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.
Construct four-dimensional quantum Yang–Mills theory for every compact simple gauge group and prove that it has a positive mass gap.
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Decide whether the vacuum of a free massive scalar field violates the CHSH inequality between bounded regions at arbitrarily large distances.
Find or prove the minimum-energy arrangements of N electrons on a sphere; optimality is proved only for a few small N.
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Find the smallest output dimension in which two quantum channels violate additivity of minimum output entropy; it lies between 2 and 182.
Prove or refute that minimum output entropy is additive when one of two channels is an arbitrary, possibly non-unital, qubit channel.
Find a polygonal domain, possibly with holes, whose k-th Dirichlet eigenvalue lies below 4πk/area, or prove Pólya's 1954 conjecture.
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Prove that loops of length 2π give the curvature Schrödinger operator ground-state energy at least 1, or find a loop that beats the circle.
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Prove that for Newtonian point masses, the initial conditions whose motion breaks down in finite time form a set of measure zero.
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Show that in two or more dimensions, −Δ + V keeps a.c. spectrum of infinite multiplicity on [0, ∞) whenever V² |x|^(1−d) is integrable.
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Prove that a nucleus of charge Z can bind at most Z + C electrons, with a constant C independent of Z.
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