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. With applications to conformal field theory in mind, we compute the second group cohomology group with real coefficients, finding cocycles extending the Bott-Thurston cocycle related to the Gelf'and-Fuks cocycle of the Virasoro algebra, and a natural relative cocycle combining the rotation number and conformal radius of a complex deformation. Complex deformations act naturally on the (infinite-dimensional) Segal moduli spaces of Riemann surfaces with analytically parametrized boundary components. These actions equip said moduli spaces 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 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.

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 $α_{ψ, G}^z$ indexed by irreducible characters $ψ$, encoding how our formula decomposes into ungraded characters arising from the Springer theory of $G$. From evidence in low rank, we conjecture that if $ψ$ is inflated from type $A$ in a particular way, then the nonzero coefficients of $α_{ψ, G}^z$ 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 $α_{ψ, G}^z$ 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. 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 fails, we introduce a calibrated weight estimation approach that combines candidate weighting models across sites and remains consistent if at least one is correctly specified. Consistency conditions are stated at the site level, ensuring that the federated estimator inherits validity from local properties. We derive a sandwich variance estimator that accounts for uncertainty in weight estimation, and 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 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 \emph{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 proofs of this classical theorem on the parabola. The first proof uses C. Fefferman's decoupling together with a decomposition into Haar wavelets. This sets the stage for the second proof in Rios and Sawyer, that uses smooth Alpert wavelets and a new decoupling method, which exploits averaging smooth Alpert projections over grids, the extraction of Dirichlet kernels, and periodic stationary phase, all of which was extended to higher dimensions in arXiv:2512.24990v7.

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.

Most of the laws of Nature involve derivatives up to the second order. Ostrogradski was the first to seek a formulation of the equations of higher-order derivatives. He extended Hamilton's equations by considering Lagrangians that depend on higher-order derivatives of generalized coordinates. The Hamilton-Ostrogradski formulation served as the basis for later studies with higher-order derivatives. However, the Hamilton-Ostrogradski formalism is rarely discussed in textbooks or the pedagogical literature. This motivated us to show how the Hamilton-Ostrogradski formalism can be applied it to the Pais-Uhlenbeck oscillator. We hope that the approach presented in this work can serve as a basis for discussion in advanced classical mechanics courses.

A polygon is equable if its area is equal to its perimeter. A pair of polygons is an amicable pair if the area of the first is equal to the perimeter of the second, and vice versa. A polygon is a lattice polygon if its vertices lie on the integer lattice. We show that amicable lattice rhombuses are actually equable.

Kernel methods are one of the cornerstones of learning-based control, modern system identification, surrogate modelling, and related fields. A key advantage of this class of learning and function approximation methods is the availability of quantitative error bounds, which in turn play a key role in guaranteeing the safety of learned controllers and related learning-based algorithms. However, these error bounds rely on a particular property of the target function -- its reproducing kernel Hilbert space (RKHS) norm -- which is usually impossible to obtain in practice. Motivated by this severe shortcoming, we present a novel sampling-based RKHS norm estimation approach with a solid theoretical foundation, leveraging very recent advances in the theory of superconvergence in kernel methods. Our method is applicable to a broad range of practically relevant function classes and requires only reasonable prior knowledge about the target function. Extensive numerical experiments demonstrate the efficacy and practical applicability of the proposed method. By providing a reliable RKHS norm estimation approach, we remove a major obstacle to the practical deployment of learning-based control algorithms.

For $α\ge 0$, let $\mathcal{W}(α)$ be the class of all analytic functions in the unit disk $\mathbb{D}$ with normalization $f(0) = 0 $ and $ f'(0) = 1 $ that satisfy the relation $Re\,\{f'(z) + αz f''(z)\} > 0$. This article aims to establish sharp bounds for logarithmic coefficients $γ_1$, $γ_2$ and $γ_3$ and logarithmic inverse coefficients $Γ_1$, $Γ_2$ and $Γ_3$ of functions in $\mathcal{W}(α)$. The sharp upper and lower bounds for $\bigl|\,γ_2 \,\bigr|-\bigl|\,γ_1\,\bigr|$ and $\bigl|\,Γ_2 \,\bigr|-\bigl|\,Γ_1\,\bigr|$ have been obtained for the class $\mathcal{W}{(α)}$. In addition, we establish sharp inequality for the second Hankel determinant of the logarithmic and inverse logarithmic coefficients for the class $\mathcal{W}{(1)}$.

