Papers

We prove local well-posedness in the Hadamard sense for the three-dimensional compressible Euler equations governing a liquid with a free surface and no surface tension. Under the Taylor sign condition and the boundary compatibility conditions, we establish existence, uniqueness, and strong continuous dependence in a hybrid Sobolev class for every $s>3$. The velocity and interface have $H^s$ regularity, while the enthalpy and velocity divergence belong to $H^{s+1/2}$ and $H^{s-1/2}$, respectively. This separation reflects the coupling of a nondegenerate acoustic equation with the free-surface and vorticity dynamics. The proof uses regularization operators that smooth the moving domain and extend the nonlinear compatibility hierarchy, together with $L^2$ and partial $H^2$ distance estimates for solutions on different domains. Regular solutions are constructed by a regularized forward Euler scheme. High-order energy estimates and frequency envelopes then yield strong convergence of smooth approximations and continuity of the data-to-solution map. Integer Sobolev indices are treated using a Lions–Magenes endpoint compatibility condition.

Hong Wang has made several tremendous contributions to harmonic analysis and geometric measure theory. On the 23rd of July 2026, she was awarded the Fields Medal. In this short article we shall focus on her work on the Kakeya problem and other problems related to the Fourier transform. We will try to put her work in historical context and present a small selection of her contributions to harmonic analysis.

The Wind Farm Layout Optimization problem consists of placing a given number of turbines within a given area so as to maximize energy production. One of its main challenges is the wake effect between turbines, which can strongly affect production and is a cumbersome function to evaluate and to embed in an optimization framework. In this paper we introduce a new endogenous wake model (EndoWake), based on the idea of treating the wind speed at every grid point as a decision variable, linked to its upwind neighbors through linear propagation constraints. This makes it possible to incorporate the wake equations directly into a Mixed-Integer Linear Programming (MILP) model. We then develop such a MILP model and, in particular, introduce a new class of valid "band" inequalities, showing that they substantially strengthen the root bound. Building on this, we design a branch-and-check scheme: the master problem retains only the site variables together with one hypograph variable per cell and scenario, integer solutions are evaluated by an exact linear-time oracle, and the optimality cuts are problem-specific "upwind cuts". Computational results are reported on a ladder of square grids under a four-sector axial wind rose, demonstrating the effectiveness of the proposed improvements.

In many statistical settings, the available data and maintained assumptions do not suffice to uniquely identify the model parameters of interest. In such cases, one can only identify sets which are guaranteed to contain the true parameters. These are often characterized through linear programs that optimize over models compatible with the observed data. These programs can be infinite-dimensional in the optimizer and the number of constraints. We provide a unified way to characterize and solve such optimization problems by phrasing them as optimal transport problems on path spaces. This allows us to regularize the problem with an entropy penalty, recasting it as a multi-marginal entropic optimal transport problem, which can be solved efficiently via Sinkhorn iterations. In addition, it allows us to establish convergence of the regularized value to the sharpest bound, derive consistency rates for a plug-in estimator, and obtain asymptotic distribution for approximate bounds. The method is general and accommodates settings ranging from instrumental variable models with continuous variables to welfare estimation in heterogeneous demand models. We verify the statistical and computational properties in simulations and provide an application to demand estimation.

We study the finite basis problem for flat semirings of the forms \(S_c(W)\), \(M_c(W)\), and \(M(W)\), where \(W\) is a nonempty set of words in a free commutative semigroup, a free commutative monoid, and a free monoid, respectively. We completely classify such flat semirings with respect to the finite basis property, allowing \(W\) to be infinite. We prove that \(S_c(W)\) is finitely based if and only if every word in \(W\) is either a cube of a letter or has length at most two, whereas \(M_c(W)\) and \(M(W)\) are finitely based if and only if \(W\) consists solely of the empty word. As applications, we recover the nonfinite basability of \(\flat(\mathbb{Z})\) and the max-plus semiring \((\mathbb{Z},\max,+)\).

