数学前沿

In this paper, we prove that an annular capillary minimal surface outside the unit ball in $\mathbb{R}^3$ with one complete embedded end and finite total curvature must be either part of the plane or part of the catenoid. The proofs are based on the use of the Weierstrass data through the Hopf differential and associated holomorphic functions. For a planar end, we control the second fundamental form, while for a catenoidal end, we analyze the Gauss map in more detail using the boundary condition. We also prove nonexistence results for annular free boundary minimal surfaces with one end asymptotic to the end of Enneper's surface, under appropriate conditions. Finally, we establish an exterior analogue of Choe's result on capillary minimal surfaces with vertices.

We prove bilinear Strichartz estimates for the Schr\"odinger equation on the hyperbolic space. These bilinear estimates are then applied, as part of an I-method argument, to establish global well-posedness for the cubic defocusing nonlinear Schr\"odinger equation on the hyperbolic plane for regularities above 4/7. The I-method is implemented in physical space using a heat-based modified energy.

Full-waveform inversion (FWI) reconstructs sound-speed fields from measured waveforms, by solving an optimization problem that minimizes the discrepancy between measured waveforms and those predicted by a numerical wave-propagation model. However, choosing suitable regularization for this optimization is challenging because it must stabilize the inversion while preserving relevant structures, prompting us to investigate whether priors learned from representative sound-speed images can improve reconstruction. In this work, we adapt plug-and-play flow matching (PnP-Flow) to FWI by alternating waveform-misfit updates with updates from a pretrained flow prior. We explore the effectiveness of PnP-Flow reconstruction by comparing it with total-variation (TV) regularization using noiseless transmission-only plane-wave data. Separate flow priors are trained on two circular-inclusion model families with constant and spatially varying backgrounds. We evaluate the method both on these in-distribution circle models and under substantial distribution shift by transferring the Gaussian-circle prior and reconstruction settings to breast phantom patches without retraining or retuning. Using the same number of waveform-misfit updates, the proposed PnP-Flow framework generally improves circle reconstruction accuracy and recovers breast lesion contrast more accurately than TV baselines.

For every field $k$, we construct a commutative coherent $k$-algebra admitting a Gorenstein projective module that is not Gorenstein flat.

We prove that the minimum Hausdorff dimension of a continuous curve joining $W_0$ to $W_1$ within the range $W([0,1])$ of a three-dimensional Brownian motion is almost surely a deterministic constant $\gamma_3\in(1,5/3]$. Moreover, we construct continuum many simple curves with pairwise distinct images that attain this minimum. Each curve has box dimension $\gamma_3$ and admits a parametrization that is $\beta$-H\"older for every $0<\beta<1/\gamma_3$. To identify $\gamma_3$, we partition Brownian time into $n$ equal intervals and form a graph by joining pairs with intersecting ranges. We show that its endpoint graph distance and diameter have a common deterministic almost sure growth exponent $\theta$ and that $\gamma_3=2\theta$. We also establish $n^{p\theta+o(1)}$ growth for their $p$-th moments, for every $p>0$.

We prove the following two statements: (1) If $X$ is a compact K\"ahler fourfold with pseudo-effective canonical bundle and no subvarieties of codimension one or two, then $K_X$ is a torsion line bundle; (2) If $X$ is a compact K\"ahler fourfold with no subvarieties of codimension one or two, then $X$ is not uniruled. Using the Beauville-Bogomolov decomposition and a characterization of Uniruled manifolds by Wenhao Ou, we get that if $X$ is a compact K\"ahler fourfold with no subvarieties of codimension one or two, then either $X$ is a torus or an irreducible holomorphic symplectic manifold.

