Papers

We introduce the dual affine Robinson-Schensted correspondence that gives a bijection between the extended affine symmetric group and tuples $(\bar{P},\bar{Q},\lambda,N)$, where $\bar{P}$ and $\bar{Q}$ are tabloids, $\lambda$ is a partition, and $N$ is an integer, subject to compatibility conditions. The construction generalizes Fomin's growth diagrams and Viennot's shadow lines for the classical Robinson-Schensted correspondence on the symmetric group, and is dual to the affine matrix ball construction as well as Shi's correspondence, in the sense that the $P$-tabloids are the same, and the $Q$-tabloids are related by affine evacuation. As a consequence, our construction also parametrizes Kazhdan-Lusztig cells in affine type $A$. We conjecture that the growth diagrams we construct admit a natural geometric realization in terms of relative positions of affine flags, similar to the interpretation given by Steinberg and van Leeuwen in the classical case.

This note is the third part of our work devoted to uniqueness sets for spaces of entire functions. Given a discrete set $\Lambda$ with angular density $\Delta$ with respect to the order $\rho$, satisfying some regularity condition, we show that there exists a type-minimizing measure $\Delta_0$ with less than $2\rho$ discrete masses. For the case $\rho=2$, the value of the critical uniqueness type is found in geometric terms.

The conventional linear scaling of beamforming gain with the number of antennas $N$ is not a fundamental physical limitation, but rather a consequence of the half-wavelength spacing that minimizes mutual coupling. Relaxing this constraint enables gains that, in the vicinity of the array axis, exceed those of uncoupled arrays. This paper shows that mutual coupling facilitates the synthesis of super-beams whose widths scale as $1/N$, in contrast to the conventional $1/\sqrt{N}$. These narrower beams achieve a quadratic gain scaling with $N$ provided antenna losses remain sufficiently small, decaying at least exponentially with growing $N$ and polynomially with decreasing spacing. Notably, this gain enhancement does not necessarily require vanishing spacings; it may also emerge for spacings slightly below half wavelength as the array aperture increases.

We present a novel approach for deriving KAM-type linearization theorems directly – and almost immediately – from the existence of the stable foliation for a renormalization operator. We give a few illustrations in dynamics in one and several complex variables, starting with a version of the classical theorem of Arnol'd and ending with a result on persistence of Herman rings in families of two-dimensional maps.

Geometric flows, where an immersed manifold evolves in time according to its own geometry, exhibit important structural properties. For example, surface diffusion dissipates surface area while conserving volume; it is desirable to preserve these properties on discretization. This has motivated a substantial body of research on structure-preserving discretizations for these flows, albeit at low order in time. In this work, we present the first discretization of geometric curvature flows (curve shortening/mean curvature flow and curve/surface diffusion) that preserves the evolution of area and volume at arbitrary order in space and time. The key idea is to introduce auxiliary variables in a particular way so that the derivation of the area dissipation law can be replicated after discretization with continuous Petrov–Galerkin in time. These auxiliary variables are indicated by a general strategy for structure-preservation in time that applies to many other problems. The proposed scheme also preserves mesh quality in the same manner as the minimal deformation rate strategy. We demonstrate its structure-preserving properties and high-order convergence on several benchmark examples.

We show the stability of a penalty-free asymmetric Nitsche's method using Nédélec edge elements for solving curl-curl-type problems with tangential Dirichlet boundary conditions imposed weakly. The main result is an inf-sup stability estimate for the asymmetric bilinear form under an isolated patch condition on the tetrahedral mesh. Applications to a curl-elliptic problem and a magnetic advection-diffusion problem are discussed.

