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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Let x_1,…,x_10 be very general points of ℙ^2, and let π:X→ ℙ^2 be the blow-up of ℙ^2 at these points. Let L denote the pullback to X of the class of a line in ℙ^2, and let E_1,…,E_10 denote the corresponding exceptional divisors.
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\.
C_3c = SD(\0,1,2,∞\;-1) is the least exponent such that one has the inequality |A stackrelG- B| ≤ max(|A|, |B|, |A stackrelG+ B|, |A stackrelG+ 2B|)^C_3c whenever A, B are finite subsets of reals and G ⊂ A × B, where A stackrelG± rB := a ± rb: a ∈ A, b ∈ B.
C_5a is the smallest constant such that Sidon sets in \1,…,N\ have cardinality N^1/2 + (C_5a + o(1))N^1/4.
The ambidextrous moving sofa constant C_41b asks for the maximum area of a sofa, as defined in C_41a that can navigate both left and right corners inside a Z-shaped corridor of width 1, where the corners are sufficiently far apart.
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.
Quantum Max Cut is the quantum analog of Max Cut. Given a graph G = (E,V), it asks for the maximum eigenvalue of H_G = Σ_(ij) ∈ E (I - X_iX_j - Y_iY_j - Z_i Z_j), where X_i, Y_i, Z_i are the Pauli matrices acting on the i'th tensor factor and trivially on all other coordinates.
For integers n ≥ 2 and t ≥ 1, an n × t partial Hadamard matrix is a matrix with entries in \± 1\ whose rows are pairwise orthogonal. Let N_n,t denote the number of such matrices. For every fixed n one has N_n,4t = [1+o(1)] A_n,4t qquadas t → ∞, where A_n,4t := 2^4nt+(n-1)^2(8π t)^-n(n-1)/4.
Let α be a nonzero algebraic number of degree d, with minimal polynomial over ℤ f(X)=a_dΠ_i=1^d (X-α_i), where a_d>0 and α_1,…,α_d are the conjugates of α. Define the Mahler measure of α by M(α) := a_dΠ_i=1^d max1,lvert α_irvert.
For each integer n ≥ 1, let S_n be the symmetric group on n letters. Over a base field k, the essential dimension ed_k(S_n) is the smallest integer d such that the general degree-n polynomial x^n + a_1 x^n-1 + ⋯ + a_n can be reduced to a d-parameter form by a Tschirnhaus transformation.
For each integer d ≥ 3, let ℓ_0(d) denote the maximal number of lines contained in a smooth surface of degree d in ℙ^3_ℂ. We define C_76 := limsup_d→∞ℓ_0(d)/d^2. The constant C_76 measures the quadratic growth rate of the maximal line count on smooth complex degree-d surfaces.
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.
Let p_n denote the n-th prime. The bounded prime gap constant is C_88a = H_1 := liminf_n → ∞ (p_n+1 - p_n), the least limit point of the sequence of gaps between consecutive primes.
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.
C_81a, Brun's Constant, is the sum of the reciprocals of the twin primes.
A Dirichlet character of level q is an arithmetic function χ that is multiplicative, is defined by a character on (ℤ/qℤ)^ast on integers coprime to q, and is 0 on integers not coprime to q.
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…
Let λ_n,2 be the random variable assigning two uniformly random binary strings of length n the length of their longest common subsequence. Then C_31a is the (well-defined) limit C_31a := lim_n → inftyE[λ_n,2]/n.
C_8 = R is the least constant such that there are no zeroes σ+it of the Riemann zeta function with lvert t rvert ≥ 2 and σ > 1 - 1/R log lvert t rvert.
The constant term of one-shot channel simulation [HJMR07], [BG14], [LEG18], [Li25] is given as (we use the definition in [Li25]) C_32=limsup_t→∞(sup_p_X,Y: I(X;Y)=t inf_p_S|X,Y: I(X;S)=0H(Y|S)-t-log_2t), where H(Ylvert S)=H(Y,S)-H(S) is the conditional entropy (in bits), and I(X;Y)=H(X)+H(Y)-H(X,Y)…
In topological graph theory, a thrackle is a drawing of a finite graph in the plane in which every pair of edges meets precisely once, either at a common endpoint or at a proper crossing.
In zero-sum theory, the Davenport constant D(G) of a finite abelian group G is defined as the smallest integer l∈ℕ such that every sequence S over G of length lvert Srvert≥ l has a non-empty zero-sum subsequence.
Let d(n) be the divisor function. The Dirichlet divisor problem concerns the error term Δ(x) := Σ_n≤ x d(n) - x(log x + 2γ -1), where γ is Euler's constant. <a href="#Tsa2010-def-Delta">[Tsa2010-def-Delta]</a> Define the divisor-problem exponent α := infBigla≥ 0: Δ(x)=O(x^a+ε) for all ε>0Bigr.
In algebraic complexity theory, for each real k ≥ 0, let ω(k) denote the exponent for multiplying an n × n^k matrix by an n^k × n matrix. We define α := supk ≥ 0 : ω(k) = 2.
Let Λ denote the von Mangoldt function. For coprime positive integers a,q, define ψ(x;q,a) := Σ_n≤ x, n≡ a (mod q) Λ(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.