We describe adapol, a Python package for fitting Matsubara Green's functions, self-energies, and hybridization functions by a compact sum of simple poles with real pole locations and matrix-valued residues. This pole-fitting step arises in many contexts, including hybridization fitting for quantum impurity solvers based on exact diagonalization, perturbation theory, and tensor networks, the compression of existing pole expansions such as the discrete Lehmann representation, diagrammatic methods based on discrete pole representations, and the analytic continuation of Matsubara Green's functions. adapol uses a modified version of the AAA rational approximation algorithm to estimate pole locations, and non-convex optimization to refine them. This procedure yields accurate and compact fits in a black-box and noise-robust manner. adapol provides a simple, self-contained interface with extensive documentation and examples, as well as an interface to the TRIQS package.

We establish exponential iteration lower bounds in the number of states for Howard's policy iteration on deterministic discounted and average-reward Markov decision processes (MDPs), with a constant number of actions per state. This rules out strong polynomiality of Howard's policy iteration in both settings. We further prove that a Dantzig counterpart of Howard's policy iteration runs in strongly polynomial time on deterministic average-reward MDPs. Together with the known strongly polynomial bound for Dantzig's rule on deterministic discounted MDPs, these results establish an exponential separation between the two algorithms in both settings. This separation between Howard's decentralized and simultaneous selfish improvements and Dantzig's coordinated selection of a single action with the largest improvement across all states reveals a “price” of algorithmic anarchy.

We consider the functional $I_{\delta,\Psi_1,\Psi_2}:X \to \mathbb{R}$ defined for any $z=(u,v) \in X$ as \begin{align*} I_{\delta,\Psi_1,\Psi_2}(z) & = \int_{\Omega} \Psi_1(\nabla u) \, dx + \int_{\Omega} \Psi_2(\nabla v ) \, dx & - \int_{\Omega} H(\delta,x,u(x),v(x)) \,dx, \end{align*} where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $\Psi_1, \Psi_2: \mathbb{R}^N \to \mathbb{R}$ are convex functions satisfying suitable conditions for $1 < p, q < N$, and the nonlinearity $H$ is allowed to exhibit both subcritical and critical growth and may depend on a parameter $\delta \in I \subseteq \mathbb{R}$. Here, the functional space setting is given by the product space $X :=W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega)$ equipped with the norm $\|z\|= \|u\|_{1,p} + \|v\|_{1,q}$ for any $z=(u,v)\in X$, where $\| \cdot \|_{1,s}$ is the usual norm in $W^{1,s}_0(\Omega)$. In this paper we prove that $I_{\delta,\Psi_1,\Psi_2}'$ is of class $(S)_+$ and we infer that each isolated critical point of $I_{\delta,\Psi_1,\Psi_2}$ has critical groups of finite type and a Poincaré-Hopf formula holds.

For each $n \in \mathbf{N}$, let $T_n$ be a uniformly random (rooted, Type I) triangulation of the sphere with $n$ vertices, viewed as a metric space equipped with its graph distance. We show that for every $\delta>0$, with probability tending to $1$ as $n \to \infty$, every embedding of $T_n$ into a separable Hilbert space has distortion at least $(\log n)^{1/4-\delta}$.

We study random polynomials of the form $R(x)=x^n+\omega_{n-1}x^{n-1}+\cdots+\omega_0$, where $\omega_0,\dots,\omega_{n-1}$ are independent, uniformly bounded integer-valued random variables, and $\omega_1,\dots,\omega_{n-1}$ have a fixed common law $\mu$. We prove (unconditionally) that, if the Rényi entropy of order $2$ satisfies $H_2(\mu)=-\log\|\mu\|_2^2>12$, then $\mathbb{P}(\operatorname{disc}(R)\text{ is a square})=O_\mu(1/\log n)$. Combined with previous results, this shows that, for such measures $\mu$ and under additional hypotheses, $R$ has full Galois group $\mathrm{Sym}(n)$ with high probability, when conditioned on $\omega_0\ne 0$.

