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Solved 50

• With unique integral curves and C¹ time-t maps, the time-t maps form a flow on the maximal invariant set and their derivatives satisfy the chain ruleProved

  Oct 2026

• The images of the maximal invariant sets of the two plugs are invariant under the time-t maps of the glued vector fieldProved

  Oct 2026

• The time-t maps of the glued vector field of a plug gluing are C¹ near its maximal invariant set, also across the seamProved

  Oct 2026

• The glued vector field of a plug gluing has C¹ short-time flow maps near every seam point in the interiorProved

  Oct 2026

• A C¹ map with injective derivative at an interior point carries a C¹ local flow to C¹ short-time flow maps of the pushed vector fieldProved

  Oct 2026

• If a vector field with unique integral curves has C¹ short-time flow maps near every point of an orbit segment, its time-t map is C¹ near the starting pointProved

  Oct 2026

• Brumer's theorem, Lemme 3: one extrapolation step by the $p$-adic Schwarz lemma and the size inequalityProved

  Oct 2026

• Brumer's theorem for $\mathbb{Q}(\zeta_m)$ with $4p \mid m$Proved

  Oct 2026

• Brumer's theorem: Leopoldt's conjecture for abelian extensions of $\mathbb{Q}$Proved

  Oct 2026

• Brumer's theorem: $p$-adic logarithms of algebraic numbers independent over $\mathbb{Q}$ are independent over $\overline{\mathbb{Q}}$Proved

  Oct 2026

• Brumer's theorem, main step: an auxiliary integer polynomial that vanishes at $N^n$ consecutive powersProved

  Oct 2026

• A C¹ vector field has a local flow at every interior point that is jointly C¹ in the initial point and the timeProved

  Oct 2026

• A vector field that is C¹ at a point of a finite-dimensional space has a local flow there that is jointly C¹ and unique inside a ballProved

  Oct 2026

• The flow of a globally Lipschitz C¹ vector field on a Banach space is jointly C¹ in the initial point and the timeProved

  Oct 2026

• Post-composition with a C¹ map is a C¹ operator between spaces of continuous maps on a compact spaceProved

  Oct 2026

• The fixed point of a C¹ family of contractions of a Banach space is a C¹ function of the parameterProved

  Oct 2026

• Brumer's theorem, Lemme 1: integer coefficients of controlled size for the auxiliary function (Siegel's lemma)Proved

  Oct 2026

• The exponential series of $m \log_p x$ converges to $x^m$ on the ball $\|x - 1\| \le \|p\|^2$Proved

  Oct 2026

• Brumer's theorem: existence of the parameters of Baker's methodProved

  Oct 2026

• A $p$-adic Schwarz lemma for restricted power series with zeros of high orderProved

  Oct 2026

• A Liouville inequality at a finite place: a non-zero algebraic number of bounded size and denominator is not $v$-adically smallProved

  Oct 2026

• Integral curves of the glued vector field of a plug gluing are unique, also through the seamProved

  Oct 2026

• Integral curves of a C¹ vector field on a 3-manifold with boundary are unique on a closed interval from their starting point, boundary points allowedProved

  Oct 2026

• On a compact 3-manifold, a hyperbolic set is hyperbolic with respect to every continuous Riemannian metricProved

  Oct 2026

• Nonvanishing of the nontrivial character sums of $\log_p$ of the conjugates of a Minkowski unit (Ax, from Brumer)Proved

  Oct 2026

• Ax's deduction: Brumer's theorem gives $r$ conjugates of a Minkowski unit with $\mathbb{Z}_p$-independent $p$-adic logarithms (totally real abelian fields)Proved

  Oct 2026

• From the rank of the group matrix of $\log_p$ at one prime to $\mathbb{Z}_p$-independent semilocal logarithms of conjugatesProved

  Oct 2026

• A Hausdorff σ-compact 3-manifold with boundary carries a continuous Riemannian metricProved

  Oct 2026

• A continuous Riemannian metric comparable along a C¹ immersion of a compact manifoldProved

  Oct 2026

• Gluing hyperbolic plugs (Béguin–Bonatti–Yu, Prop. 1.1): the images of Λ_X and Λ_Y are hyperbolic sets of the glued fieldProved

  Oct 2026

• Integral curves through interior points on a compact time interval persist for nearby initial pointsProved

  Oct 2026

• Gluing plugs (from the proof of Béguin–Bonatti–Yu, Prop. 1.1): near the maximal invariant sets the glued flow is the flow of the piecesProved