The seminal work of Roth, Sönmez, & Ünver shows that the Edmonds-Gallai structure theorem for non-bipartite matching can be leveraged to yield a randomized algorithm to match patient-donor pairs in kidney exchange with extraordinarily strong properties. This breakthrough led to randomized polynomial-time algorithms to find a maximum-cardinality matching maximizing individual fairness objectives--measured by the probability that nodes are matched--such as Nash social welfare. But the exchanges allowed in practice go beyond cardinality matching, generalizing to weighted variants and allowing structures such as paths and 3-cycles. We show that strongly polynomial algorithms guaranteeing the same fairness properties can be obtained in weighted settings for matching and 2-paths. While even maximum cardinality coverage with cycles and paths of length at least three is NP-hard, we provide a general result showing that any optimization subroutine (for whichever structure is allowed) can be bootstrapped using a polynomial number of calls to yield a mechanism that has analogous fairness properties to those obtained for matching. We complement these theoretical results with computational results, both on well-studied synthetic data-sets and on samples drawn from real data, that demonstrate the striking advantages of adding fairness considerations to more general kidney-exchange mechanisms.

We develop a Galois descent approach to finite-field Fourier spectra over an arbitrary finite base field. Let $\mathbb K=\mathbb F_q$ and $\mathbb L=\mathbb F_{q^m}$. If a Fourier transform is applied to a $\mathbb K$-valued vector, then its spectrum is not an arbitrary element of $\mathbb L^n$: it satisfies the Frobenius consistency relation \[ V_s^q=V_{qs \bmod n}. \] We prove a general Galois-descent theorem for Fourier transforms on finite abelian groups, characterize the one-dimensional spectra as products of subfields indexed by $q$-cyclotomic classes, and show that the orbit-seed representation is optimal in base-field coordinates. For arbitrary vectors in $\mathbb L^n$, we study a two-stage representation $g=f+h$, where $f$ is class-consistent and $h$ is a residual. The residual optimization separates over cyclotomic classes. We give exact support minimization, a symbol weight enumerator for the class-consistent code, recovery guarantees, global covering-radius formulas, random residual tail bounds, and entropy-type lower bounds. We also discuss implementation consequences for trace decompositions, normal bases, canonical subfield embeddings, and sparse-polynomial residual backends.

Let \(A\) be a central simple algebra over a field \(F\) with index \(n\) and let \(\mathrm{SB}_r(A)\) denote the \(r\)-th generalized Severi--Brauer variety associated with \(A\). We prove that the Chow group of zero cycles of degree zero \(\mathrm{A_0}(\mathrm{SB}_r(A))\) is \((d, n/d)\)-torsion where \(d = (r,n)\). Our approach reduces the general case to division algebras of prime power index and yields several new instances in which \(\mathrm{A_0}\) is trivial, together with sharper torsion bounds in general.\\ We also show that if \(F\) is a local or global field, then \(\mathrm{A_0}(\mathrm{SB}_r(A))=0\). Since Severi--Brauer flag varieties are stably birational to generalized Severi--Brauer varieties, these results extend to them, yielding corresponding torsion bounds and vanishing results for \(\mathrm{A_0}(X)\), where \(X\) is stably birational to \(\mathrm{SB}_r(A)\).

This paper studies the Chow and cohomology rings of the Hacking moduli stack $\mathcal{P}^{\mathrm{H}}$ of plane quartics. We construct a smooth proper Deligne--Mumford stack resolving the Calabi--Yau wall crossing between the KSBA and K-moduli compactifications for plane quartics via stack-theoretic weighted blowups. Its coarse moduli space is, up to normalization, the fiber product of the natural diagram relating the KSBA, K-moduli, and boundary polarized Calabi--Yau compactifications. From this, we compute the Poincaré polynomial of $\mathcal{P}^{\mathrm{H}}$, show that the cycle class map is an isomorphism with rational coefficients, and determine generators and relations for its Chow ring in terms of tautological classes. Analogous results are established for the GIT and K-moduli stacks.