We formalize a ramification theory for finite covers of knot exteriors. Given a knot group $G_K$ and a finite-index subgroup $U\le G_K$, we define meridional inertia subgroups $U\cap g\langle m\rangle g^{-1}$ and the global ramification subgroup $M_U\triangleleft U$ as their normal closure. We then analyze $M_U$ from three complementary viewpoints: (1) finite quotients, where $U/M_U$ is shown to be the universal “maximal meridionally unramified” quotient of $U$; (2) profinite completions, where we identify the closed ramification subgroup $\widehat M_{\widehat U}$ as the closed normal subgroup generated by closed inertia and prove that meridian-preserving isomorphisms of profinite completions preserve inertia and ramification; (3) cohomology, where “unramified” $H^1$-classes (discrete and profinite) are characterized as those vanishing on all inertia subgroups, in direct analogy with number-theoretic inertia conditions in Galois cohomology.

In 1991 Bollobás and Frieze found the threshold for the emergence of a spanning triangulation of a triangle in the binomial random graph, up to a logarithmic factor. In this paper, we find the threshold probability for the emergence of a spanning triangulation of a $k$-gon for any $3\leq k\leq n$, up to a constant factor.

The Canonical Polyadic (CP) tensor decomposition is a well-known method for interpretable analysis of high-dimensional data. Recently, the Generalized CP method (GCP) was introduced by Hong and Kolda to allow for flexible choice of the loss function in the optimization problem defining the CP model, enabling more interpretable decompositions of strongly non-Gaussian data such as count or binary data. Furthermore, Kolda and Hong introduced a version of GCP that leverages randomization and stochastic optimization to address scalability to large, sparse data sets. In this work, we take these ideas a step further and consider synchronous and asynchronous algorithms for parallel GCP tensor decomposition through the GenTen software package, exploiting both shared and distributed memory parallelism. We build on shared memory parallel CP decomposition algorithms utilizing Kokkos for portability across CPU and GPU architectures to support the random sampling and stochastic optimization methods required by GCP. We then couple this approach to the well-known medium-grained distributed memory parallelism scheme developed for traditional CP decompositions through MPI, providing a synchronous, hybrid MPI+Kokkos, parallel GCP decomposition capability. Finally, we propose an asynchronous distributed parallelism approach building on related techniques for federated learning to achieve even better scalability to large data sets. We study the effectiveness of the proposed synchronous and asynchronous approaches vis-a-vis computational cost and accuracy on synthetic and publicly-available real-world datasets of varying sizes, dimensions, and sparsity patterns using several loss functions.

We study the connection problem for a class of linear differential equations of order $N$ closely related to the Baxter equation of the quantum Toda chain. The space of solutions is $N$-dimensional and several linearly independent solutions decay at each singularity, leading to a rich structure of boundary value problems. We derive the weakest quantization conditions compatible with decaying behavior at both singularities, and formulate these conditions in terms of the associated monodromy data. In doing so, we prove the quantization conditions predicted by the topological string/spectral theory duality for a family of deformed Schrödinger equations. More generally, our results point to a hierarchy of spectral problems interpolating between the minimal conditions studied here and the maximally decaying boundary conditions of the $N$-particle quantum Toda chain.

Optimization problems such as the NP-complete 3-SAT provide an important benchmark for the difficult task of finding ground-states in strongly correlated many-body systems with rugged energy landscapes. The study of random 3-SAT problems as Ising spin Hamiltonians in statistical physics has yielded major insights including the existence of a satisfiability phase transition, and the prediction of a critical parameter line of particularly hard instances. Yet, progress on solving those instances has been scarce for several decades. Here, introducing the Target-SAT (TSAT) algorithm, we roughly triple the tractable problem sizes in the hardest regime, with an even greater improvement in a vast range of neighboring regions. By leveraging statistical information hidden in the combinatorial constraints of the problem, TSAT is actively guided in its stochastic local search toward a target within the relevant parameter space. Our analysis also explains why established local search algorithms are limited to relatively small system sizes due to a vast low-energy trap. Furthermore, we characterize the aforementioned critical line in terms of a dominant additional complexity barrier, whose exponential scaling is quickly overcome by TSAT only in the surrounding parameter space. With TSAT, the lead in solving the hardest known random satisfiability problems returns to the realm of stochastic local search algorithms.

