Papers

Schreyer's algorithm is usually the fastest way to determine a (typically non-minimal) free resolution of a finitely presented module over a polynomial ring. We adapt a refined version of Schreyer's algorithm to compute free resolutions over the exterior algebra, relying on relative Groebner bases. We illustrate the use of our algorithm for the computation of cohomology of coherent sheaves over projective space, which by the BGG correspondence can be computed via free resolutions over the exterior algebra. Schreyer's method relies on a tree traversal and thus has the potential for parallel computations. We report on ongoing work on a massively parallel implementation, observing that Gnawali's parallel approach over polynomial rings can be carried over to our setting.

For orientation-preserving planar harmonic mappings, postcomposition by real-affine mappings induces automorphisms on the space of complex dilatations. Motivated by this fact and the lack of natural affine-invariant geometric quantities in classical distortion theory, we introduce the affine circumradius and affine diameter associated with the image of the complex dilatation. The exponentiated affine circumradius gives a characterization of minimal quasiconformal distortion under affine normalization. Optimal affine balancing is unique up to similarity and yields canonical harmonic mappings with centrally symmetric dilatation. Using a three-point support principle and a sharp hyperbolic Jung theorem, we establish universal sharp two-sided bounds for these invariants and provide their pseudohyperbolic reformulations. For canonically balanced mappings, we prove sharp second-order estimates for pre-Schwarzian derivatives, showing that affine normalization cancels leading-order conformal discrepancies. We develop an affine-stable hierarchical classification for families of harmonic quasiconformal mappings, including normality and an invariant ordering. This framework unifies extremal distortion problems and connects classical analytic function theory with the distortion theory of harmonic quasiconformal mappings.

We prove an analog of the Ax-Schanuel theorem for the Drinfeld $j$-function in odd characteristic. Roughly speaking, if the graph of $\boldsymbol{j}\colon\Omega^n\rightarrow\mathbb{A}^n_{\mathbb{C}_\infty}$ and its derivatives has an atypical intersection $\mathcal{V}$ with an algebraic variety, then $\mathcal{V}$ projects to a weakly-special subvariety in $\mathbb{A}^n_{\mathbb{C}_\infty}$. More generally, we prove a positive characteristic analog of the differential Galois theoretic Ax-Schanuel theorem of Blázquez-Sanz, Casale, Freitag and Nagloo. Our main theorem for $\boldsymbol{j}$ follows by applying this result to a suitable foliation. A theorem of Pink allows us to conclude that the special varieties in the sense of Blázquez-Sanz et al. in this context agree with the classical weakly-special subvarieties in the Drinfeld sense.

We prove long-time strong Sobolev instability for a class of quasilinear Schrödinger equations on $\mathbb T^2$ arising, via the Madelung transform, in the description of Euler-Korteweg capillary fluids. More precisely, for every $s\ge 7$, arbitrarily small $\mu>0$, and arbitrarily large $\mathcal K>0$, we construct a solution such that $$ \|u(0)\|_{H^s}<\mu, \qquad \|u(T)\|_{H^s}>\mathcal K $$ at some finite time $T$, for which we provide an explicit exponential upper bound. The instability mechanism is generated by a finite-dimensional resonant Toy Model associated with the cubic semilinear part of the equation, in the spirit of Colliander-Keel-Staffilani-Takaoka-Tao. The main technical difficulty is to justify this dynamics over the long time scale of the energy transfer in the presence of quasilinear derivative losses. We achieve this by combining a weak Birkhoff normal form with paradifferential calculus, microlocal symmetrization, and modified energy estimates. Our result provides a rigorous quasilinear realization of a resonant energy-transfer mechanism, and suggests a connection with weakly turbulent dynamics in capillary fluids.

{Let \(K\subseteq\mathbb R_+^E\) be a closed convex cone and \(P=K\cap[0,1]^E\). For a linear objective with positive optimum, we study the intersection of the coordinate supports of all optimal solutions.} We prove that this common support is a transversal of a natural clutter associated with the positive directions of the cone and introduce a decreasing hierarchy of bottleneck families that stabilizes after at most \(|E|-1\) levels. In the polyhedral setting, the first level consists exactly of the sets carrying feasible upper-bound dual multipliers, while every set carrying an optimal multiplier belongs to the stabilized family. {The breadth of the join-semilattice generated under union by the zero sets of the nonzero vertices gives an objective-independent upper bound on the universal stabilization depth;} this bound is exact for simplicial truncations. Every possible depth occurs, and the \(|E|-1\) bound is sharp. Membership in the second level is co-NP-complete and fixed-parameter tractable in the size of the candidate bottleneck. Under upper-box-integrality and the integer decomposition property, the hierarchy collapses at level two, and its stabilized members are exactly the carriers of optimal upper-bound multipliers. When \(P\) is integral, the common optimal support is the union of the supports of all optimal dual solutions. Applications include bipartite matchings, maximum-order cycle subdigraphs, binary circulations, balanced hypergraph matchings, interval hypergraphs, and maximum-weight closures.

