Papers

In this paper, we study a problem posed by Matsumura on the extension of pluricanonical forms in Kähler families with a relatively nef canonical bundle. We give an affirmative answer in the smooth case, and for one-parameter degenerations under the additional assumption that the relative canonical bundle restricts to a big line bundle on an irreducible component of the central fiber. In particular, we confirm Siu's conjecture on the invariance of plurigenera for a smooth Kähler family with one fiber admitting the nef canonical bundle. The key step in the smooth case is to construct a semipositively curved singular metric on the relative canonical bundle with the integrability required for Cao's $L^2$ extension theorem, using Păun's twisted form of Schumacher's curvature formula and a uniform Monge–Ampère estimate obtained by adapting the capacity method of Boucksom–Eyssidieux–Guedj–Zeriahi. For one-parameter degenerations, we propagate bigness from an irreducible component of the central fiber to nearby smooth fibers and then apply Matsumura's relative Kawamata–Viehweg vanishing theorem on a projective modification to obtain the desired extension.

We provide generalized linearization theorems for skew-product flows on abstract Banach bundles over locally compact metric spaces, allowing for a unified treatment of uniform and nonuniform, exponential and strong exponential dichotomies. This is accomplished by constructing an operator whose fixed points are exactly the conjugacies that are bounded perturbations of the identity, and by giving criteria under which this operator is a contraction. Under stronger assumptions we refine the argument to obtain Hölder regularity of the conjugacy. Finally, we apply our results to recover known results for non-autonomous differential equations and to obtain new linearization results for control-affine systems on manifolds.

We compute $K$-theoretic invariants for four three-dimensional operator systems to illustrate the general theory. We find that in some cases the invariants distinguish the operator system from its $C^*$-envelope, while in others they do not. We also explore extensions of the theory.

We compute $K$-theoretic invariants for several finite-dimensional operator systems. These include the Toeplitz operator system and its dual (the Fejér-Riesz operator system), as well as operator systems associated to tolerance relations, aka graph operator systems. A variety of techniques is applied, adapted to each of these cases, ranging from a generalization of Carathéodory's factorization result for block Toeplitz matrices, to Wiener-Hopf factorization and Schur complements. In most instances we find that the invariants coincide with their analogues for the $C^*$-envelope, except for the $K_1$-invariants of the Fejér-Riesz operator system.

We investigate a recently introduced generalisation of determined directions in affine planes. A direction $(d)$ in an affine plane is a point of the line at infinity in the projective completion, and hence corresponds to a parallel class. A (multi)set $S$ of points is equidistributed from direction $(d)$ if all lines from that parallel class intersect $S$ in the same number of points. In this paper, we give a construction of point (multi)sets that are inequidistributed from exactly 3 directions in any finite translation plane, and prove that all such (multi)sets arise from this construction, generalising a result of Kiss and Somlai in Desarguesian planes of prime order. We also use ideas from algebraic graph theory to prove results on directions associated with a pair of point sets $S$ and $T$. If for every direction $(d)$ either $S$ or $T$ is equidistributed from $(d)$, we prove that $|S \cap T| = |S||T|/q^2$, where $q$ is the order of the affine plane. In the appendix, we give a combinatorial alternative approach to proving this equality, which can be of independent interest. We also introduce the notion of directions cross-determined by a pair of sets $S$ and $T$, and prove that $|S| |T| \leq q^2$ if $(S,T)$ cross-determines at most half of the directions, where equality forces $S = T$.

We describe one explicit simple graph G on 32 vertices whose adaptive, normal, linear, entropic, and submodular widths are pairwise distinct. We compute all these widths exactly, except for the entropic width, where we only give a lower and upper bound. We use Ingleton's inequality and the Zhang-Yeung inequality for upper bounds, and give explicit constructions of modular, normal, linear, entropic, and non-entropic polymatroids for lower bounds.

