [-1/4,1/4]. This is the maximum of lvert f\f rvert over t, for signed f.
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 2: B_b(-2) = 1.
The Flint Hills series summing csc(n)^2 / n^3 from n=1 to ∞ converges. (Note that we 0-index the series below.)
De Giorgi's conjecture holds in dimension n ≤ 8.
For any 0<α<1, let f(α,n)=1/log nΣ_1≤ k≤ n(1/2- α k). Does f(α,n) have an asymptotic distribution function? In other words, is there a non-decreasing function g such that g(-∞)=0, g(∞)=1, and lim_n→ ∞lvert α∈ (0,1): f(α,n)≤ crvert=g(c)?
What is the infimum of |x ∈ ℝ : |f x| < 1| over all nonconstant monic polynomials f such that all of its roots are real and contained in [-1,1]?
For x_1,…,x_n∈ [-1,1] let l_k(x)=Π_i≠ k(x-x_i)/Π_i≠ k(x_k-x_i), which are such that l_k(x_k)=1 and l_k(x_i)=0 for i≠ k. Let x_1,x_2,…∈ [-1,1] be an infinite sequence, and let L_n(x) = Σ_1≤ k≤ nlvert l_k(x)rvert, where each l_k(x) is defined above with respect to x_1,…,x_n.
Let C>0. There exists ε>0 such that if n is sufficiently large the following holds. For any x_1,…,x_n∈ [-1,1] there exist y_1,…,y_n∈ [-1,1] such that, if P is a polynomial of degree m<(1+ε)n with P(x_i)=y_i for at least (1-ε)n many 1≤ i≤ n, then max_x∈ [-1,1]lvert P(x)rvert >C.
If p(z) ∈ ℂ[z] is a monic polynomial of degree n, is the length of the curve z ∈ ℂ : lvert p(z)rvert = 1 maximised when p(z) = z^n - 1? We require n ≥ 1. For n = 0 the only monic polynomial is 1, whose level set is all of ℂ, while z^0 - 1 = 0 has an empty level set.
Is there some constant c > 0 such that, for all large enough n and all polynomials P of degree n with coefficients in -1, 1, max_|z|=1 |P(z)| > (1 + c) √(n)?
For n≥ 1 fix some sequence of n distinct numbers x_1n,…,x_nn∈ [-1,1]. Let ε=ε(n)→ 0.
Does there exist, for every α ∈ [0, 1], a ring in ℝ with Hausdorff dimension α?
Let a=(a_1,…,a_n) be points, ordered consecutively, around the circle (normalised to have circumference 1).
Let f(z) ∈ ℂ[z] be a monic non-constant polynomial. Can the set z ∈ ℂ : |f(z)| ≤ 1 be covered by a set of closed discs the sum of whose radii is ≤ 2?
Chowla's cosine problem If A⊂ ℕ is a finite set of positive integers of size N > 0 then is there some absolute constant c>0 and θ such that Σ_n∈ Acos(nθ) < -cN^1/2?
Let f be a transcendental entire function. What is the greatest possible value of liminf (fun r : ℝ => ratio r f) atTop?
If f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that nₖ / k → ∞, is it true that f assumes every value infinitely often?
Let f(z)=Σ_0≤ k≤ n ε_k z^k be a random polynomial, where ε_k∈ -1,1 independently uniformly at random for 0≤ k≤ n. Is it true that, if R_n is the number of roots of f(z) in z∈ ℂ : lvert zrvert ≤ 1, then R_n/n/2→ 1 almost surely?
Does there exists an entire non-zero transcendental function f : ℂ → ℂ such that for any sequence n₀ < n₁ < ..., z | ∃ k, iteratedDeriv (n k) f z = 0 is dense.
Erdős Problem 995: For every lacunary sequence (n_k) of integers and every f ∈ L^2([0,1]) with ∫_0^1 f = 0, is it true that for almost all α, Σ_k < N f(α n_k) = o (N √(loglog N))?
Does there exists a positive constant C such that for all f ∈ L²[0,1] and all lacunary sequences n, if ‖f - fₖ‖₂ = O(1 / log log log k ^ C), then for almost every x, lim ∑ k ∈ Finset.range N, f (n k • x)) / N = ∫ t, f t ∂t?
Four exponentials conjecture Let x_0, x_1 and y_0, y_1 be ℚ-linearly independent pairs of complex numbers, then some e^x_i y_j is transcendental.
Conjecture 1.3 (the × p, × q conjecture): the only atomless Borel probability measure on T which is both T_p- and T_q-invariant is the Lebesgue measure.
Goodman's conjecture. For every p-valent normalised function f on the unit disk and every n > p, the n-th coefficient is bounded by the Goodman bound: |b_n| ≤ Σ_k=1^p 2k (n+p)!/(p-k)! (p+k)! (n-p-1)! (n^2-k^2) |b_k|.
Suppose that A is an open subset of [0, 1]^2 with measure α. Are there four points in A determining an axis-parallel rectangle with area gt c α^2?
Assume for n>1, f:ℝ^n→ℝ^n is a bijection, where ℝ^n is equipped with the standard topology. Does the connectedness of (the induced power set map) f imply that of f^-1?
Gerver's sofa is the unique sofa that attains the sofa constant, up to a rigid motion. The motion is needed: horizontalHallway is (-∞, 1] × [0, 1], so a leftward translate of any moving sofa is again one, obtained by sliding right and then following the original motion.
Are e and π algebraically independent?
[KLM2023, Problem 7.1] asks whether a bounded, measurable, nowhere dense subset Ω ⊂ ℝ^d of positive measure can be spectral. The answer is known to be negative for d = 1, so the dimension is restricted to d ≥ 2, where the problem is open.
Rule 30 Prize, Problem 1 (non-periodicity). The center column of Rule 30 is not eventually periodic: there is no positive period p and threshold N past which the column repeats with period p.
Problem 4.2. Let Ω ⊂ ℝ be a finite union of three or more intervals. If Ω weakly tiles its complement, must it also tile its complement properly?
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