Near-field electromagnetic focusing is central to 6G communication, sensing, and integrated sensing and communication (ISAC) systems. However, for a fixed aperture, the resolution-constrained usable range of conventional single-layer transmissive metasurfaces is far shorter than the classical Rayleigh distance. This discrepancy stems not from fundamental near-field physics limitations, but from inadequate wavefront control, implementation imperfections, and the quadratic degradation of axial resolution with distance.To quantify this gap, we distinguish the Rayleigh distance from the engineering-usable near-field distance (UNFD), defined as the maximum range where predefined focusing gain and resolution requirements are jointly satisfied. Under identical aperture, feed, and input power constraints, we investigate how stacked intelligent metasurfaces (SIMs) extend UNFD via cascaded wavefront shaping. A unified framework combining effective-phase-distance and discrete Green's-function operator perspectives is developed, interpreting multilayer SIM design as an operator approximation problem for ideal near-field focusing.We derive low-complexity analytical models revealing an inherent distance-resolution dilemma: lateral resolution degrades linearly with distance, while axial resolution degrades quadratically, making axial performance the dominant bottleneck. Multilayer stacking mitigates this by improving wavefront curvature matching and reducing residual phase error. Engineering correction factors for practical imperfections and a higher-order phase framework for extreme near-field operation are incorporated. Numerical simulations confirm that increasing layer count pushes UNFD closer to the Rayleigh limit, but gains saturate as accumulated losses offset control flexibility benefits.

Interpolation and smoothing using cubic and generalized splines are fundamental tools in data analysis and statistical modeling. Recently, fast computational algorithms were developed for natural $L$-splines of order four, which arise as piecewise solutions to the differential operator $L_{\xi}^2 = (\frac{d^2}{dt^2} - \xi^2)^2$. In this paper, we extend this mathematical framework to the important case of clamped (or complete) boundary conditions, where the first derivatives at the interval endpoints are prescribed. We explicitly construct the governing linear system for the interpolation problem and mathematically prove that the resulting tridiagonal matrix is strictly row diagonally dominant, thereby guaranteeing its invertibility and the numerical stability of the fast algorithm. The proposed method is implemented in MATLAB. Furthermore, the developed clamped $L$-splines provide a foundation for constructing multivariate clamped polysplines, which serve as a promising alternative to Physics-Informed Neural Networks (PINNs) for solving partial differential equations in Mathematical Physics.

What is the maximum, over all 2-colourings of the edges of the $n$-dimensional hypercube $Q_n$, of the minimal number of times a path between a vertex $v$ and its antipode $\bar{v}$ changes colour? A conjecture of Norine, in a form due to Feder and Subi, states that this maximum should be 1. The previous best-known upper bound on the number of colour changes was $(\tfrac{5}{16} + o(1))n$ due to Kirchweger, Peitl, Subercaseaux, and Szeider. We improve this bound and answer a question of Leader and Long by finding a geodesic path with at most $(\tfrac{\pi}{2} + o(1))\sqrt{n}$ colour changes. In fact, we show that this is the expected number of colour changes for a uniformly random start vertex. This is optimal (up to the constant) when the start vertex is chosen uniformly at random.

We establish sharp inequalities for two-dimensional systolic invariants of metrics with positive scalar curvature: the $2$-systole and the spherical $2$-systole of compact Kähler manifolds, and the stable $2$-systole of Riemannian metrics on a general class of $\mathrm{spin}^c$ manifolds and their products. These bounds attain equality precisely for complex projective space $\mathbb{CP}^n$ equipped with the Fubini–Study metric, and admit further refinements for Fano manifolds which distinguish the complex quadric, cubic, and quartic with their canonical Kähler–Einstein structures. We also obtain an algebraic characterization of manifolds admitting Kähler metrics with non-negative total scalar curvature, which implies Gromov's rational-essentialness conjecture for Kähler metrics. Finally, we prove uniform bounds for the stable $2$-systole of $\mathrm{spin}^c$ manifolds under a general essentialness condition, as well as for the Gromov width, volume, and higher stable systoles of Kähler manifolds.