We investigate Lie groups endowed with left-invariant Lorentzian metrics of nonzero constant sectional curvature. We first revisit Heintze's theory and obtain a characterization of the Lie algebras of Riemannian Lie groups of negative constant curvature. We also extend classical results of Nomizu and Barnet to the pseudo-Riemannian setting by constructing a large family of Lie groups carrying incomplete left-invariant metrics of constant sectional curvature. We then describe Lorentzian Lie algebras of nonzero constant curvature according to the causal nature of their center and derived ideal. This description is complete except when the derived ideal is Lorentzian, for which we obtain a complete classification in dimension four. We show that every left-invariant Lorentzian metric of nonzero constant curvature on \(\mathrm{SL}(2,\mathbb R)\) is bi-invariant and complete. Moreover, a semisimple Lie group admits a complete left-invariant Lorentzian metric of nonzero constant curvature if and only if it is locally isomorphic to \(\mathrm{SL}(2,\mathbb R)\). We also characterize \(\mathrm{SL}(2,\mathbb R)\) through the existence of a noncentral spacelike left-invariant Killing vector field. Finally, we classify Lorentzian Lie algebras of nonzero constant curvature in dimensions at most four.

We study the inversion of the geodesic ray transform from finitely many local averages of its data. Under suitable geometric assumptions, we prove Lipschitz stability on any fixed finite-dimensional reconstruction space when the measurement partition is sufficiently fine. We develop convergent reconstruction algorithms based on Steepest Gradient Descent and Conjugate Gradients and implement them using piecewise constant finite element spaces. Numerical experiments in 2D and 3D illustrate the influence of geometry and measurement discretization on reconstruction quality.

We study a reduced multigrid Schur preconditioner for the fictitious-domain method with distributed Lagrange multipliers applied to elliptic interface problems. The method replaces the costly background solve by a multigrid approximation and treats the resulting correction only in the multiplier coupling space through a Woodbury formula. Numerical experiments show a marked improvement over the baseline preconditioner and consistently rapid convergence under mesh refinement and coefficient contrast.

In this article, we analyze a recently introduced kinetic model based on Hegselmann-Krause-type interaction dynamics describing consensus-based image segmentation in the mean-field regime, where particle density depends on spatial coordinates and the normalized gray level (feature) value. The model is formulated as an evolutive nonlocal partial differential equation featuring a non-local spatial drift term involving a bounded confidence interaction kernel, depending on spatial and feature proximity, and a spatial diffusion term degenerating in the feature variable representing aleatoric uncertainties. Establishing well-posedness on a bounded domain, essential for practical image processing, presents significant analytical challenges: the lack of uniform control of space derivatives due to kernels with jump discontinuities and degenerate spatial diffusion, complex non-standard boundary terms under homogeneous Neumann conditions, and a lack of compactness arising from the absence of derivatives with respect to the feature variable. To address these difficulties, we introduce a three-level regularization scheme. Specifically, we smooth the interaction kernel, apply structural boundary regularization to the non-local interaction term to simplify boundary conditions, and incorporate artificial diffusion in spatial and feature variables to resolve degeneracy. We prove the well-posedness and regularity of the regularized system via a finite difference scheme and study the asymptotic limit as regularization parameters vanish, proving the existence of weak/distributional solutions of the original model. This approach advances the mathematical theory of kinetic models for image segmentation by extending the framework of distributional solutions to bounded domains and non-smooth interaction kernels with jump discontinuities.

We develop a pressure-free virtual element method for the surface Stokes problem on polygonal approximations of closed surfaces of arbitrary genus. The method extends the planar VEM Stokes complex to discrete surfaces, yielding a nonconforming complex with an exactly tangential, $H(\operatorname{div})$-conforming velocity space, commuting interpolation, and the correct discrete cohomology. To obtain the weak consistency needed for optimal convergence, we introduce an edge correction that preserves the complex without adding degrees of freedom; the resulting construction reduces to the original complex in the planar setting. The virtual formulation requires no penalties, and a local divergence-preserving reconstruction provides a computable pressure-robust load. We also construct explicit macroelement representatives whose energy and load induce an alternative formulation on the VEM space, algebraically equivalent to a direct macroelement Galerkin method. The original virtual formulation and the macroelement method are stable and attain optimal first-order convergence in the broken $H^1$ norm and second-order convergence in the $L^2$ norm for the velocity. Numerical experiments illustrate these convergence rates, the role of the edge correction, and pressure robustness.

