Papers

We prove a new extension principle for the spherically symmetric Einstein–Maxwell–charged scalar field system. Rather than imposing conditions for mass ratio $2m/r$ as in Christodoulou's small-mass-ratio criterion or as in An–Chen–Singh's proof [1] with $2m/r<3/8$, here we work through a proposal by An-Tan [2] with finite Doppler exponent (blueshift). By conducting two successive blow-up analyses, followed by a stationary comparison and a rigidity argument, we establish the uniform boundness of $\alpha'$ with $\alpha'=\partial_v\alpha/\partial_vr$ and $\alpha=\partial_v(r\phi)/\partial_vr$. This provides the desired extension principle in An-Tan [2] and completes the picture of weak cosmic censorship developed therein. As we were about to release our preprint, Cicortas communicated to us that together with his collaborators, they concomitantly used a different approach to obtain the corresponding extension principle with finite blueshift (Doppler exponent) for the uncharged Einstein-scalar field system. Hence, we decided to put the current paper as an appendix to An-Tan [2], thereby completing the proof of the weak cosmic censorship for the Einstein-Maxwell-charged scalar field system. At the same time, we are working with Cicortas and his collaborators to write a new preprint containing both approaches for the Einstein-scalar field system.

We present a new quantitative, scale-critical $C^1$ extension principle for the spherically symmetric Einstein–scalar field system. We derive estimates for solutions in the causal past of a possibly singular endpoint $\mathcal{O}$ of the center. Assume that there is no trapped surface in this region and assume that the Doppler exponent (blue shift) along each incoming null cone is uniformly bounded. We then prove that the solution can extend regularly across $\mathcal{O}$ provided the mass ratio $\mu=2m/r$ satisfies $\sup\mu<\frac38$. This improves the small-mass-ratio condition in Christodoulou's extension principle by an explicit threshold of order one. Our proof has three new ingredients. First, we derive several new uniform scale-critical pointwise estimates for the scalar field and its derivatives. Second, we obtain refined estimates in an exterior self-similar domain within the triangular causal past region. Third, we establish a family of higher-order Morawetz-type identities and identify the ranges of $\mu$, with which there are coercive spacetime estimates. The unweighted identity yields the threshold $\mu\leq\frac14$, while employing extra self-similar weight raises the threshold to $\mu<\frac38$ and provides a weighted control, which implies that the solution's BV norm along an outgoing null cone is small, so that Christodoulou's theory of bounded variation solutions gives the extension.

We study the category of constructible $p$-Kummer-étale sheaves on the log adic generic fiber $(\mathrm{X},\mathcal{M}_{\mathrm{D}})_\eta$ for the stratification induced by a horizontal normal crossings divisor $\mathrm{D}$, showing that it is antiequivalent to a category of finitely generated unit Laurent crystals on the log prismatic site. We give a local description of finitely generated unit crystals on a framed log affine in terms of $(\varphi,\Gamma)$-modules, and show pushforward compatibility for strict log smooth proper morphisms.

We establish topological recursion for the spin enumeration of hypermaps, or, in the dual language, spin dessins d'enfants. This problem has remarkable features distinguishing it from other applications of topological recursion to enumerative combinatorics. First, the spectral curve is elliptic rather than rational, as in most cases studied before. Second, it contains degenerate ramification points, and one has to apply a version of degenerate topological recursion. Finally, the meromorphic correlator differentials have "moving" poles on the antidiagonals between pairs of distinct arguments, and one has to adjust the loop equations to this kind of degeneracy as well. The computations involve concepts of different kinds, such as BKP integrability, Gaussian integrals, the x-y swap, and symplectic duality.

In this short note, using OpenAI's resolution of the Solomon–Yau conjecture, we extend our joint work with Pedro Gaspar to all dimensions $n\geq2$. We construct eternal Allen–Cahn flows on $S^{n+1}$ connecting approximations of any prescribed least-area Clifford hypersurface and equatorial sphere. We also construct infinitely many eternal integral Brakke flows from least-area Clifford hypersurfaces to equatorial spheres.