We study rate-independent systems in which the evolution is obtained by incremental minimization of elastic energy and one-homogeneous dissipation under constraints on the admissible inelastic increments. Such constraints arise naturally in several models of inelastic materials, including shear-transformation-zone models, intermittent plasticity, and quantized plasticity, where evolution proceeds through prescribed elementary activation events rather than arbitrary plastic increments. We establish the existence of a continuous-time limit for the corresponding constrained incremental scheme under suitable assumptions. As an application, we formulate a simplified shear-transformation-zone model consisting of localized activation events on a lattice embedded in a two-dimensional elastic medium. We prove that this model generates shear-band formation through the successive activation of neighboring transformation zones. More precisely, we show that every admissible evolution remains localized on a growing line segment in the infinite-domain setting and that the same mechanism persists on sufficiently large bounded domains. These results provide a rigorous mathematical framework for constrained activation models and show that restrictions on admissible elementary events alone constitute a mechanism for localization in rate-independent systems.

A complete classification is established for continuous, SL($n$) contravariant, and translation invariant tensor valuations defined on the Sobolev space $W^{1,p}(\mathbb R^n)$. When these valuations are further assumed to be homogeneous, the classification reveals that they are precisely the Fisher information tensors, which constitute a higher-order generalization of the Fisher information matrix.

This paper investigates the construction of full-systematic permutation codes capable of correcting multiple deletions under two complementary models, namely symbol-invariant deletions (SIDs), where surviving symbol values are preserved, and permutation-invariant deletions (PIDs), where the surviving sequence is standardized to a permutation. For any fixed integer $t \ge 1$ and all sufficiently large message lengths $n$, our proposed encoders map any message permutation of length $n$ to a codeword by inserting distinct redundancy symbols while strictly preserving the sequence order of the original message symbols. The proposed constructions correct up to $t$ deletions using $7t-1$ redundancy markers for PIDs and $4t$ redundancy markers for SIDs, achieving redundancies of $(7t-1)\log n + O_t(1)$ bits and $4t\log n + O_t(1)$ bits, respectively. Both code families are uniformly constructible, encodable, and decodable in $n^{O(t)}$ time. The underlying framework stores an inner deletion-correcting syndrome in the relative positions of redundancy markers via an algebraic outer code based on integer moments and residual graph coloring. We further extend this framework to fixed-composition and strictly $\lambda$-regular multipermutations, proving that the PID and SID channels coincide whenever the common multiplicity satisfies $\lambda > t$.

We argue that dividing codec design into Coding for Machines (CfM) and Coding for Humans (CfH) is a misleading distinction for deciding what information a codec may discard. Receiver identity does not determine admissible information loss. The required rate depends on task scope, including the predictions to support, their losses and tolerated risks, the encoder observation, and the permitted decoding procedures. Notably, a machine task may have a higher minimum rate than a restricted human decision. Rate savings on selected machine tasks apply only to the stated requirements, not to an intrinsic ordering by receiver type. We extend source and feature coding to finite task families, derive when restricting the encoder observation preserves the minimum rate, and show that equality between source and split-feature coding rates can no longer hold as the task scope expands.

This work aims to investigate the long-time asymptotic behavior of solutions to the Cauchy problem for the Heisenberg ferromagnet equation with the Schwartz initial data. Utilizing the gauge transformations, spectral analysis and the inverse scattering method, we prove that the solutions to the Heisenberg ferromagnet equation can be expressed in terms of the solutions to a matrix Riemann-Hilbert problem formulated in the complex $k$-plane. Various Deift-Zhou contour deformations and the motivation behind them are given. By applying the nonlinear steepest descent method to the associated matrix-valued Riemann-Hilbert problem, we obtain the exact leading-order asymptotic formulas and uniform error estimates for solutions to the Cauchy problem of the Heisenberg ferromagnet equation.

A message drawn from a small alphabet helps a teammate only if it keeps the distinctions that change that teammate's next decision. We show that scoring messages by action values averaged over the receiver's situation can erase exactly these distinctions, and we propose RAVEN (Receiver-conditioned Action-Value ENcoding), which trains a four-symbol, one-step-delayed channel to preserve each receiver's centered action-value profile within the receiver's own context. The sender never needs to know that context: the receiver decodes every symbol with its private information. We give two estimators of this target. With a teacher, offline RAVEN selects the codebook that exactly minimizes an empirical conditional distortion and distills it into a frozen sender; we bound the resulting codebook-selection error and one-step decision loss. Without a teacher, online RAVEN aligns, inside a QMIX learner, the deployed symbol pathway with a training-only continuous reference that shares its routing. Against five recent communication methods on eight navigation settings, offline RAVEN attains the highest return in seven, and removing receiver conditioning forfeits 83% of its communication gain. Online RAVEN raises predator-prey capture success from 53.2% to 96.0% over the same QMIX backbone without communication, and on SMAC and MPE it attains the best mean normalized score of 14 methods, including methods that exchange kilobit messages. Every RAVEN message costs 2 bits, 12-1,024x fewer than those of NDQ, CACOM and ExpoComm on navigation.