The Scrooge ensemble associated with a density matrix $\rho$ is the maximally random ensemble whose average density matrix is $\rho$. It has attracted growing interest for its applications to quantum many-body dynamics and quantum information. The intrinsic statistical properties of the Scrooge ensembles are encoded in their higher-order moments, yet they are nontrivial to analyze and prepare for they feature a non-polynomial Haar integrand, blocking direct use of Haar integration machinery. Existing physical mechanisms to sample multiple copies of states from the Scrooge ensembles, meanwhile, typically require costly postselection, rendering them inefficient for large system size. In this work, we develop efficient and accurate methods for analyzing and implementing Scrooge moments. We first introduce a polynomial approximation to the Scrooge $k$-th moments whose error is tunable down to a threshold that improves from inverse polynomial to exponential in $1/||\rho||_\infty$ over previous results. Building on our approximation, we then give the first efficient quantum algorithm that estimates observable expectation values using $\widetilde{\mathcal{O}}(k^3/\epsilon^5)$ copies of $\rho$ for any target error $\epsilon$ above this threshold. We further provide a construction of a block encoding of the Scrooge $k$-th moment when the preparation circuit of $\rho$ is known. Of independent interest could be a key subroutine in our algorithms, which efficiently implements projection onto the $k$-fold symmetric subspace for $\rho^{\otimes k}$ using $\mathcal{O}(k^3/\epsilon^2)$ samples of $\rho$, avoiding the naive $k!$ postselection cost of such a projection. These results provide new tools for both the theoretical analysis and practical prediction of properties of Scrooge ensembles, with potential applications to quantum many-body dynamics and quantum information processing.

For a field $F$ we study the coinvariants for the $SL_n(F)$-action on the double Steinberg module $St_n(F) \otimes St_n(F)$ and show they have a rich algebraic structure: for $n = 2$ it is the Grothendieck-Witt group of $F$, and for all $n$ they assemble to a graded nonunital $\mathbb{Z}[F^\times]$-algebra, whose rational (underived) indecomposables may be expressed in terms of the augmentation ideal of the Grothendieck-Witt group. We then explain, building on work of Galatius-Kupers-Randal-Williams, that the special linear groups $SL_n(F)$ assemble to an $E_\infty$-algebra in a suitable functor category, whose $E_2$-homology groups have a vanishing line of slope 2 and on the critical line are given by the double Steinberg coinvariants.

We study the problem of optimally placing a large number of small absorbing traps to minimize the mean first-passage time (MFPT) of particles diffusing in a heterogeneous medium with space-dependent diffusivity $D(x)$ and prescribed initial particle distribution $\omega(x)$. In two and three dimensions we identify two distinct regimes, depending on how strongly the trap ensemble depletes the particles. In the weak trapping regime (fewer traps) the optimal trap density $\rho$ is the arithmetic average $\rho = \frac12(\mu+\omega)$ of the normalized inverse diffusivity $\mu \propto 1/D$ and the initial particle distribution $\omega$. In the strong trapping regime (more traps), it is proportional to their geometric average, $\rho \propto \sqrt{\mu \omega}$. We revisit the classical Green's function approach and show that it applies to weak trapping only; in 2D this requires the trap size to be exponentially small in the number of traps, so that most applications of interest fall in the strong trapping regime instead. We therefore develop a homogenization approach that captures both regimes. A special choice of initial particle distribution is $\omega \propto 1/D$, which corresponds to the equilibrium particle distribution in the absence of traps according to the Itô interpretation. In this case we show that the optimal trap distribution is the stationary particle distribution itself, regardless of the trapping strength, and this makes the MFPT constant throughout the domain; the same trap distribution also minimizes the worst-case capture time. Finally, in one dimension we find instead $\rho\propto\left(\mu \omega\right) ^{1/3}$, and a similar analysis covers space-dependent drift and thin domains of variable cross-section. Direct numerical optimization confirms the analytical results.

