This paper initiates the study of matrix-weighted martingale inequalities. In particular, we introduce the martingale A p A_p condition for matrix weights and establish matrix-weighted Doob maximal inequalities for vector-valued martingales. The proof mainly relies on the idea of principal sets and new sparse dominations. Our approach allows to obtain the optimal dependence of the constant on the characteristic of the matrix weight involved.
Let R R be a unital ring satisfying the invariant basis number property, that every stably free R R -module is free, and that the complex of partial bases of every finite rank free module is Cohen–Macaulay. This class of rings includes every ring of stable rank 1 1 (e.g., any local, semi-local or Artinian ring), every Euclidean domain, and every Dedekind domain O S \mathcal {O}_S of arithmetic type where | S | > 1 |S| > 1 and S S contains at least one noncomplex place. Extending recent work of Galatius–Kupers–Randal-Williams and Kupers–Miller–Patzt, we prove that the sequence of general linear groups GL n ( R ) \operatorname {GL}_n(R) satisfies slope- 1 1 homological stability with Z [ 1 / 2 ] \mathbb {Z}[1/2] -coefficients.
A vanishing theorem for uniformly RC k k -positive Hermitian holomorphic vector bundles is established. It turns out that the holomorphic tangent bundle of a compact complex manifold equipped with a positive k k -Ricci curvature Kähler metric is uniformly RC k k -positive. Two main applications are presented. The first one is to deduce that spaces of some holomorphic tensor fields on such Kähler or more generally Kähler-like Hermitian manifolds are trivial, generalizing some recent results. The second one is to show that a compact Kähler manifold whose holomorphic tangent bundle can be endowed with either a uniformly RC k k -positive Hermitian metric or a positive k k -Ricci curvature Kähler-like Hermitian metric is projective and rationally connected.
Let F \mathfrak {F} be a nonarchimedean local field of residual characteristic p p , and let G G denote the group of F \mathfrak {F} -points of a connected reductive group over F \mathfrak {F} . For an open compact subgroup U \mathscr {U} of G G and a unital commutative ring k k , we let X U \mathbf {X}_{\mathscr {U}} denote the space of compactly supported k k -valued functions on G / U G/\mathscr {U} . Building on work of Ollivier–Schneider, we investigate the graded E x t \mathrm {Ext} -algebra E U ∗ ≔ E x t G ∗ ( X U , X U ) o p E_{\mathscr {U}}^* ≔\mathrm {Ext}_G^*(\mathbf {X}_{\mathscr {U}},\mathbf {X}_{\mathscr {U}})^{\mathrm {op}} . In particular, we describe the Yoneda product, an involutive anti-automorphism, and (when k k is a field of characteristic p p and U \mathscr {U} has no p p -torsion) a duality operation. We allow for the reductive group to be nonsplit, and for the open compact subgroup U \mathscr {U} to be non-pro- p p . Specializing further to the case G = SL 2 ( Q p ) G = \operatorname {SL}_2(\mathbb {Q}_p) with p ≥ 5 p \geq 5 and a coefficient field of characteristic p p , we obtain more precise results when U \mathscr {U} is equal to an Iwahori subgroup J J or a hyperspecial maximal compact subgroup K K . In particular, we compute the structure of E J ∗ E_J^* as an E J 0 E_J^0 -bimodule, obtain an explicit description of the center Z ( E J ∗ ) \mathcal {Z}(E_J^*) of E J ∗ E_J^* , and construct a surjective morphism of algebras Z ( E J ∗ ) ⟶ E K ∗ \mathcal {Z}(E_J^*) \longrightarrow E_K^* (analogous to the compatibility between Bernstein and Satake isomorphisms in characteristic 0). From this we deduce the (somewhat surprising) fact that E K ∗ E_K^* is not graded-commutative, contrary to what happens for almost all ℓ \ell -modular characteristics.
As a result of 33 intercontinental Zoom calls, we characterise big Ramsey degrees of the generic partial order. This is an infinitary extension of the well known fact that finite partial orders endowed with linear extensions form a Ramsey class (this result was announced by Nešetřil and Rödl in 1984). Towards this, we refine earlier upper bounds obtained by Hubička based on a new connection of big Ramsey degrees to the Carlson–Simpson theorem and we also introduce a new technique of giving lower bounds using an iterated application of the upper-bound theorem.
This paper is a continuation of our recent work [Adv. Math. 469 (2025), Paper No. 110230] concerning the capillary Minkowski problem. We propose, in this paper, a capillary $L_p$-Minkowski problem for $p\in \mathbb{R}$, which seeks to find a capillary convex body with a prescribed capillary $L_p$-surface area measure in the Euclidean half-space. This formulation provides a natural Robin boundary analogue of the classical $L_p$-Minkowski problem introduced by Lutwak [J. Differential Geom. 38 (1993), no. 1, 131--150]. For $p>1$, we resolve the capillary $L_p$-Minkowski problem in the smooth category by reducing it to a Monge--Ampère equation with a Robin boundary condition on the unit spherical cap.
The authors provide a corrected proof of Theorem 5 in their paper which appeared in Trans. Amer. Math. Soc. 368 (2016), 8499–8518. This result characterizes Fuss-Catalan sequences that are moments of a probability distribution with compact support in [ 0 , ∞ ) [0,\infty ) .
Under certain hypotheses, we prove a loop space decomposition for simply-connected Poincaré Duality complexes of dimension n n whose ( n − 1 ) (n-1) -skeleton is a co- H H -space. This unifies many known decompositions obtained in different contexts and establishes many new families of examples. As consequences, we show that such a looped Poincaré Duality complex retracts off the loops of its ( n − 1 ) (n-1) -skeleton and describe its homology as a one-relator algebra.
It is shown that, for a free and minimal Z d \mathbb {Z}^d -action on a compact Hausdorff space X X , the transformation group C*-algebra C ( X ) ⋊ Z d \mathrm {C}(X)\rtimes \mathbb {Z}^d always has stable rank one, i.e., the invertible elements are dense. Moreover, the C*-algebra C ( X ) ⋊ Z d \mathrm {C}(X)\rtimes \mathbb {Z}^d is also shown to be classified by the Elliott invariant if, and only if, the strict order of its Cuntz semigroup is determined by the traces. That is, it satisfies the Toms-Winter conjecture. In fact, for any free and minimal Γ \Gamma -action on X X , where Γ \Gamma is a countable discrete amenable group, if ( X , Γ ) (X, \Gamma ) has the uniform Rokhlin property and Cuntz comparison of open sets, then the C*-algebra C ( X ) ⋊ Γ \mathrm {C}(X)\rtimes \Gamma is shown to have stable rank one and to satisfy the Toms-Winter conjecture.