We study the behaviour of partially de Rham Selmer groups of $B$-pairs along abelian $p$-adic Lie extensions. In the cyclotomic case we compare them with the universal norms studied by Berger. For analytic Lubin–Tate $(\varphi,\Gamma)$-modules we prove that the $\Psi=1$ eigenspace is coadmissible of the expected rank over the algebra of analytic distributions and establish a comparison result between its Lie-algebra coinvariants and the projective limit of the partially de Rham Selmer groups of the attached $B$-pair.

A loose hypertree is a hypergraph whose incidence graph is a tree. Let $b\ge3$, $n\equiv1\pmod{b-1}$, and $s:=(n-1)/(b-1)$. We determine the largest $t$-intersecting families of loose spanning $b$-uniform hypertrees on $[n]$, simultaneously for every $1\le t\le s$, provided $n$ exceeds an explicit constant depending only on $b$. The extremal families consist of all such hypertrees containing a fixed $t$-edge loose forest whose component sizes are as equal as the divisibility condition permits. In particular, when $bt\le n$, the extremal forests are matchings. We also prove a Hilton–Milner-type bound: if $n\ge C_b(t+1)$, where $C_b$ depends only on $b$, every nontrivial $t$-intersecting family has at most $(s+1)r_{t+1}(n)$ members. Here $r_j(n)$ is the maximum possible number of loose spanning $b$-uniform hypertrees containing a fixed $j$-edge loose forest. For fixed constants $b,t$, this bound is asymptotically sharp and is $O_{b,t}(n^{2-b})r_t(n)$. The proofs combine exact forest enumeration with the intersection theorems of Kupavskii for spread families.

Let $E\to B$ be a smooth $S^n$ bundle with oriented Riemannian total space $(E,g)$, where $n\ge2$. We prove K-area$(E,g)\le4\pi n(n-1)/\sigma$ whenever every fibre has intrinsic scalar curvature at least $\sigma>0$, independently of the dimension of $B$. In particular, unit round fibres give K-area$(E,g)\le4\pi$. For arbitrary metrics on two-sphere fibres, K-area is bounded by the supremum of their areas. We also obtain stronger bounds from fibrewise scalar spectra.

For a meromorphic function $f$ in the complex plane, we determine when $f$ and $f^{(k)}$ are algebraically dependent. This happens precisely when $f$ belongs to the class $W$ of elliptic functions and their degenerates, or satisfies a linear differential equation $f^{(k)} + af + b = 0$ with constant coefficients. The case of functions having a pole was settled previously. We complete the proof by treating entire solutions.

We investigate the notion of holomorphic $p$-contact structure, that we introduced recently in the scalar case with H. Kasuya and L. Ugarte and in the line-bundle case with K. Broder, in the context of a holomorphic family of compact complex manifolds equipped with a holomorphic family of holomorphic line bundles. Firstly, we show that the existence of a holomorphic $p$-contact structure is deformation-open when a non-jumping assumption is made on the $\bar\partial$-cohomology of bidegree $(p,\,0)$. This ought to become a source of further examples of holomorphic $p$-contact manifolds. Secondly, we obtain a Berndtsson-type (semi-)positivity result for the holomorphic vector bundle over the base induced, as a direct image, by the line bundle on the total space that supports the p-contact structures. Thirdly, we introduce a moduli space of relative holomorphic p-contact structures and we show that it is Hausdorff and open in a certain quotient topological space.

The category of all topological spaces and homotopy classes of maps is cartesian closed. In fact, the $(\infty, 1)$-category of all topological spaces and continuous maps, localised at homotopy equivalences, is cartesian closed. We model the exponential object using a topological space whose points are continuous maps equipped with putative neighbourhood filters. In general, this topological space will not even be $T_1$-separated, but the evaluation map induces a natural weak homotopy equivalence witnessing the universal property.