We show that the space of vacuum initial data sets on a closed manifold often has many non-trivial homotopy groups. The starting point is a result of the second named author, which constructs non-trivial elements in the homotopy groups of the space of initial data sets satisfying the strict dominant energy condition, together with a later result in joint work with Bernd Ammann showing that these elements often persist when the strictness assumption is dropped. In this work, we use a parametrized version of the conformal method to show that these elements may also be represented by maps into the space of vacuum initial data sets. This requires two results that may be of independent interest: metrics admitting conformal Killing vectors can be removed from the space of metrics without changing its weak homotopy type, and over the remaining metrics the York decomposition can be carried out in families, the TT-tensors forming a trivial Hilbert bundle. To our knowledge, this is the first result on the global topology of this space.

The classical linear quadratic regulator (LQR) minimizes the expected cumulative return but fails to account for performance variability, rendering it inadequate for risk-aware applications. To address this, we introduce the variance of the cumulative return as a risk measure in LQR. We derive the first exact closed-form expression for the variance of the discounted in?finite horizon return within the discrete-time Distributional LQR framework, for i.i.d. disturbances with symmetric probability densities. For Gaussian disturbances, this expression elegantly simplifies to a form dependent only on the disturbance covariance. Leveraging these theoretical foundations, we formulate a mean-variance optimal control problem that explicitly manages the trade-off between expected return and performance variability. To address the resulting non-convex optimization problem, we propose a novel adjoint gradient descent algorithm for a penalized formulation of the original problem, and establish that all iterates remain stabilizing and converge to a stationary point of the penalized objective. The effectiveness of this framework and the inherent risk-performance trade-off? are demonstrated through numerical experiments.

We prove a permanental analogue of Koteljanskii's inequality. If $A$ is an inverse $M$-matrix that becomes symmetric after a positive diagonal similarity, then $\mathrm{per}(A_{S\cup T})\,\mathrm{per}(A_{S\cap T})\ge\mathrm{per}(A_S)\,\mathrm{per}(A_T)$ for all $S,T\subseteq[n]$, where $A_S$ is the principal submatrix indexed by $S$. The proof expresses permanents as moments of a complex Gaussian vector and uses Ginibre's correlation inequality. Finally, an explicit counterexample shows that symmetry cannot be dropped.

We construct non-flat non-Hermitian Yang–Mills connections with zero Einstein constant on smoothly trivial rank-two bundles over compact Kähler surfaces, with positive harmonic metrics and stable induced and adjoint holomorphic bundles. Thus vanishing Chern classes do not force flatness even under stability on both sides. The local model comes from a known complex anti-self-dual ansatz; the stable descents yield explicit global moduli phenomena. For a fixed stable bundle, the Kaledin–Verbitsky map contracts a complex line containing flat and non-flat points in the connected component of the Hermitian–Einstein connection. Contraction also occurs in the two-sided stable locus. The pair of holomorphic projections has a fibre containing both flat and non-flat points. An energy identity and a spectral-gap estimate quantify local flatness with the induced holomorphic structure fixed.

Haah's 3D cubic code was introduced as a candidate for a self-correcting quantum memory in three dimensions, using the absence of string logical operators to obstruct thermal errors. It has been a long-standing open question whether it is a self-correcting quantum memory. Even though the memory time was explored numerically, no analytical upper bound was known. Here, we settle this question rigorously and show that the cubic code thermalizes rapidly at every positive temperature. Our proof relies on the framework of [Commun. Math. Phys. 407, 195 (2026)] to turn a decay-of-correlations condition into a modified logarithmic Sobolev inequality for the Davies generator. To the best of our knowledge, this is the first analytic proof of rapid mixing for all positive temperatures for any fracton code.