A family of three distinct sets A,B,C is a 3-sunflower (or Δ-system) if A∩ B = A∩ C = B∩ C. A family of sets is sunflower-free if it contains no 3-sunflower (equivalently, no sunflower of any size ≥ 3). Let [n]:=\1,2,…,n\ and let f(n) denote the maximum size of a sunflower-free family F⊆ 2^[n].
Let overlineℚ be the set of all algebraic numbers. The naïve height h : overlineℚ → ℝ is defined as follows. Let α ∈ overlineℚ and let P(x) be an irreducible primitive polynomial with integers coefficients such that P(α)=0. Let n be the degree and a be the leading coefficient of P(x).
Let p_n denote the n-th prime and, for m ≥ 1, write H_m := liminf_n → ∞ (p_n+m - p_n) for the least limit point of the gaps between primes m apart.
Let H be an infinite-dimensional complex Hilbert space and let B(H) be the Banach algebra of bounded operators on H, equipped with the operator norm.
C_17 is the limit (if it exists) of R(k)^1/k as k → ∞, where the diagonal Ramsey number R(k) is the smallest integer n such that every red/blue colouring of the edges of the complete graph K_n contains a monochromatic copy of K_k.
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 E⊂ ℝ^2 be a planar set. The Favard length of E is defined by Fav(E) := 1/π∫_0^π lvert Proj R_θ Ervert dθ, where Proj is orthogonal projection to the horizontal axis and R_θ is rotation by angle θ.
A convex body K⊂ℝ^d is hollow (lattice-free) with respect to a lattice Λ if int(K)∩Λ=∅. <a href="#CS2019-hollow-def">[CS2019-hollow-def]</a> For a hollow body, the lattice width is w(K) := min_u∈ℤ^d∖0 (max_x∈ Ku· x-min_x∈ Ku· x).
Let f:\-1,1\^n→\-1,1\ be a Boolean function with Fourier expansion f(x)=Σ_S⊆[n]hat f(S)χ_S(x). Its spectral entropy is H(hat f^2) := Σ_S⊆[n]hat f(S)^2log_21/hat f(S)^2, and its total influence is Inf(f) := Σ_S⊆[n]hat f(S)^2 lvert Srvert.
Let r(N) be the maximum size of a subset A⊂\1,…,N\ with no non-zero square differences a-b=n^2.Then C_4b is the least constant such that r(N) ≤ N^C_4b+o(1).
Let N(t) := \#(m,n)∈ℤ^2: m^2+n^2≤ t^2 be the number of integer lattice points inside the (closed) disk of radius t centered at the origin. The Gauss circle problem is to find the smallest exponent θ such that, for every ε>0, N(t) = π t^2 + O(t^θ+ε).
C_43 is defined as the infimum of the ratio of the length of the Steiner Minimal Tree to the length of the Euclidean Minimum Spanning Tree over all finite sets of points V ⊆ ℝ^2: C_43 = inf_VL_S(V)/L_M(V), where L_S(V) and L_M(V) denote the lengths of Steiner Minimal Tree and Minimum Spanning Tree…
We define C_56 = δ_2 to be the smallest real number δ ≥ 0 such that the following uniform bound toward the Generalized Ramanujan Conjecture holds.
Let f be a convex function with 1-Lipschitz gradient. We assume black-box access to the function and its gradient. Gradient descent will converge to a global minimum with an appropriate choice of _step size_ s: x_k+1 := x_k - s· ∇ f(x_k). In general, s can be chosen to vary with the step k.
C_39=H_3 is the Hadwiger covering number in dimension 3, which can also be formulated in terms of illumination of the boundary.
C_33=A(2) is the Ihara constant over 𝔽_2. <a href="#DM2013-def-Aq">[DM2013-def-Aq]</a> For each integer g≥ 1, let N_2(g) := maxbigl\#X(𝔽_2) : X/𝔽_2 a smooth projective geometrically integral curve of genus gbigr. <a href="#DM2013-def-Nqg">[DM2013-def-Nqg]</a> Then A(2) := limsup_g→inftyN_2(g)/g.
Let G = (g_ij) be an M × N random matrix with independent standard Gaussian entries, and let Z(G) := | σ ∈ -1,1^N : G σ ≥ 0 coordinatewise |. This is the zero-margin binary (or Ising) perceptron. Write M = ⌊ α N ⌋. Define C_80 to be the infimum of all α > 0 such that ℙ(Z(G) > 0) → 0 as N → ∞.
C_3b = SD(\0,1,∞\;-1) is the least exponent such that one has the inequality |A stackrelG- B| ≤ max(|A|, |B|, |A stackrelG+ B|)^C_3b whenever A, B are finite subsets of reals and G ⊂ A × B, where A stackrelG± B := a ± b: a ∈ A, b ∈ B.
C_24 is the Komlós discrepancy constant (often denoted K). For a real matrix A∈ℝ^m× n, define its (sign) discrepancy by disc(A) := min_x∈-1,1^n ‖Ax‖_∞. For each n≥ 1, define the dimension-n Komlós discrepancy K_n := supdisc(A): A∈ℝ^n× n and ‖A_ast j‖_2≤ 1 for all columns j.
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.