We present a "myopic" Tutte polynomial for graphs on $\mathbb{R}P^2$ which takes only nullhomologous spanning subgraphs as input. It recovers the generalized Krushkal polynomial and Drobotukhina's analogue of the Jones polynomial for alternating, nullhomologous links in $\mathbb{R}P^3$. We use this myopic Tutte polynomial to prove an analogue of the Kauffman-Murasugi-Thistlethwaite Theorem, relating the Jones polynomial of an alternating link to certain refinements of the crossing number. Finally, we construct a spanning tree model for the Khovanov homology of nullhomologous links, mirroring work by Champanerkar-Kofman and Wehrli for links in $S^3$. For alternating links, we use our model to prove that the Khovanov homology in $\mathbb{Z}/2\mathbb{Z}$ coefficients is determined entirely by the Jones polynomial and signatures of the link.

Motivated by the computational challenges of large-scale Cox regression, we study stochastic minimization of LogSumExp objectives over large sets. Mini-batch normalizer estimates generally yield biased gradients. We instead use a softplus surrogate that introduces one auxiliary scalar per normalizer and admits unbiased single-sample gradients. For smooth convex LogSumExp objectives, we prove an $O(T^{-1/2})$ averaged objective bound, improving the previous $T^{-1/4}$ analysis. With a strongly convex regularizer on the original variable, we also obtain a last-iterate squared-error rate of $\widetilde{O}(T^{-1})$ without strong convexity in the auxiliary variables. For Cox regression, the normalizers are defined over nested risk sets. We exploit this structure by grouping neighboring failures and sharing one auxiliary variable per group. The resulting compressed objective admits uniform score and curvature bounds that control the errors from grouping and softplus approximation. Together with the general optimization result, these bounds give a mean-square rate of $T^{-4/5}$, up to logarithmic factors, relative to the full Cox solution. The compressed estimator also matches the full estimator's asymptotic distribution. Experiments on synthetic and real survival datasets with slowly decreasing risk sets show a favorable performance relative to stochastic baselines.

We consider the wave equation on a manifold of dimension d>1 with smooth strictly convex boundary, with Neumann boundary condition. We construct a sharp local in time parametrix for the Neumann wave equation near glancing, extending the classical Melrose-Taylor construction for the Dirichlet problem to the Neumann boundary condition. Our construction is based on the microlocal framework and the parametrix developed in [10] for the Dirichlet problem, together with the Melrose-Taylor argument for solving the transport equations under more general, affine-type boundary conditions. Once the Neumann parametrix is constructed, the dispersive and Strichartz estimates follow the analysis developed in [10] for the Dirichlet problem. We therefore only recall the main ingredients of that argument. In particular, the fixed time decay rate for the Green function exhibits the same loss of 1/4 with respect to the boundaryless case, associated with swallowtail type singularities in the wave front set, and this decay is optimal. Moreover, the corresponding Strichartz estimates are obtained by balancing lossy long time estimates at a given incidence with short time ones with no loss: for d=3, this heuristically means that, on average, the decay loss is only 1/6 .

Dynamic approaches to minimize the classical curvature-elasticity free energy for a fluid lipid membrane, are extended towards a Surface Beris–Edwards–Helfrich model. This extension explicitly treats the liquid crystal structure of the membrane, which results from the lipid molecules, that are, on average, oriented normal to the surface. Locally varying liquid crystal order leads to local variations in bending rigidities and surface viscosity and thus influences the shape evolution. This provides a multiscale coupling between the local membrane mechanics on the level of the lipids and the mesoscopic length and time scales of biological functions. The model is derived using the Lagrange–d'Alembert principle. We provide a numerical algorithm to solve the equations in the one-constant approximation and demonstrate the impact on the emerging equilibrium shapes, which not only depend on the specified conserved surface area and enclosed volume, as for the classical curvature-elasticity free energy, but also the elastic parameter. Exploring the dynamic evolution shows a tight coupling of tangential flow, shape evolution and liquid crystalline order. We further point to extensions towards lipid phase separation and asymmetric lipid bilayers.

This paper is devoted to questions concerning rational quadrilaterals and rational tetrahedra. In both cases there is a finite group which acts on the set. In the case of rational quadrilaterals it was discovered by Kurt Heegner; in the second case there are two variants, one again due to Heegner the other a consequence of an involution introduced by Tullio Regge. These allow one to construct a number of solutions to the appropriate diophantine equation from given ones. The original question was posed by Kummer. There is an intimate connection with the theory of the tetrahedroid and so Kummer surfaces.