We enumerate all families of involutions avoiding a classical pattern of length four together with 4321. While the enumeration of permutations avoiding two classical patterns is by now well understood, the case of pattern-avoiding involutions presents additional structural constraints that require dedicated techniques. In particular our main enumerative tool is Biane's bijection between involutions and labelled Motzkin paths. The paper provides exact bivariate generating functions (taking into account length and number of descents), and structural descriptions for all except one family. In the remaining case we provide a functional equation for the generating function and express it as an explicit continued fraction. We make extensive use of experimental and software-supported methods.

We show that circle-valued spectral values in Tate Rabinowitz Floer homology give rise to quantized energy levels, which recover the quantum mechanical spectrum of a system of uncoupled harmonic oscillators.

The input-to-state stability of a class of infinite-dimensional second-order port-Hamiltonian systems on a one-dimensional spatial domain is analyzed under nonlinear dynamic boundary feedback and boundary disturbances. Using energy methods in the port-Hamiltonian framework, we obtain sufficient conditions that guarantee uniform input-to-state stability of the closed-loop system.

We introduce a simple reaction-network model of a quorum-sensing population that couples the cell density $x$ to the signal density $w$. In the four-channel cell-signal network studied here, scaling signal production and removal by the same factor $r$ leaves all deterministic equilibria, and their stability types, unchanged. Nevertheless, we show that $r$ shifts the quasipotential barrier $\Delta V(r)$ for rare transitions towards extinction, and thus, under metastable exit assumptions, the mean time to reach a fixed neighbourhood of the extinction state on the exponential scale $e^{N\Delta V(r)}$. We compute the barrier by minimization of the path action with the signal retained as a fluctuating coordinate, and we check it against mean extinction times obtained without the action solver, i.e., from the backward equation of the jump process on a truncated lattice and from exact stochastic simulation. Over a range of $r$, the minimum-action barriers satisfy $\Delta V(r)=\Delta V_\infty+O(1/r)$, where $\Delta V_\infty$ is obtained by eliminating the signal first, and at $r=1$ the barrier is 55% larger than $\Delta V_\infty$. A formal first-order expansion attributes this increase to the lag of the signal, of which the fluctuations of the signal remove a quarter. The saddle barrier $\Delta V(r)$ is also the least action needed to enter the basin of extinction, whereas the density threshold can be crossed more cheaply, by paths which keep the signal high and are usually followed by recovery. The mean extinction times of the backward equation, which include failed attempts, have a prefactor which tends to a constant, and they grow at a rate within $10^{-5}$ of $\Delta V(r)$ at every tested rate.

We construct smooth, non-locally-conformally-flat metrics on the closed ball for which the boundary Yamabe equation admits $L^\infty$-unbounded sequences of positive solutions. The background metric and the prescribed scalar and boundary mean curvatures are fixed along each sequence. For zero scalar curvature and positive boundary mean curvature, such examples exist with umbilic boundary in every dimension $N\ge22$ and with nonumbilic boundary in every $N\ge15$. For positive scalar curvature and minimal boundary, the corresponding ranges are $N\ge21$ and $N\ge15$. For every $N\ge9$, examples with either umbilic or nonumbilic boundary also exist when the scalar curvature is $N(N-1)$ and the prescribed boundary mean curvature is below a negative threshold depending on $N$. The construction uses polynomial metric perturbations and corrected bubbles on the half-space, followed by conformal compactification to the ball. We evaluate or estimate the correction solving the linearized Neumann or Robin boundary problem and its contribution to the quadratic term of the reduced energy. We prove that this quadratic term has negative, nondegenerate local extrema with respect to tangential translations and scale in every stated dimension.

Every non-linear plane Cremona map can be decomposed into quadratic maps, and the minimum number of quadratic maps required is called its quadratic length. It is known that plane Cremona maps of degree 3 have quadratic length either 2 or 3. In this paper, we study the quadratic length of plane Cremona maps $\varphi$ of degree 4. Recall that either $\varphi$ is de Jonquières, i.e. it has a base point of multiplicity 3, or it is not de Jonquières. In the latter case, $\varphi$ has quadratic length 2, whereas in the former case we prove that $\varphi$ has quadratic length 3, 4, or 5, and we classify those maps that reach the upper bound.

