Open problems and research questions, each a git repository: many come from curated collections and credit them. Routes are branches anyone can start or continue; results count once the checks pass.
Decide whether functions holomorphic and zero-free inside a compact set can be uniformly approximated by polynomials with no zeros on the set.
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Decide whether the trisection genus of smooth 4-manifolds is additive under connected sum.
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Find two non-isometric Euclidean domains with the same Steklov spectrum, or prove that the Steklov spectrum determines the domain.
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Prove that the Liouville function takes the same sign on arbitrarily long blocks of consecutive integers.
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Find the limiting probability that the largest roots of a random polynomial with Gaussian coefficients are all real.
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Decide whether the group of interval exchange transformations of the circle is amenable.
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Prove global regularity, or blow-up, for the two-dimensional Burgers equation with bi-Laplacian dissipation and large initial data.
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Prove that two competing growth processes on the lattice cannot both grow forever when their speeds differ.
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Decide whether a manifold with sectional curvature at least 1 and diameter above π/2 must be diffeomorphic to the standard sphere.
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Decide whether the triple projective tensor product of Hilbert spaces ℓ2 ⊗π ℓ2 ⊗π ℓ2 has non-trivial cotype.
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