We study discrete-time quantum trajectories of a finite-dimensional system in a disordered environment, where the instrument applied at each step is determined by an invertible measure-preserving dynamical system. For general quantum instruments on a standard Borel outcome space, we construct the quenched probability law on sequences of measurement outcomes for a fixed realization of the disorder and a given initial state. Under a quenched Doeblin condition with a uniform deterministic minorization constant and an environment-dependent minorizing probability measure, we establish the existence and uniqueness of an equivariant family of random posterior laws and identify its barycenter as the unique dynamically stationary state for the associated non-selective channel cocycle. If the environment is ergodic, we prove a pathwise ergodic theorem valid for every measurable choice of initial state: for almost every realization of the disorder, time averages of posterior states converge almost surely under the corresponding quenched law to the disorder average of the unique dynamically stationary state. For an ergodic environment, we establish quenched variance asymptotics and, when the asymptotic variance is positive, a central limit theorem and standardized Berry–Esseen bounds for additive functionals generated by bounded measurable real-valued functions of the environment and posterior state. These results hold for every measurable initial state and almost every disorder realization. We then show that the stationary annealed joint process of instruments and posterior states inherits the $\alpha$-mixing of the instrument process, up to an exponentially decaying term. Finally, we provide classes of examples involving perfect and imperfect measurements, including models with continuous or finite outcome spaces, that satisfy the standing Doeblin condition.

We introduce quantum tilted walks, a quantum algorithmic framework for solving exact combinatorial optimization problems. The framework applies an average of powers of a tilted Hamiltonian that biases the discriminant matrix of a base Markov chain (mixer) with the objective function. Our starting point is quantum short-path algorithms, which prepare the ground state of such a Hamiltonian and obtain super-quadratic speedups over exhaustive search for certain combinatorial optimization problems. Recently, Le Gall and Tamaki~(arXiv:2604.12131) developed a classical conditioning-and-search algorithm for weighted MAX-E$k$-LIN2 and weighted MAX-$k$-CSP. Under the same assumptions, their algorithm is only sub-quadratically slower than quantum short-path algorithms. Consequently, existing short-path algorithms do not establish a super-quadratic speedup over this stronger classical baseline. For maximization problems, we give conditions under which tilted walks increase the amplitude on the target state with high objective value when initialized from a starting state with lower objective value. This framework captures conditioning-and-search and yields super-quadratic speedups over it for the same problems. While our framework recovers quantum short-path algorithms as a special case, it neither requires ground-state preparation nor initialization in the ground state of the base mixer. We demonstrate these advantages on a synthetic optimization problem for which tilted walks achieve a super-quadratic speedup whereas the short-path algorithms do not.

We prove that the Kazhdan–Lusztig polynomial of a loopless sparse paving matroid is unimodal. When the rank is at least five and the corank is positive, we show that the coefficient sequence is strictly unimodal except for three explicitly determined parameter triples, each giving two equal maximal coefficients. We also determine the modes and classify the degenerate loopless sparse paving matroids of positive rank and corank.

We investigate the Casimir effect of a massive, self-interacting real scalar field with quartic coupling in a $(3+1)$-dimensional Hořava-Lifshitz-type theory, subject to rough boundaries obeying Dirichlet boundary conditions. Using the effective action and generalized $\zeta$-function approach, we derive the renormalized effective potential, incorporating anisotropic scaling, boundary roughness, and temperature in the low-temperature regime. We determine the topological mass at one-loop order and the Casimir energy up to two loops. In the massless limit, the topological mass and the two-loop Casimir contribution exhibit infrared divergences for odd values $z>1$, associated with the absence of an intrinsic mass scale, while the one-loop Casimir energy remains finite. For $z=1$, the standard relativistic Dirichlet result is recovered in the zero-temperature and vanishing-roughness limits.

We present a universal entanglement distillation protocol that asymptotically achieves the distillable entanglement of any unknown bipartite state in a fixed finite dimension. That is, its ebit yield attains every rate below the optimum that would be available if the state were known, with vanishing trace-distance error. Our approach combines tomography by local operations and classical communication with an adaptive distillation routine, using a local robustness lemma to control finite-sample estimation errors. We also characterize worst-case distillation from a family of states: the optimal uniform rate is the minimum distillable entanglement on the family's closure. These results show that ignorance of the state need not incur an asymptotic rate penalty.