We approach the question of complexification of the diffeomorphism group of the circle by considering real-analytic maps from the circle into the punctured complex plane with winding number +1. Such complex deformations form an infinite-dimensional manifold with partially defined inversion and composition operations, smooth in the sense of Frölicher structures, and with Lie algebra relations at the identity given by the Witt algebra. We compute their second group cohomology with real coefficients. It is generated by two cocycles: One cocycle extending the Bott-Thurston cocycle related to the Gelf'and-Fuks cocycle of the Virasoro algebra, and another Euler class cocycle combining mean rotation and conformal radius. Complex deformations act naturally on the (infinite-dimensional) Segal moduli spaces of Riemann surfaces with enumerated and analytically parametrized boundary components, equipping them with smooth Frölicher structures. We prove a Virasoro uniformization theorem: The tangent spaces of the Segal moduli spaces are spanned by vector fields induced by the Witt algebra. Finally, we relate the actions of complex deformations to Fenchel-Nielsen coordinates and Schiffer variation on finite-dimensional moduli spaces of bordered hyperbolic surfaces with one marked point on each boundary component.

The proliferation of generative artificial intelligence has given rise to an interactive learning environment, where model parameters are continuously updated using not only data generated by natural processes, but also synthetic outputs produced by other models. This paradigm introduces two major challenges: (1) training data are no longer drawn exclusively from the target population, undermining a core assumption of classical statistical learning, and (2) model training processes become inherently correlated, as models interact with one another through repeated exposure to each other's synthetic outputs in a potentially complex manner. Establishing reliable statistical inference in such structured interactive learning environments therefore remains an important open problem. In particular, there is growing concern about model collapse, a phenomenon in which the performance of generative models progressively degrades as they are trained on synthetic data produced by earlier model generations. Prior work on model collapse primarily focuses on a single model trained on its own output, failing to capture model performance in multi-model interactive settings. In this work, we fill this gap by investigating the performance of generative models in an interactive learning environment with general interaction patterns. In particular, we formalize model interactions using directed graphs and show that the occurrence of model collapse depends critically on the topology of the interaction graph. We further derive an explicit necessary and sufficient condition characterizing when model collapse occurs, and establish finite-sample results for linear regression and asymptotic guarantees for general M-estimators. We support our theoretical findings through extensive numerical experiments.

In this paper, we present a novel numerical framework for solving a specific biological reaction-diffusion-advection system of cancer growth in three dimensions (3D) using particles of variable mass. We adopt empirical particle measures to represent cell density and dynamically construct the concentration fields of multiple related chemical species throughout the 3D domain. Efficient interaction between the particles and the spatial grid is achieved through a Particle-in-Cell (PIC) algorithm, while diffusion in space is solved rapidly using a spectral method. We demonstrate that for this particular system, the rate of change of particle mass remains bounded over finite time intervals. Furthermore, in addition to the inherent positivity preservation of cell density guaranteed by the empirical particle measures, the concentrations constructed by the algorithm are also unconditionally positivity-preserving on the spatial grid. Moreover, we present a rigorous error analysis for the proposed method, and numerical experiments confirm the theoretical convergence rates. To the best of our knowledge, this is the first numerical work to solve this system in three dimensions, wherein a rapid spread of cells driven by haptotactic flux is observed, similar to the behavior documented in the two-dimensional case.

For any connected complex reductive group $G$ and element $z$ of its Weyl group $W$, we use work of Lusztig and Abreu-Nigro to compute the graded $W$-character of the intersection cohomology of any closed Lusztig variety for $z$ over the regular semisimple locus of $G$. We relate the resulting formula to unipotent Lusztig varieties, giving a new geometric model for unicellular LLT polynomials. We then consider Laurent polynomials $\alpha_{\psi, G}^z$ indexed by irreducible characters $\psi$, encoding how our formula decomposes into ungraded characters arising from the Springer theory of $G$. From evidence in low rank, we conjecture that if $\psi$ is inflated from type $A$ in a particular way, then the nonzero coefficients of $\alpha_{\psi, G}^z$ are positive and unimodal. This offers an answer to a 1993 question of Haiman about generalizing a conjecture he posed for symmetric groups. We also prove that the matrix formed by the $\alpha_{\psi, G}^z$ is partially triangular, and that their positivity and unimodality properties are stable under inclusions of Levi subgroups.