For $\beta\ge0$, let $A_1^{(\beta)}$ be the class of weights satisfying $$[w]_{A_1^{(\beta)}}:= \sup_{0<\varepsilon\le1} \varepsilon^\beta[w]_{A_{1+\varepsilon}}<\infty.$$ These classes satisfy $A_1\subsetneq A_1^{(\beta)}\subsetneq\bigcap_{p>1}A_p$ for $\beta>0$. For every $\alpha\ge0$, we prove that $w\in A_1^{(\beta)}$ if and only if $$w\left(\left\{M_{L(\log L)^\alpha}f>\lambda\right\}\right) \le C\int_{\mathbb{R}^n} \frac{|f(x)|}{\lambda} \left[ \log\left(e+\frac{|f(x)|}{\lambda}\right) \right]^{\alpha+\beta} w(x)\,\mathrm{d}x.$$ The least constant $C$ is comparable to $[w]_{A_1^{(\beta)}}$. Also, we obtain corresponding characterizations for Calder\'on--Zygmund operators and their commutators.

The multiplicative ergodic theorem (MET) describes the asymptotic growth rates of $q$-volumes under the iterates of an operator cocycle, i.e., compositions of operators driven by a stationary process, organizing the underlying space into equivariant filtrations or splittings into so called Oseledets subspaces. For operator cocycles in finite dimensions, and more generally in Hilbert spaces, one can control the geometry of the Oseledets subspaces given by the MET via the convergence rates of finite-time volume growth data. This makes the MET "effective" for matrix and Hilbert-space operator cocycles. We extend this effective theory to cocycles of Banach-space operators, obtaining quantitative angle estimates that are new even in the finite-dimensional setting. We take a sequential approach \`{a} la Raghunathan and Ruelle, allowing us to separate the geometric argument of the MET from measure-theoretic considerations. This permits weaker cocycle measurability hypotheses for the non-invertible and semi-invertible METs on a broad class of Banach spaces, including nonseparable ones. As an application, we establish the MET for transfer operator cocycles on $\operatorname{BV}[0,1]$ associated with jointly measurable random Lasota-Yorke maps.

Let $Y$ be a smooth projective threefold with $H^2(Y, \mathcal{O}_Y)=0$ and unobstructed deformations. We define reflection groups $W_Y^{sm}$ and $W_Y^{big}$ canonically associated to $Y$, acting on the nef, resp. movable cone of a very general deformation of $Y$. If $-K_Y$ is nef, we show that $\mathrm{Nef}(Y)\cap\mathrm{Big}(Y)$ (resp. $\overline{\mathrm{Mov}}(Y)\cap\mathrm{Big}(Y)$) is a fundamental domain for the action of the corresponding group on the big part of the generic cone. This generalizes work of Wilson and Lutz on Calabi--Yau threefolds. The group $W_Y^{big}$ recovers and gives a deformation-theoretic explanation for the Weyl group action on divisor classes of point blowups of $\mathbb P^3$ studied by Dolgachev--Ortland and Mukai.

This article revisits earlier work by the last author together with Kay Magaard. We correct mistakes in results about the Sylow 2- and 3-structure of groups that act with fixity 3 and one result about possible components, we improve some results and we add details.

We study optimal control of non-linear graphon dynamics in which the interaction kernel itself is the control. Motivated by opinion formation in social networks, we ask how changing who influences whom can steer a population toward a desired outcome. We develop a framework using interaction kernels with bounded variation in the receiver label, providing compactness while accommodating finite networks. We prove existence of an optimal pair and show that finite-agent approximations recover the continuum optimal value under successive refinement of the network and initial data, with subsequential convergence of suitably chosen discrete optimizers. Numerical experiments use neural networks to approximate optimal controls for bounded-confidence and graphon Kuramoto dynamics, illustrating how changes in interactions can prevent fragmentation and promote synchronization.

