The week’s notable results across mathematics — number theory, combinatorics, geometry, analysis — and the AI-assisted proofs now joining them, from arXiv, Quanta and mathematicians’ own blogs.
Ten of twelve arXiv papers this week report new theorems in combinatorics, number theory and algebraic geometry. The most worth reading are the exact asymptotics for the fractional chromatic number of uncrowded hypergraphs and the paper that answers two questions of Felten on joint deformations of manifolds and bundles. Two weaker items are marked as dropped.
Pairs (X,E) have unobstructed deformations when X is a Fujiki manifold with torsion canonical bundle and H²(X, End⁰E)=0, extending theorems of Li–Pan and Iacono–Manetti to the non-Kähler case. Every strict projective Calabi–Yau manifold of dimension at least three admits a simple bundle with obstructed joint deformations, which answers two questions of Felten.
For degree n covers C→E of elliptic curves, fields L of degree 2, 3, 4 and 5 with a point generating L have density 0 among fields of that degree. Degrees 4 and 5 are new, using work of Bhargava–Shankar–Wang and McGown–Thorne–Tucker, and 8 bielliptic modular curves X₀(N) have no such points for 100% of quadratic fields.
The open substacks of the stack of genus-one Gorenstein curves with n marked points that have a proper good moduli space are exactly the Bozlee–Kuo–Neff stacks M̄₁,ₙ(c). For pure c of level m, a numerical criterion shows these spaces are asymptotically almost never projective as n grows.
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Real progress on the six open Millennium Prize Problems — the Riemann hypothesis, P vs NP, Navier–Stokes, Yang–Mills, Hodge and Birch–Swinnerton-Dyer: serious papers, partial results, AI-assisted advances, and what experts say about claimed solutions.
Where the P vs NP question stands: circuit and proof-complexity lower bounds, barriers, meta-complexity, geometric complexity theory, new reductions and hardness results, and how the community reads each claimed proof.