In this article, we aim to further clarify certain subtle aspects of processes that exhibit long memory in the second-order sense. We construct a long-memory stochastic sequence, in the sense that the series of absolute autocovariances diverges, whose spectral density has an almost everywhere unboundedly divergent Fourier series. This suggests that the Fourier series of the spectral density of a generic long-range dependent process, one for which nothing is known except that its autocovariances are not absolutely summable, should be handled with great care. On the other hand, it is known that if one assumes regularly varying behavior for the autocovariances or the spectral density, along with suitable conditions on the associated slowly varying function, then the Fourier series of the spectral density converges everywhere, except possibly at 0. The process we construct can easily be simulated, and we compare its empirical and theoretical autocovariances.

The design of physics-augmented neural networks (PANNs) for the purposes of constitutive modeling has received considerable attention as of late for a variety of material behaviors. Here, we revisit the classical framework of isotropic incompressible hyperelasticity in light of recent advances in the study of constitutive inequalities. We show that polyconvexity implies true-stress-true-strain monotonicity for a large class of incompressible strain-energy functions. The resulting elastic law obeys the physically reasonable Legendre-Hadamard (or ellipticity) condition as well as the notion of increasing stress with increasing strain. These results then inform the architecture of four distinct PANNs which are subsequently calibrated to three different sets of experimental data each. We show that different PANN parametrizations - satisfying the same constitutive constraints a priori - have varying approximation power for the description of material behavior. Moreover, even when distinct parametrizations perform comparatively well within the calibration regime, they show pronounced differences in extrapolation. This observation motivates a critical discussion about the predictive power of PANNs which also has implications for the modeling of more complex material behavior by virtue of neural networks.

We study local quasihelix and generalized quasihelix properties of several Gaussian Volterra processes with tempered, power-weighted, and logarithmic kernels, including tempered fractional Brownian motions and generalized fractional Brownian motion-type processes. These properties depend significantly on the values of the parameters involved, and we consider all possible cases in detail.

We study a time-dependent non-Hermitian extension of the Schütte-Da~Providência spin-boson Hamiltonian with complex couplings. A time-dependent Dyson map containing a squeezing transformation maps the model, in an admissible bounded regime, to a Hermitian Hamiltonian with real instantaneous energy spectrum. The squeezing contribution generates a dilatation term allowing the Hermitian partner to be interpreted as a fixed-domain representation of a system with moving boundaries. While the fixed-boundary Hermitian model conserves $Q=N-S_0$ and forbids transitions between sectors differing by two bosonic quanta, the boundary motion opens such channels. For closed boundary protocols with constant background parameters the first-order integrated transition amplitude vanishes, reflecting the unitary nature of constant squeezing. Nontrivial transition control arises when the non-Hermitian parameter varies during the boundary motion, changing the dressed basis and allowing boundary-induced transitions to be suppressed or enhanced by coherent interference.

A version of the Law of the Iterated Logarithm for smooth functions in the upper-half space is proved. As a consequence, we show that certain size conditions on the gradient and the gradient of the laplacian of a smooth function, lead to self-improvement growth properties. The results are applied in situations where harmonicity is not present.

Extending Enriques' characterization to algebraically closed fields of characteristic $p \geq 7$, we show that every smooth projective surface $X$ with $h^1(X, \mathcal{O}_X) = 2$ and $p_1(X) = p_2(X) = 1$ is birational to an Abelian surface. This characterization fails if $p \leq 5$, and we give a sharp alternative.

We study $\ell$-colourings of $(v,k,λ)$-BIBDs (balanced incomplete block designs) where, within each block, one colour is absent and each of the $\ell-1$ other colours appears exactly $\frac{k}{\ell-1}$ times. We establish several necessary conditions for such colourings to exist. We also use these coloured BIBDs to provide new necessary conditions for the existence of Hadamard matrices, affine planes, and twin prime powers.

We numerically approximate the Tomita-Takesaki modular operator for local subalgebras of the 1+1-dimensional massive Majorana field. Our method works at the one-particle level with a discretisation of time-0 data in position space. The local subspaces we consider are associated with one double cone and with the disjoint union of two double cones. In order to avoid boundary effects, we primarily choose the overall spacetime to be a cylinder; different choices of boundary conditions (antiperiodic and periodic) are considered. We compare our numerical results to known analytic expressions in the massless case. It turns out that the modular operator has a non-trivial dependence on the mass. In the case of two double cones, the modular generator does not only have ''local'' contributions (supported on the diagonal in configuration space) but also ''bilocal'' terms (connecting the two double cone regions); we find the latter to be less prominent at higher masses, in line with expectations.