We obtain new integral representations, expressed as contour integrals in the complex Fourier plane, for the solution of fully nonhomogeneous interface problems for the linearized Cahn-Hilliard equation with arbitrary initial data on the line and general interface conditions prescribed at the origin. Cahn-Hilliard-type models emerge in applied mathematics in connection to a spectacular variety of phenomena of mathematical physics, continuum mechanics, chemistry and biology. A novel implementation of Fokas' unified transform method is in force herein for a fourth-order operator for the first time, with particular challenges arising due to the nature and the generality of the problems under consideration. Our explicit formulae directly lend themselves to exploration of the solution's qualitative properties such as regularity and asymptotic behavior. This work is also useful in the investigation of well-posedness for nonlinear counterparts as well as in the study of free-boundary and diffuse-interface problems.

This work develops a rate-distortion-based approach to stochastic Chase decoding of algebraic codes over binary memoryless symmetric (BMS) channels, replacing the heuristics traditionally used to determine flip probabilities with information-theoretically grounded flipping rules. In particular, we reinterpret stochastic Chase decoding as a random-coding construction for error-pattern covering codes. Our approach builds on the framework of Nguyen et al., who introduced a rate-distortion formulation of multiple-attempt decoding for Reed-Solomon codes over nonbinary channels. In their formulation, erasure patterns are generated so as to align with, and thereby mask, hard-decision errors. We adapt this framework to the design of bit-flip probabilities for Chase decoding over BMS channels. This yields an explicit characterization of the asymptotically optimal bit-flipping rule, together with the expected list size required to ensure that the transmitted codeword appears in the decoding list with high probability. Moreover, for binary and quaternary symmetric channels, we demonstrate that the optimal bit-flipping rule, determined by exhaustive search, closely matches the information-theoretic rule even at short block lengths.

We investigate quantum periods and toric Landau-Ginzburg models under divisorial contractions of terminal Fano threefolds. Let $g:Y \rightarrow X$ be a divisorial contraction between $\mathbb{Q}$-factorial Fano threefolds with ordinary terminal singularities and $E$ be the exceptional divisor. Assuming that the center of the contraction is either a smooth point, a terminal quotient point, a point of type cA/n, or a smooth curve with singularities of type cA or cA/n, we prove the regularized period identity $$ \lim_{r\to+\infty}\hat{G}_{Y,rE}(t)=\hat{G}_X(t) $$ where $\hat{G}_{Y,rE}(t)$ and $\hat{G}_X(t)$ are the regularized quantum periods of $(Y,rE)$ and $X$ respectively. This gives a mirror approach to the computation of the Sarkisov links and higher syzygies of central models of dimension 3.

Privacy constraints have driven the rise of federated learning (FL), which enables multi-site analyses without sharing individual participant data. Existing FL estimators largely assume complete data, whereas multi-site studies often face missingness. We develop a framework for FL with missing data, identifying conditions under which the complete case (CC) estimator is preferred over the inverse probability weighting (IPW) estimator. For settings where the CC estimator leads to bias, we introduce a calibrated weight estimation approach that combines candidate weighting models across sites and remains consistent if at least one is correctly specified at each site; we further show that pooling many weighting candidate models with redundant information degrades the calibrated estimator, so a small set is preferable. Consistency conditions are stated at the site level, ensuring that the federated estimator inherits validity from site-level properties. We prove consistency and derive a sandwich variance estimator that accounts for uncertainty in the outcome model, and in both the estimated weighting models and the calibration step. Additionally, we show that all estimators require only one or a few communication rounds, making them practical under real-world data-governance constraints. We illustrate the framework by evaluating risk factors for 90-day mortality among patients with pleural infections treated with intrapleural enzyme therapy.