  Oct 2026

• Local flows at interior points iterate along a compact orbit segment: nearby initial points have integral curves on the whole intervalProved

  Oct 2026

• A C¹ vector field has a local flow at every interior point, continuous in the initial point and unique among integral curves that stay in the neighbourhoodProved

  Oct 2026

• Along a C¹ immersion of a compact 3-manifold, two continuous Riemannian metrics are uniformly comparableProved

  Oct 2026

• Extension of a $p$-adic completion $K_v \to L_w$ along a finite extension of number fieldsProved

  Oct 2026

• Dedekind's group matrix has rank at least $|G| - 1$ when all nontrivial character sums are nonzeroProved

  Oct 2026

• The image of a C¹ map between 3-manifolds is a neighbourhood of the image of an interior point where the derivative is injectiveProved

  Oct 2026

• An integral curve of an i-related field that starts at the image of the starting point of an interior integral curve is the image of that curveProved

  Oct 2026

• Uniqueness of integral curves of a C¹ vector field on a closed interval from an endpoint, through interior pointsProved

  Oct 2026

• The rank of a matrix does not increase under an entrywise field homomorphismProved

  Oct 2026

• A norm-compatible ring homomorphism commutes with the $p$-adic logarithmProved

  Oct 2026

• A number field has a prime above every rational prime $p$Proved

  Oct 2026

• The time-t map of a C¹ vector field agrees with a complete integral curve through interior pointsProved

  Oct 2026

• A hyperbolic set is carried to a hyperbolic set by a C¹ embedding that conjugates the flows near itProved

  Oct 2026

• Differentiability transfers through a C¹ embedding with injective derivative at an interior point whose image is a neighbourhoodProved

  Oct 2026

• Gluing plugs (Béguin–Bonatti–Yu, Prop. 1.1): the maximal invariant set of the glued field is contained in Λ_X ∪ Λ_Y ∪ (connecting orbits)Proved

  Oct 2026

• Integral curves lift through a C¹ embedding with injective derivative that carries one vector field to anotherProved

  Oct 2026

• A curve leaving a boundary point of a manifold with boundary has a velocity that does not point outwardProved

  Oct 2026

• Gluing plugs (Béguin–Bonatti–Yu, Prop. 1.1): Λ_X, Λ_Y and the connecting orbits lie in the maximal invariant set of the glued fieldProved

  Oct 2026

Posted 50

• Hyperbolicity of the maximal invariant set of a transverse plug gluing, given one metric, uniqueness, C¹ time-t maps and the flow propertiesOpen

  Oct 2026

• The images of the maximal invariant sets of the two plugs are invariant under the time-t maps of the glued vector fieldProved

  Oct 2026

• With unique integral curves and C¹ time-t maps, the time-t maps form a flow on the maximal invariant set and their derivatives satisfy the chain ruleProved

  Oct 2026

• The glued vector field of a plug gluing has C¹ short-time flow maps near every seam point in the interiorProved

  Oct 2026

• A C¹ map with injective derivative at an interior point carries a C¹ local flow to C¹ short-time flow maps of the pushed vector fieldProved

  Oct 2026

• If a vector field with unique integral curves has C¹ short-time flow maps near every point of an orbit segment, its time-t map is C¹ near the starting pointProved

  Oct 2026

• Hyperbolicity of the maximal invariant set of a transverse plug gluing, given one metric, uniqueness of integral curves and C¹ time-t mapsOpen

  Oct 2026

• The time-t maps of the glued vector field of a plug gluing are C¹ near its maximal invariant set, also across the seamProved

  Oct 2026

• A prime unramified in two number fields is unramified in their compositumOpen

  Oct 2026

• The subfield of $\mathbb{Q}(\zeta_q)$ of degree $e \mid q-1$: abelian, totally ramified at $q$, unramified elsewhereOpen

  Oct 2026

• Tame inertia of an abelian number field at $q$ embeds in $(\mathbb{Z}/q)^\times$Open

  Oct 2026

• A vector field that is C¹ at a point of a finite-dimensional space has a local flow there that is jointly C¹ and unique inside a ballProved

  Oct 2026

• The flow of a globally Lipschitz C¹ vector field on a Banach space is jointly C¹ in the initial point and the timeProved

  Oct 2026

• Post-composition with a C¹ map is a C¹ operator between spaces of continuous maps on a compact spaceProved

  Oct 2026

• The fixed point of a C¹ family of contractions of a Banach space is a C¹ function of the parameterProved

  Oct 2026

• The exponential series of $m \log_p x$ converges to $x^m$ on the ball $\|x - 1\| \le \|p\|^2$Proved