We study irrevocable maximum-cardinality matching in trees revealed by successive leaf attachments, with a known horizon and an exogenous growth law that is misspecified or unknown. For deterministic affine attachment forecasts with nonnegative degree reinforcement, the optimal threshold policy loses at most twice the cumulative expected conditional total-variation error relative to an online oracle knowing the actual growth law. This follows from a unit-span property of the Bellman continuation score and has no additional horizon factor. A four-vertex example attains the coefficient two for the specified deterministic policy, and a two-model argument gives a lower bound linear in the model-error budget for arbitrary policies under general misspecification. For uniform-preferential attachment, the local error has an exact expression through the leaf count. When its constant mixture parameter is unknown, we estimate it from the same growing tree and update the threshold policy at geometric times. A parameter-sensitivity bound for individual Bellman prices and uniform degree-moment estimates yield expected regret $O(\sqrt{n}\log^2 n)$, using $O(n^2\log n)$ arithmetic operations and $O(n)$ stored entries. The exact minimax rate remains open.

In the adjacent transposition shuffle on a cycle, $n$ distinct cards are placed at the vertices of a cycle, and adjacent cards swap positions according to independent Poisson clocks of rate 1. We prove that this Markov chain exhibits total variation cutoff at time $n^2\log n/(8\pi^2)$, with a window of order at most $n^2\log\log n$.

We compute the Morse index of several complete nonorientable minimal surfaces immersed in Euclidean three-space. In particular, we prove that the Meeks minimal Möbius band has Morse index two, which is the least possible Morse index for such nonorientable minimal surfaces, by previous works of Ros and Chodosh-Maximo. This is the first known minimal surface with index two. We show that the López minimal Klein bottle has Morse index three. As a consequence of this computation, we extend results in the literature and observe that any complete minimal surface immersed in Euclidean three-space with total curvature $8 \pi$ has Morse index three. We also compute the Morse index of the Oliveira family of minimal Möbius bands in terms of their total curvature and show that every non-negative integer is the Morse index of a complete minimal surface immersed in Euclidean three-space. We show that five is a sharp lower bound for the Morse index of any minimal oriented double cover in Euclidean three-space, and equality is attained by the double cover of the Meeks minimal Möbius band. Finally, we prove results towards the classification of complete minimal surfaces with Morse index two immersed in Euclidean three-space, obtaining topological and geometric restrictions on such surfaces.

In the present paper, we extend the results in the sequence of works \cite{KS-GCM1} \cite{KS-GCM2} \cite{KS:Kerr} by Sergiu Klainerman and the author, \cite{GKS22} by Elena Giorgi, Sergiu Klainerman and the author, and \cite{Shen} by Dawei Shen from the slowly rotating regime $|a|\ll m$ to the full subextremal range $|a|<m$, thereby completing the proof of the Kerr stability conjecture. The restriction $|a|\ll m$ enters in fact only through the estimates for wave equations and for the Bianchi system established in \cite{GKS22}. To remove this restriction, we crucially rely on the two companion papers \cite{MaSz24} \cite{MaSz26} by Siyuan Ma and the author, in which we prove energy-Morawetz estimates respectively for the inhomogeneous scalar wave equation and for inhomogeneous Teukolsky equations on perturbations of Kerr with $|a|<m$. Additionally, some of the results of the present paper are proved in our companion paper \cite{Sze}.

The bosonic quantum entropy power inequality bounds the entropy of a beam-splitter output in terms of the input entropies. We determine its complete equality class among independent inputs of finite mean photon number: for any number of modes and any transmissivity strictly between zero and one, equality in either the linear or the exponential form holds if and only if the inputs are Gaussian with the same covariance matrix, allowing arbitrary displacements. The main step shows that equality for one output forces the two outputs to be independent: a thermal auxiliary converts equality into preservation of mutual information by a fixed noisy channel, and a two-outcome instrument shows that preservation of mutual information requires a product state with the reference. The quantum Darmois–Skitovich theorem then gives Gaussianity, while a central-limit argument reduces exponential equality to linear equality.