Let $S$ be a family of sequences decreasing to zero. A set $E$ in a metric space is $S$-dominated if, for every $s\in S$, there exists a countable cover $\{E_n\}$ of $E$ such that $diam E_n<s_n$ for every $n$. We continue the study of families of dominated sets initiated in the previous paper. We look, e.g., into Cartesian products of dominated sets with strong measure and interaction of microscopic and porous sets. Based upon a tight relationship of Hausdorff measures and dominated sets, cardinal invariants of the ideals of dominated sets are determined and a related problem of Kwela is resolved.

We study when concave weighted power means of positive semidefinite quadratic forms admit sharp quadratic upper bounds. The starting point is the scalar fact that each such mean is the lower envelope of its supporting affine majorants. After the scalar variables are replaced by quadratic forms, the resulting bounds may cease to be extremal among quadratic majorants. We prove that sharpness is preserved under a lower-envelope convexity condition on the joint numerical range. In that case, a nonlinear inequality involving power means of quadratic forms is certified, with no gap, by a single linear matrix inequality. The condition is weaker than convexity of the full augmented range and holds automatically for two quadratic forms. This yields a sharp Peter-Paul-type majorant for the whole concave power-mean scale, including Yuan's lemma and the weighted geometric mean as special cases. We also derive variational formulas for the extrema of sums of power means of quadratic forms on the unit sphere; under the same lower-envelope condition, the maximization becomes an exact convex eigenvalue minimization. A finite-horizon control application illustrates how the certificate can be used constructively.

We prove a continuous-time Carleson embedding theorem with constant $e$ for a system of square-integrable continuous martingales whose quadratic covariations mimic the generalised Cauchy–Riemann relations. The argument is based on a Bellman function and a multidimensional Itô formula. As an application we transfer the estimate to the upper half-space via the Gundy–Varopoulos representation and obtain a Carleson embedding, still with constant $e$, for the vector consisting of a function and its Riesz transforms in arbitrary dimension.

We present a new lower bound on the number of lattice points of all totally unimodular polytopes, generalizing previous lower bounds on contingency tables, integer flows, and beyond. Our bound is based on the Gurvits capacity convex optimization problem, and thus our result implies an efficient deterministic algorithm for approximate counting of the lattice points up to an explicit exponential factor. We achieve our bounds by showing that the associated generating polynomials fit into a new general class of log-concave polynomials called VLC ("variable-wise log-concavity”). This also implies a conjecture of Ferroni and Higashitani on the evaluations of the Ehrhart polynomials of unimodular polytopes. The essential ingredient for these results is the resolution of Barvinok's log-concavity conjecture for contingency tables on lines, which was proven using ChatGPT 6 Astra. We conjecture a generalization of Barvinok's conjecture, which we believe will lead to stronger and more general bounds.

Traditional simulations of parametric partial differential equations (PDEs) rely on repetitive computations for each parameter, which makes high-fidelity design impractical. Neural operators address this issue by learning solution operators, accelerating parameter-space mapping by orders of magnitude. Recent Transformer-based neural operators attempt to capture global dependencies, but often at the cost of quadratic attention complexity. Transolver resolves this problem by projecting physical states into a reduced slice space for attention computation. Although fast, this projection sacrifices fine spatial information. Moreover, by operating in this reduced space with shared weights across attention heads, it may constrain the model's flexibility, thereby limiting its capacity to capture complex phenomena. To address these issues, we propose the Cluster Attention Neural Operator (CANO), which reformulates attention via a novel cross-attention mechanism that dynamically clusters queries while preserving full-resolution keys and values. This avoids slice compression loss and removes weight-sharing limits. At the same time, the model remains fast without losing global interactions. Empirically, CANO achieves state-of-the-art performance across canonical PDE benchmarks, covering fluid and solid dynamics (e.g., Navier-Stokes, Airfoil, Plasticity), irregular unstructured geometries (e.g., Pipe Turbulence, Composites), and long-term temporal rollouts. Across solid deformation and turbulent flow benchmarks, CANO achieves lower errors than baselines and exhibits strong geometric adaptability and temporal consistency.