The Riemannian Penrose inequality is proven in arbitrary dimensions for smooth complete asymptotically flat manifolds with nonnegative scalar curvature and compact outer-minimizing minimal boundary allowing singular sets of Hausdorff dimension at most n-8, with equality only for Riemannian Schwarzs
On the Riemannian Penrose inequality in dimensions less than eight
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Mode stability without symmetry assumptions is proved for self-similar wave map blowups in all dimensions d ≥ 4.
Strict good reduction of a rational map over a local field equals the residual morphism being finite étale of degree d over a nonempty open set outside the post-critical locus, implying unramified extensions along forward orbits via orbital arboreal representations.
Quantitative stability estimates bound |λ_k(Ω) - λ_k(Θ)| by C(d,k) times (λ_2(Ω) - λ_2(Θ)) to the power α = α_d/(d+1)^2 (with α=1/2 when λ_k(Ω) ≥ λ_k(Θ)), for Θ the union of two equal balls.
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Riemannian Penrose inequality in all dimensions
The Riemannian Penrose inequality is proven in arbitrary dimensions for smooth complete asymptotically flat manifolds with nonnegative scalar curvature and compact outer-minimizing minimal boundary allowing singular sets of Hausdorff dimension at most n-8, with equality only for Riemannian Schwarzs
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Mode stability of self-similar wave maps without symmetry in higher dimensions
Mode stability without symmetry assumptions is proved for self-similar wave map blowups in all dimensions d ≥ 4.
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A Dynamical N\'eron--Ogg--Shafarevich Criterion via Orbital Arboreal Representations
Strict good reduction of a rational map over a local field equals the residual morphism being finite étale of degree d over a nonempty open set outside the post-critical locus, implying unramified extensions along forward orbits via orbital arboreal representations.
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Quantitative stability control of the full spectrum of the Dirichlet Laplacian by the second eigenvalue
Quantitative stability estimates bound |λ_k(Ω) - λ_k(Θ)| by C(d,k) times (λ_2(Ω) - λ_2(Θ)) to the power α = α_d/(d+1)^2 (with α=1/2 when λ_k(Ω) ≥ λ_k(Θ)), for Θ the union of two equal balls.