Given a finite point set satisfying condition $\mathcal{A}$, the subset selection problem asks, how large of a subset satisfying condition $\mathcal{B}$ can be extracted? In this paper, we make progress on three instances of subset selection problems in planar point sets. Let $n,s\in\mathbb{N}$ with $n\geq s$, and let $P\subseteq\mathbb{R}^2$ be a set of $n$ points, where at most $s$ points lie on the same line. Firstly, we select a general position subset of $P$. This problem was proposed by Erd\H{o}s under the regime when $s$ is a constant. For $s$ being non-constant, we give new lower and upper bounds on the maximum size of such a subset. In particular, we show that in the worst case such a set can have size at most $O(n^{5/6+o(1)}/\sqrt{s})$ when $3\leq s\leq n^{1/3}$ and $O(n/s)$ when $n^{1/3}\leq s\leq n$. Secondly, we select a monotone general position subset of $P$, that is, a subset in general position where the points are ordered from left to right and their $y$-coordinates are either non-decreasing or non-increasing. We present bounds on the maximum size of such a subset. In particular, when $s=\Omega(\sqrt{n})$, our upper and lower bounds differ at most by a logarithmic factor. Lastly, we select a subset of $P$ with pairwise distinct slopes. This problem was initially studied by Erd\H{o}s, Graham, Ruzsa, and Taylor on the grid. We show that for $s=O(\sqrt{n})$ such a subset of size $\Omega((n/\log{s})^{1/3})$ can always be found in $P$. When $s=\Theta(\sqrt{n})$, this matches a lower bound given by Zhang on the grid. As for the upper bound, we show that in the worst case such a subset has size at most $O(\sqrt{n})$ for $2\leq s\leq n^{3/8}$ and $O((n/s)^{4/5})$ for $n^{3/8}\leq s=O(\sqrt{n})$. The proofs use a wide range of tools such as incidence geometry, probabilistic methods, the hypergraph container method, and additive combinatorics.

We observe that the $2$-norm distance $d_{U,2}$ between the unitary orbits of normal elements in a $\mathrm{II}_1$ factor $\mathcal{M}$ is equal to the $2$-Wasserstein distance between the spectral measures induced by the trace $\tau_\mathcal{M}$. Using classification and optimal transport theory, we deduce an analogous $2$-norm equation for normal operators $x$ and $y$ in simple, separable, unital, nuclear, $\mathcal{Z}$-stable $\mathrm{C}^*$-algebras that are either monotracial, or real rank zero with finitely many extremal traces, provided that $\sigma(x)=\sigma(y)$ is convex. Consequently, $d_{U,2}$ equips the set of approximate unitary equivalence classes of contractive normal elements of $\mathcal{M}$ with the structure of a compact length space. The same is true of the set of equivalence classes of embeddings into the Jiang-Su algebra $\mathcal{Z}$ of classifiable tracial $2$-Wasserstein spaces over compact, convex planar domains.

Modelling qualitative uncertainty in formal argumentation is essential both for practical applications and theoretical understanding. Yet, most existing work focuses on abstract models of argumentation under uncertainty. Following a recent trend in the literature, we tackle the open question of studying plausible instantiations of these abstract models. To do so, we root the uncertainty of arguments in their components in structured models: rules and premises. Our main conceptual contribution is to provide an intuitive notion of comparative expressivity suitable for both abstract and structured formalisms. Roughly, one formalism for arguing with qualitative uncertainty is at least as expressive as another if the former can generate the same set of abstract argumentation frameworks (completions) as the latter. On the technical side, we prove negative and positive results comparing the expressivity of abstract and structured models of argumentation with uncertainty. These results concern incomplete abstract argumentation frameworks and their extension with dependencies on the abstract side, and ASPIC$^+$, assumption-based argumentation (ABA) and deductive argumentation on the structured side.

We introduce Controlled Stochastic Activity Networks (Controlled SANs), extending classical SANs with explicit control actions and policies for stochastic discrete-event and real-time systems. The framework combines nondeterminism, probabilistic branching, and stochastic timing through controlled automata, controlled probabilistic automata, and controlled stochastic automata. At the nondeterministic level, we distinguish structurally available controls from behaviorally admissible transitions and establish a constructive modeling-power result: every decidable, effectively finitely branching controlled automaton with finite activity and control sets and a finite nonempty initial-state set admits a computable, well-behaved controlled activity network realization whose reachable restriction is effectively isomorphic to the original. We define memoryless, finite-memory, stack-augmented, and tape-augmented policies, investigate their language-theoretic expressiveness, and develop behavioral equivalences including bisimulation and stochastic isomorphism. At the probabilistic level, contingent activity-indexed control assignments and an activity-generation law yield a discrete-time Markov decision process (DTMDP) representation; conversely, every conventional DTMDP admits a single-activity representation. Under Markovian assumptions, controlled stochastic automata admit continuous-time Markov decision process (CTMDP) projections, and conventional CTMDPs admit single-activity representations. Randomized-policy correspondences follow the adopted product-form convention. These results provide a semantic basis for verification, reward optimization, policy synthesis, and potential applications to reinforcement learning, safety-critical control, and agentic AI.