While sensitivity analysis for linear programs with parametric right- hand side vectors is well established, cases involving parametric constraint matrices remain less explored. To address this gap, we investigate linear programs where the constraint matrix depends linearly on a scalar parameter. Our goal is to compute the minimum and maximum optimal values of these parametric linear programs within a fixed parameter range. This corresponds to solving two bilevel programs. We show that both decision versions of these optimization problems are NP-complete by proving both NP-hardness and membership in NP. We additionally identify three sufficient conditions on the parametric matrix under which the bilevel programs are no longer NP-hard. We introduce a reformulation for both problems, together with a Farkas-based certificate for the lower-level feasibility condition, and compare the execution time for solving them on standard benchmark instances. The experiments highlight a practical dichotomy. Although both bilevel optimizations are NP-hard, computing the parameter value corresponding to the minimum of the linear program is fast, whereas computing the value corresponding to the maximum is not. We relate this asymmetry to the parametric equality constraints, whose free dual variables leave the relaxation of the maximum reformulation unbounded.

We study the parabolic Anderson model $\partial_t u=\Delta u+\beta u\,\dot W_\alpha$ on an unbounded Ahlfors $d_h$-regular Dirichlet metric measure space whose heat kernel satisfies two-sided sub-Gaussian estimates with walk dimension $d_w$. The noise is white in time and, in space, either white ($\alpha=0$) or colored through the Riesz kernel of $(-\Delta)^{-2\alpha}$, where $0\le 2\alpha<d_h/d_w<1+2\alpha$. No recurrence is assumed. We prove existence, uniqueness and Hölder regularity of the mild Skorohod solution for bounded initial conditions. For nonzero nonnegative bounded initial conditions, the solution is positive and its moment Lyapunov exponents exist for every real $p\ge1$, independently of the spatial point and the initial condition. For $p\ge2$, these exponents have order $p\,(\beta^2(p-1))^{d_w/((1+2\alpha)d_w-d_h)}$, with upper and lower bounds matching up to constants.

We develop a dual-inertial framework for convex optimization that introduces inertia in dual coordinates while retaining the Bregman proximal-type oracles. For unrestarted iterations, uniformly bounded positive directional variation gives a last-iterate $\cO((1+\log k)/k^2)$ rate, improving to $\cO(k^{-2})$ under additional weighted summability. A computable safeguarded algorithm gives a global $\cO(N^{-1})$ objective-gap bound for the best accepted point, where $N$ counts all proximal oracle calls. Under local strong-convexity and kernel regularity conditions, all three safeguarded methods restart finitely and attain a final-epoch last-iterate $\cO(N^{-2})$ rate. Numerical experiments compare objective accuracy and computational cost on entropy regression and nonnegative least squares. Comparative numerical experiments illustrate the performance advantages of the proposed methods.

For the fundamental problem of prediction from expert advice, Freund asked whether one could compete with a comparator that switches $k$ times among only $m$ experts. For a discrete set of experts, this problem was resolved by Bousquet and Warmuth (2020), who gave an efficient algorithm with optimal regret matching the information-theoretic limit. We study the analogous problem in online convex optimization, where the action space is continuous. An inefficient algorithm can be derived by discretizing the action space and applying the Bousquet–Warmuth algorithm, giving regret of $\widetilde{O}(md+k)$, while the best known efficient algorithm for this problem has regret $\widetilde{O}(dk)$. We show that this computational inefficiency is inherent, establishing a separation between the information-theoretic and polynomial-time regret achievable for this problem. Specifically, under the hypergraphic planted-clique conjecture, we show that for every polynomial-time algorithm there exist instances for which it incurs at least $(k\sqrt d)^{1-o(1)}$ regret.

Let $G$ be the terminal graph of the random triangle-removal process started from $K_n$. We prove that $e(G)/n^{3/2}\to1/(2\sqrt2)$ in $L^2$. More generally, we obtain the analogous result for any fixed strictly $k$-balanced $k$-graph. Our main tool is a more general theorem on random greedy matchings in nearly regular uniform hypergraphs with few short cycles under a stability assumption on the survival equations.