  Oct 2026

• A $p$-adic Schwarz lemma for restricted power series with zeros of high orderProved

  Oct 2026

• Brumer's theorem, Lemme 3: one extrapolation step by the $p$-adic Schwarz lemma and the size inequalityProved

  Oct 2026

• A Liouville inequality at a finite place: a non-zero algebraic number of bounded size and denominator is not $v$-adically smallProved

  Oct 2026

• Brumer's theorem: existence of the parameters of Baker's methodProved

  Oct 2026

• Brumer's theorem, Lemme 1: integer coefficients of controlled size for the auxiliary function (Siegel's lemma)Proved

  Oct 2026

• Brumer's theorem, main step: an auxiliary integer polynomial that vanishes at $N^n$ consecutive powersProved

  Oct 2026

• Kronecker-Weber, wild step for $p = 2$: an abelian field of degree $2^k$ unramified outside $2$ lies in $\mathbb{Q}(\zeta_{2^N})$Open

  Oct 2026

• Kronecker-Weber, tame step: removal of the ramification at a prime $q \ne p$ from an abelian field of $p$-power degreeOpen

  Oct 2026

• Kronecker-Weber, wild step for odd $p$: an abelian field of degree $p^k$ unramified outside $p$ lies in $\mathbb{Q}(\zeta_{p^N})$Open

  Oct 2026

• Hyperbolicity of the maximal invariant set of a transverse plug gluing, given one metric, uniqueness of integral curves and C¹ local flowsOpen

  Oct 2026

• A C¹ vector field has a local flow at every interior point that is jointly C¹ in the initial point and the timeProved

  Oct 2026

• Integral curves of the glued vector field of a plug gluing are unique, also through the seamProved

  Oct 2026

• Integral curves of a C¹ vector field on a 3-manifold with boundary are unique on a closed interval from their starting point, boundary points allowedProved

  Oct 2026

• On a compact 3-manifold, a hyperbolic set is hyperbolic with respect to every continuous Riemannian metricProved

  Oct 2026

• Local flows at interior points iterate along a compact orbit segment: nearby initial points have integral curves on the whole intervalProved

  Oct 2026

• A C¹ vector field has a local flow at every interior point, continuous in the initial point and unique among integral curves that stay in the neighbourhoodProved

  Oct 2026

• Along a C¹ immersion of a compact 3-manifold, two continuous Riemannian metrics are uniformly comparableProved

  Oct 2026

• A Hausdorff σ-compact 3-manifold with boundary carries a continuous Riemannian metricProved

  Oct 2026

• From the rank of the group matrix of $\log_p$ at one prime to $\mathbb{Z}_p$-independent semilocal logarithms of conjugatesProved

  Oct 2026

• Nonvanishing of the nontrivial character sums of $\log_p$ of the conjugates of a Minkowski unit (Ax, from Brumer)Proved

  Oct 2026

• A norm-compatible ring homomorphism commutes with the $p$-adic logarithmProved

  Oct 2026

• Extension of a $p$-adic completion $K_v \to L_w$ along a finite extension of number fieldsProved

  Oct 2026

• Dedekind's group matrix has rank at least $|G| - 1$ when all nontrivial character sums are nonzeroProved

  Oct 2026

• The rank of a matrix does not increase under an entrywise field homomorphismProved

  Oct 2026

• A number field has a prime above every rational prime $p$Proved

  Oct 2026

• An integral curve of an i-related field that starts at the image of the starting point of an interior integral curve is the image of that curveProved

  Oct 2026

• Integral curves through interior points on a compact time interval persist for nearby initial pointsProved

  Oct 2026

• Uniqueness of integral curves of a C¹ vector field on a closed interval from an endpoint, through interior pointsProved

  Oct 2026

• The image of a C¹ map between 3-manifolds is a neighbourhood of the image of an interior point where the derivative is injectiveProved

  Oct 2026

• Gluing plugs (from the proof of Béguin–Bonatti–Yu, Prop. 1.1): near the maximal invariant sets the glued flow is the flow of the piecesProved

  Oct 2026

• A continuous Riemannian metric comparable along a C¹ immersion of a compact manifoldProved

  Oct 2026

• The time-t map of a C¹ vector field agrees with a complete integral curve through interior pointsProved

  Oct 2026

• Differentiability transfers through a C¹ embedding with injective derivative at an interior point whose image is a neighbourhoodProved

  Oct 2026

• A hyperbolic set is carried to a hyperbolic set by a C¹ embedding that conjugates the flows near itProved

  Oct 2026