The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. We provide two separate conditions for a finite horizon billiard to admit equilibrium states. The first one -- namely \emph{negligible singularities} -- only concerns the billiard table and implies upper semi-continuity for the metric pressure map of $T$. The condition of negligible singularities is shown to be generic (dense $G_\delta$) among billiard tables with $C^{3+\alpha}$ boundaries, in the $C^{3+\alpha}$ topology, and implies the sparse recurrence condition for the null potential. Compactness of the set of $T$-invariant measures is proved. For finite horizon Sinai billiards, we provide a bound on the defect of upper semi-continuity of the metric pressure map. From it, a condition of existence of equilibrium states only involving potentials is deduced. In both situations, for the potentials $g \equiv 0$ and $g = -h_{\rm top}(\Phi^1) \tau$, we recover and extend the recent results of the existence of measures of maximal entropy (MME) for both the map $T$ and the flow $\Phi^t$, due to Baladi and Demers for $T$, jointly with Carrand for $\Phi^t$. Upper semi-continuity of the metric entropy is shown. Assuming negligible singularities, equidistribution of the periodic orbits with respect to the unique MME is proved. A bound on the topological tail entropy is given.

We consider the left translation action of an element $a$ on a finite-volume semisimple homogeneous space $G/\Gamma$, where $\operatorname{Ad}(a)$ is diagonalizable over $\mathbb{R}$. Assuming exponential mixing, we prove a sub-Gaussian large-deviation estimate for compactly supported smooth test functions. As an application, we prove effective uniqueness of the measure of maximal entropy. We also obtain quantitative pointwise equidistribution for Haar-almost every point on each full expanding horosphere.

Sierksma's conjecture asserts that every affine map $a\colon\Delta_{(d+1)(r-1)}\longrightarrow\mathbb{R}^d$ has at least $((r-1)!)^d$ unordered Tverberg partitions into $r$ parts. Until now, lower bounds growing with $d$ were known only when $r$ is a prime power. As one of our main results we prove that for every integer $r\geq3$, every decomposition $r=kq+s$ into $q=p^\alpha$ a prime power, $k\geq1$ and $s\geq0$, and every affine map $a$, the number $\operatorname{Tv}(a)$ of unordered Tverberg partitions satisfies $$\operatorname{Tv}(a)\ \geq\ \frac{q}{r!}\,\Big(\frac{q}{\alpha+1}\Big)^{\big\lceil\frac{(r-1-2s)(d+1)}{2}\big\rceil}.$$

We study a variational discretization in time of Aubry--Mather theory for Tonelli Lagrangians on the flat torus. We define discrete Ma\~n\'e, Aubry and Mather sets, establish their compactness and inclusion properties, and investigate their convergence as the time step tends to zero. We prove upper-limit results for the Ma\~n\'e sets and, when the continuous Aubry set has finitely many static classes, for the Aubry sets. We obtain full Kuratowski and Hausdorff convergence of the Aubry sets when this set is a single equilibrium or periodic orbit, and, under additional assumptions, a finite union of equilibria. Every weak limit of discrete Mather measures is a continuous Mather measure. For equilibrium Aubry sets, we give sufficient conditions for exact preservation of Mather sets and measure families at small time steps. A mechanical example illustrates these results and exhibits exact preservation of the Aubry set as well.

In this work, we focus on deriving the absolute moments of the general ized hyperbolic (GH) distribution via its mean-variance mixture structure. Conditioning on the GIG mixing variable reduces the problem to Gaussian moment calculations and known GIG power moments, avoiding direct integration with Bessel functions. The variance gamma (VG), normal inverse Gaussian (NIG), hyperbolic, Normal, and Student t distributions emerge as natural special or limiting cases of the generalized hyperbolic family. This approach offers a substantially simpler and more transparent derivation, facilitating both implementation and verification.