Momentum is a classical tool for accelerating convex optimization and a standard component of modern large language model training. Yet recent empirical work has questioned the usefulness of first-moment momentum in reinforcement learning with verifiable rewards (RLVR). We study this question through a bandit formulation of RLVR with bias-corrected exponential moving average (EMA) momentum, where the EMA coefficient $\beta$ controls how quickly the weights on past gradients decay. When policy gradients are evaluated exactly, we prove that increasing $\beta$ within a range that prevents overshoot strictly accelerates optimization. When gradients are estimated from sampled responses, a positive expected gain can be obscured by sampling fluctuations. We show that as the learning rate $\eta$ decreases, the expected gain with fixed $\beta$ vanishes faster than these fluctuations, whereas increasing $\beta$ to keep $\eta/(1-\beta)$ within the safe regime preserves a nonvanishing gain that eventually dominates them. Bandit simulations quantitatively validate these predictions. RLVR experiments on Qwen3-4B-Base using SGD with EMA and AdamW support our principle of choosing larger coefficients in practical LLM post-training. On Qwen2.5-32B with AdamW, simply increasing $\beta_1$ improves mean validation accuracy by nearly $2$ percentage points over the conventional choice, demonstrating the principle's effectiveness at a larger model scale.

The resetting of stochastic processes has become a key mechanism appearing in many research fields, such as searching for a target, the transmission control protocol, biology, chemistry, finance, insurance, etc., with the main interest concerning the first passage time (FPT) by the process under resetting. In this paper, we focus on the exit time $\mathcal{T}_L$ from the interval $(-L, L)$ of Lévy flights (LF) under resetting, but we also consider spectrally negative Lévy processes (SNLP). As the main result, we find the asymptotics of the mean first passage time (MFPT) as $L$ tends to $+ \infty$. For LF with $\alpha \in (0, 2)$, the scaling behaves as $L^\alpha$ regardless of the reset rate and the asymmetry of the initial condition, a behaviour completely different from that for the known case of Brownian Motion ($\alpha=2$). For the symmetric $\alpha$-stable case with $\alpha \in (0, 2)$, we also found that taking $r \to \infty$ with fixed $L$ yields a finite MFPT, which therefore may be used as a valid approximation. For completely asymmetric stable processes, we show that the scaling is very different from that for LF, and point out a method to obtain the exact distribution of FPT under resetting. The analytical calculations are verified by numerical simulations of the processes under resetting.

A perfect 1-factorization of the complete graph $K_{2t}$ is a partition of its edges into perfect matchings such that the union of any two of them is a Hamilton cycle. Kotzig conjectured in 1963 that $K_{2t}$ has a perfect 1-factorization for every $t \ge 2$, but apart from some infinite families and a list of sporadic orders the conjecture is open, and $92$ was one of the nine orders up to $100$ for which no perfect 1-factorization was known. In this paper we construct two non-isomorphic perfect 1-factorizations of $K_{92}$.

For a graph $G$, let $\mathrm{tw}(G)$, $\mathrm{tree}\text{-}\alpha(G)$, and $\mathrm{tree}\text{-}\chi(G)$ denote its treewidth, tree-independence number, and tree-chromatic number, respectively. We prove that every nonempty graph $G$ satisfies $\mathrm{tw}(G)+1\leq\mathrm{tree}\text{-}\alpha(G)\mathrm{tree}\text{-}\chi(G)(1+\lfloor\log_2(\mathrm{tree}\text{-}\alpha(G))\rfloor)$. This improves the quadratic bound of Koutsoutis, Krause, Liu, Redzic, and Ueckerdt. For every pair of integers $n,k\geq2$, we construct a graph $G$ with $\mathrm{tree}\text{-}\alpha(G)=n$, $\mathrm{tree}\text{-}\chi(G)=k$, and $\mathrm{tw}(G)=\Omega(nk\log_2 n)$. Thus the upper bound is best possible up to a constant factor.

In this article, we provide a complete answer to one of the most classical conjectures in differential geometry (of foliations). More precisely, we present a complete positive answer to the classical conjecture, due to Meeks III, Pérez, and Ros (2008), that states that every foliation on $\mathbb{R}^{n+1}$ by (possibly varying) CMC hypersurfaces is a foliation by minimal hypersurfaces. Moreover, if $n\leq 6$, such a CMC foliation must consist of parallel hyperplanes.