We develop a theory of exact transmission of finite-dimensional $C^*$-algebras through quantum channels. These algebras describe hybrid classical-quantum information, allowing the dimension of the quantum system to depend on the classical message. Allowing arbitrary encodings and decodings yields a transmission set of algebra types, ordered by embedding, that unifies zero-error information theory with operator-algebraic error correction. We ask if this set admits a dominating algebra into which every transmittable algebra embeds. Our central finding is that domination can fail even in small dimensions. In its absence, several incomparable maximal algebras can describe different optimal uses of the same channel, forcing the user to select one depending on the operational task and the type of information to be preserved. We introduce hybrid capacities that reconstruct the only possible dominating algebra type and present a complete finite characterization of domination using minimal forbidden algebra types. We identify dominating algebras for channels whose operator systems are graph-isomorphic to $*$-algebras, including highly divisible channels, and for channels with zero one-shot zero-error quantum capacity. Under tensor products, we prove that joint coding can produce new algebra types, giving an algebraic analogue of superadditivity from Shannon theory. Domination can fail for the joint use of two channels, even when both channels separately admit dominating algebras. Finally, we construct a channel whose $n$-fold tensor powers have doubly exponentially many maximal algebra types, attaining the largest possible growth scaling for finite-dimensional channels. Consequently, new transmittable algebra types appear at arbitrarily large block-lengths for this channel, so its full transmission structure cannot be generated from any finite collection of bounded-block-length codes.

We review the discrete theory of differential-geometric Poisson brackets as introduced by B. A. Dubrovin, and we present a complete proof of their characterisation in the non-degenerate case. We use this characterisation to show several novel examples of such structures, providing a complete classification for non-degenerate four-dimensional differential-geometric Poisson brackets under the additional assumption of the Lie algebra to be of quasi-Frobenius type. We also present a complete classification of quasi-Frobenius filiform N-graded Lie algebras, including two families in arbitrary dimensions.

This version contains the introduction and conclusion of my PhD thesis, "Freely Generated Categorical Structures and Automatic Differentiation", together with its English and Dutch summaries. The full thesis consists of an introductory chapter, six joint research papers, and a concluding chapter, developed during my PhD studies at Utrecht University under the supervision of Gabriele Keller and Matthijs Vákár. The research papers are available separately and are not reproduced here. The introduction presents the scope of the thesis, explains the contributions of the six papers and their connections, and introduces the categorical foundations of our approach. The guiding idea is that programming languages, viewed as freely generated categorical structures, provide a principled setting for constructing structure-preserving program transformations and proving their correctness. Automatic differentiation supplies the central application: we study forward- and reverse-mode differentiation for expressive typed languages, including higher-order functions, recursive types, iteration and partiality. The semantic requirements of these transformations also motivate independent mathematical results on free distributive and extensive categories, cartesian closedness, and Grothendieck constructions. The conclusion brings these contributions together, discusses their limitations, and outlines further directions. Throughout, the thesis develops a dialogue between theory and practice: categorical semantics guides the construction of reliable and practically useful program transformations, while the demands of computation lead to new categorical structures and results.

We construct a smooth complex projective rational surface whose automorphism group is discrete and not finitely generated. It is obtained by blowing up a point on a branch curve of a rational quotient of a product Kummer surface. Every point with transcendental coordinate gives such a surface. The proof uses leading jets along the branch curve, the complete family of blowups and the cubic intersection of the family, which are not being considered in relevant earlier works.

For any pseudo-Euclidean space $V \simeq \mathbb{R}^{p, q}, p \geq 3$ and module $W$ over the even Clifford algebra $\text{Cl}^{0}(V)$, V. Córtes constructed noncompact homogeneous quaternion pseudo-Kahler space. In particular, such homogeneous space is always Einstein and for $p=3$ it is Riemannian. Based on this approach we construct the series of Riemannian Einstein homogoneous spaces with negative scalar curvature associated with $V\simeq \mathbb{R}^{p, q}, p\equiv 3 \pmod{4}$ and an irreducible $\text{Cl}^{0}(V)$-module $W$.