We study the random conductance model on the lattice $\Z^d$, i.e. we consider a linear, finite-difference, divergence-form operator with random conductances $a$. We allow the conductances $a$ to be unbounded and degenerate. Assuming the conductances satisfy a spectral-gap inequality, we establish sharp bounds on the spatial growth of correctors, together with a quantitative relation between the stochastic integrability of the correctors and that of $a$.

We propose a notion of scalar curvature lower bounds in a three-dimensional Riemannian manifold endowed with a $C^0$ metric based on the monotonicity of the Hawking mass along the inverse mean curvature flow. We present a stability theorem for continuous Riemannian metrics with nonnegative scalar curvature in such IMCF sense.

We study the Hermitian hull-variation problem for vector rank-metric codes. Except for one parameter pair, we show that the Hermitian hull dimension of such a code can be reduced to any smaller value within its equivalence class, and in particular every such code is equivalent to a Hermitian LCD code. We then address the existence of maximum rank distance (MRD) codes with prescribed Hermitian hull dimension. To this end, we introduce the notion of a scaled trace-self-dual basis of a finite field extension, which exists in all cases, and use it to construct Hermitian self-orthogonal generalized Gabidulin codes for every prime power. Combined with the hull-variation theorem, this yields MRD codes attaining every admissible Hermitian hull dimension.

We develop a motivic framework for Feynman integrals of one-loop graphs in momentum space. Its advantage compared to the already existing framework in Feynman representation is that it naturally includes graphs with cuts. To each such graph, we associate a motivic local system over the space of generic kinematics. Our construction is functorial with respect to the natural operations on graphs: edge contraction and cutting. We compute the weight-graded pieces of the motivic local systems. They are Tate twists of quadratic Artin motives associated with maximally cut quotient graphs. We also derive a formula for the (co)action of the de Rham motivic Galois group, expressed in terms of cut quotient graphs.

The Fourier extension conjecture of E. Stein was proved in the plane in 1970 by C. Fefferman, see also Zygmund and Carleson and Sjölin, with simplifications given by other authors later on, in particular by L. Hörmander and T. Tao. We discuss yet two more arguments for this classical theorem on the parabola. The first argument uses C. Fefferman's decoupling together with a decomposition into Haar wavelets. This sets the stage for the second argument whose point of departure is the bilinear characterization of Tao, Vargas and Vega, and relies on the bilinear interplay with the classical wave packet constructions and discrete characterizations with an induction on scales. Each of the above two arguments rely on some form of Fefferman's convolution decoupling and the special nature of the critical planar index 4 as a positive even integer, but the second might be more amenable to weighted generalizations.

Independent component (IC) models are a standard tool for representing multivariate data in statistics, signal processing, and machine learning. Despite the extensive use of IC models, much less attention has been given to goodness-of-fit tests for assessing their compatibility with data. We develop the first goodness-of-fit test for IC models that is supported by a theoretical validity guarantee when the data dimension and sample size diverge proportionally. This is made possible by the fact that the test does not rely on a pre-whitening step, which often limits the applicability of other goodness-of-fit tests in high dimensions. Our theoretical analysis is complemented with numerical experiments that demonstrate the test's size control and power under a range of conditions. In addition, we provide examples involving gene-expression data to illustrate that the test has potential for effective diagnostic use in practice.

This paper establishes the projective equivalence between the Knizhnik-Zamolodchikov connection and the Hitchin connection in genus 0 with at least 3 marked points. The Knizhnik-Zamolodchikov connection is defined on the sheaf of conformal blocks in the Tsuchiya-Ueno-Yamada model of conformal field theory. The Hitchin connection is defined on the Verlinde bundle via geometric quantisation of the moduli space of flat connections. Pauly's isomorphism establishes the equivalence of these two vector bundles. The main theorem of this paper is that the isomorphism intertwines these two connections up to a scalar-valued one-form. In addition, this theorem is used to construct a Hitchin connection through an auxiliary metaplectic correction. As a corollary of the main theorem, this construction of the Hitchin connection is projectively unique and projectively flat.