Radical parametrizations describe algebraic varieties by means of nested radicals of polynomials, and their theory relies on three varieties associated with a parametrization, namely, the incidence, tower and radical varieties. In this paper we revisit this framework from a computational point of view and show that it is governed by the radical tower rather than by the particular parametrization. We prove that the tower variety depends only on the tower and that the incidence variety is always irreducible, so that no prime decomposition is needed. Moreover, we show how to replace a tower by an isomorphic one whose tower variety is defined by the equations of the tower, so that it can be computed without elimination. We introduce universal radical parametrizations and we use them to settle, under a natural hypothesis on the coefficients, an open case in the transformation of differential equations with radical coefficients into equations with rational coefficients. We show that radical subtowers form a lattice, which yields minimal radical parametrizations, and we give algorithms to compute them. We also prove that dominance between towers induces a unique dominant rational map between their tower varieties, through which unirationality is transferred. Finally, we study towers over subfields of $\mathbb{C}$ and, in the real case, relate the real points of the tower variety to the real domains of the associated parametrizations.

Most of qualitative properties of finite element (FE) approximations for various PDE models, in particular the validity of the discrete maximum principles (DMPs), are usually guaranteed by certain geometric properties (e.g. acuteness/nonobtuseness) of FE meshes used. In this work, we revisit the linear FE discretization of the Poisson equation and discuss, in the context of DMPs, the representation of the stiffness matrix entries in terms of the edge lengths and cotangents of the dihedral angles of the tetrahedral meshes, thus avoiding the common use of face areas and element volumes. Using this type of entry representation, certain structural properties of meshes, and simple trigonometric relations, as proposed in the two-dimensional case by Strang and Fix, we show that DMPs still hold (as the stiffness matrices remain $M$-matrices) for many tetrahedral configurations with obtuse dihedral angles. Such a relaxation of the standard acuteness/nonobtuseness conditions can be vital in practical calculations, as these angle properties are difficult to preserve in global and local mesh refinements. Several representative examples are presented.

We review the role of relative entropy in quantum field theories, where the local von Neumann algebras are generically of type III. Consequently, only few exact results are known, and even bounds are difficult to obtain. As a new result, we prove that the relative entropy between a locally squeezed state and the Minkowski vacuum is finite for free fermions.

The first major step in the theory of graph minors of Robertson and Seymour was the grid theorem, a fundamental duality result describing that any graph of sufficiently high treewidth must contain a large grid as a minor. Since this pioneering work, multiple grid theorems have been established in structural graph theory, linking various width parameters of combinatorial objects with the non-existence of grid-like obstructions. In particular, Kawarabayashi and Kreutzer [STOC'15] proved a variant of the grid theorem for directed graphs that links directed treewidth with the non-existence of topological minors of large cylindrical walls. On the other hand, to study immersions of undirected graphs $-$ another containment relation that focuses on edge-disjointness, rather than vertex-disjointness $-$ Wollan [JCTB'15] introduced a parameter, tree-cutwidth, and proved that it is dual to immersions of large walls. In this work, we present a version of Wollan's parameter for directed graphs, the directed tree-cutwidth. The main result is a grid theorem for directed tree-cutwidth, establishing that it is dual to directed immersions of large cylindrical walls.

We study joint recovery of the load (the number of active users) and message-count vector in type-based unsourced random access with very short messages. Each user's $k$-bit message activates one of $M=2^k$ orthogonal resources; coherent collisions therefore produce a noisy, unnormalized histogram of the transmitted messages. For arbitrary load and message priors, we propose an exact joint maximum-a-posteriori (MAP) decoder. Known-load allocation is separable and convex; our contribution is to show that the solutions for all candidate loads are nested prefixes of one greedy chain. Scoring these prefixes with the Bayesian posterior terms jointly estimates the load and message-count vector in $O(M+K_{\max}\log M)$ time. We compare the proposed joint MAP decoder with marginal MAP, three scalar-count two-stage receivers, oracle-load MAP, a sum-only receiver, and a soft-pilot receiver. For $K\sim\operatorname{Unif}\{0{:}60\}$ at 2 dB, the proposed decoder cuts the no-activity false-alarm probability from $25.9\%$ to $3.5\%$ relative to the Bayesian-count two-stage receiver and, conditioned on $K\ge1$, reduces normalized count error by $21.4\%$. The advantage concentrates under idle or sparse uncertain activity, whereas support-count processing approaches joint MAP when the possible loads are well separated. We also prove that the sum-only statistic is insufficient and quantify prior and amplitude mismatch. The model targets synchronized, power-controlled, very-short payloads rather than general URA.

