Banach-Mazur Rotation Problem
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.
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Open problems
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.
26 shown
Ben Green's Open Problem 35
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 = ∞.
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Ben Green's Open Problem 94
Let A ⊂ R be a set of positive measure. Does A contain an affine copy of 1, 1/2, 1/4, . . . ?
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Bloch and Landau constants
Ahlfors and Grunsky also conjectured in [AG37] that this upper bound is the precise value of the Bloch constant.
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Brennan's Conjecture
Brennan's conjecture, part 1: B(-2) = 1.
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Conjectures about the Mandelbrot and Multibrot sets
The MLC conjecture, stating that the mandelbrot set is locally connected.
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Convergence of the Flint Hills and Cookson Hills series
The Flint Hills series summing csc(n)^2 / n^3 from n=1 to ∞ converges. (Note that we 0-index the series below.)
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De Giorgi's conjecture
De Giorgi's conjecture holds in dimension n ≤ 8.
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Erdős Problem #1038
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]?
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Erdős Problem #1133
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.
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Erdős Problem #243
Let a_1 < a_2 < … be a sequence of integers such that lim_n→∞ a_n/a_n-1^2 = 1 and Σ 1/a_n ∈ ℚ. Then, for all sufficiently large n ≥ 1, a_n = a_n-1^2 - a_n-1 + 1.
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Erdős Problem #509
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?
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Erdős Problem #513
Let f be a transcendental entire function. What is the greatest possible value of liminf (fun r : ℝ => ratio r f) atTop?
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Erdős Problem #517
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?
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Erdős Problem #906
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.
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Erdős Problem #996
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?
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Exponentials conjectures and theorems
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.
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Furstenberg's times p, times q conjectures
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.
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Goodman's conjecture on coefficients of p-valent functions
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|.
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Green's Open Problem 85
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?
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Mathoverflow 235893
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?
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Moving Sofa Problem
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.
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Open questions on irrationality of numbers
Are e and π algebraically independent?
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Spectral sets and weak tiling
[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.
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The Rule 30 Prize Problems
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.
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Weak tiling problems
Problem 4.1. Let Ω ⊂ ℝ be a finite union of intervals and ν a weak tiling measure for Ω. Must supp(ν) have bounded density?
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