We establish a Kaplansky-Walker type lemma for idempotent complete additive categories with direct sums. It will be used in compactly generated triangulated categories to provide a version of Kaplansky's theorem on projective modules and an isomorphism-refinement result for direct sums of countable objects.

Given $0\le m<n$ and a positive Radon measure $\mu$ on $\mathbb R^n$ with at most polynomial growth, assume that it has finite $m$-dimensional Gaussian discrepancy, defined as $$\mathcal D(\mu):=\limsup_{R\to\infty}\Bigl[R^2\sup_{x\in\mathrm{spt}\mu}|G_R^\mu(x)-1|\Bigr],$$ where $G_R^\mu(x):=(4\pi R^2)^{-m/2}\int_{\mathbb R^n} e^{-|z-x|^2/(4R^2)}\,d\mu(z)$. Provided that $\mu$ admits a flat blow-down $\mathcal H^m\llcorner V$ (an assumption that cannot be dropped entirely), we prove that there exists a finite (smallest) $0\le\delta<\infty$, called the width of $\mu$, such that $$\mathrm{spt}\mu\subseteq \bar B_\delta(V+x_0),\quad \delta^2\le C(m,n)\mathcal{D}(\mu),$$ where $\bar B_\delta(V+x_0)$ denotes the closed $\delta$-neighborhood of a suitable translate of $V$. Note that the exponent is optimal because of scale invariance. In particular, when $\mathcal D(\mu)=0$, the support of $\mu$ is included in an $m$-dimensional affine plane. We use this fact to give a simpler proof of Preiss's rectifiability theorem: while still based on Gaussian averages of $\mu$, this new proof avoids showing uniqueness (and hence conicality) of blow-downs of $m$-uniform measures, as well as studying Gaussian-weighted moments beyond second order. The strategy of proof consists of an anisotropic blow-down of $\mu$, followed by an analysis of the resulting Young measure, using a rigidity lemma involving the heat semigroup.

We prove two height inequalities for rational points on abelian varieties over function fields of characteristic 0. The first is the GCD inequality, which shows that the height with respect to a closed subscheme of an abelian variety of codimension at least 2 is much smaller than the usual height function. This can be viewed as a compact and geometric analogue of a theorem of Levin. The second is a special case of the (geometric) abc conjecture proposed by Vojta, which compares the truncated counting function and the usual counting function. This is a compact analogue of a theorem of Guo, Nguyen, Sun and Wang, and also resolves a conjecture of Buium, which was only known for traceless abelian varieties or constant abelian varieties previously. The proofs of these inequalities follow a common three-step pattern: from microscopic local estimates to local estimates, and then to global estimates. Building on the canonical Xie-Yuan heights, we control discrete counting functions by continuous integrals of Betti forms. The main new tools are uniform Nevanlinna-type estimates for families of linear entire curves on abelian varieties; combined with covering arguments based on Besicovitch theory and averaging, these handle overlaps and gaps precisely.

This paper addresses the computation of persistent homology for filtrations of cellular cosheaves over a fixed cell complex, where the algebraic data vary along the filtration while the underlying topological structure remains unchanged. We extend methods from discrete Morse theory to this setting, establishing conditions under which chain complexes can be reduced while preserving the associated persistence module. Building on these results, we introduce "FilteredScythe", an algorithm that identifies Morse matchings compatible with all stages of the filtration and performs reductions using Schur complements, based on the exist "Scythe" algorithm. We prove the correctness of these reductions and establish bounds on the algorithm's time and space complexity.