For every integer $n\ge2$, we prove that every countable group embeds in a group of type $\mathrm{FP}_n$. We also construct a group of type $\mathrm{F}_n$ containing a copy of every recursively presented group. Consequently, a finitely generated group embeds in a group of type $\mathrm{F}_n$ if and only if it is recursively presented. This answers questions of Fournier-Facio and Zaremsky, and confirms a suggestion of Gromov. An appendix combines the construction with the controlled-filling argument in a subsequently released OpenAI preprint to prove that every countable group embeds in a two-generator group of type $\mathrm{FP}_\infty$.

We construct a correspondence between ordinary irreducible characters of height zero above the generalized Dade-Glauberman-Nagao correspondence for blocks with central defect. The correspondence is equivariant under group and Galois automorphisms and satisfies a block relation between $\mathcal{H}$-triples. For each pair of corresponding characters, one pair of associated projective representations realizes matching factor sets, equal central scalars, synchronized mixed comparison functions and compatible block correspondences for every intermediate group. The construction uses the multiplicity algebra over the group algebra of the central defect subgroup, a normalized integral lift of a residual semilinear realization, and a graded corner isomorphism. It treats all central-character fibers, including those with nontrivial central character. We also obtain a form suited to normal $p$-sections in an Alperin-McKay-Navarro reduction.

In this paper, we first construct two graphs $\mathcal{F}(l,m,t,s)$ and $\mathcal{G}_4(l,m,t,s)$. Then we introduce the graph $\mathcal{F}_n(l,m,t,s)$ and the operator $E_{\mathcal{G}_4(l,m,t,s)}$, where $\mathcal{F}_n(l,m,t,s)$ is defined by identifying the vertex $c$ of $n$ copies of $\mathcal{F}(l,m,t,s)$, and $E_{\mathcal{G}_4(l,m,t,s)}$ is defined by replacing each edge of $G$ with $\mathcal{G}_4(l,m,t,s)$, for any simple finite undirected graph $G$. By using the theory of Lorentzian polynomials, we prove that the independence polynomials of the graphs $\mathcal{F}_n(l,m,t,s)$ and the image graphs of $E_{\mathcal{G}_4(l,m,t,s)}$ are log-concave, respectively. As applications, our results not only make progress on the conjecture of Alavi, Malde, Schwenk and Erdős, but also generalize the results of Bendjeddou and Hardiman.

Emerging optical-network applications increasingly use received data for inference and control rather than exact source reproduction, creating an opportunity to trade bit-level fidelity for greater transmission reach and efficiency. We propose an end-to-end optical semantic communication system for joint image classification and reconstruction over a nonlinear wavelength-division multiplexed (WDM) fiber channel. The system maps each image directly into a fixed-length sequence of channel symbols that preserves task-relevant information, without explicit source compression or channel coding. Experiments on the MNIST dataset cover launch powers from -9 to +3 dBm, fiber lengths up to 800 km, and 16-, 64-, and 256-Quadrature Amplitude Modulation (QAM) formats. At 0 dBm, classification accuracy remains between 98.92% and 99.31% across all tested link lengths and modulation orders, while requiring fewer transmitted symbols than a Low-Density Parity-Check (LDPC)-coded JPEG baseline at every tested modulation order. These results show that semantic communication can simultaneously extend optical reach and reduce transmission resources by conveying only task-relevant information.

We prove Petrunin's smoothing conjecture in all dimensions: every compact Euclidean polyhedral space without boundary and with nonnegative Alexandrov curvature is a Gromov-Hausdorff limit of smooth Riemannian orbifolds with geometrically nonnegative curvature. More strongly, the approximating metrics are positive-time slices of a single orbifold Ricci flow whose metric initial condition is the given polyhedral space. The proof rests on a new short-time existence and regularization theory for Ricci flow that, under two-sided volume bounds and a generalized segment inequality, replaces pointwise lower curvature control by small scale-invariant integral control of the defect from a preserved curvature cone. Although it allows arbitrarily large pointwise violations, this theory yields an existence time and positive-time curvature estimates independent of the initial upper curvature bound. As a further application, it gives rigidity consequences for manifolds with small integral curvature defect.

We address the numerical approximation of a scalar conservation law with a highly oscillatory flux, in the linear setting. Taking advantage of the intrinsic numerical viscosity of the scheme employed that yields a parabolic regularization of the original hyperbolic equation, we use homogenization theory to derive an effective equation which in turn we solve numerically. Our numerical experiments show the performance of the approach on the example of the Lax-Friedrichs scheme.

This work establishes global well-posedness for the three-dimensional defocusing energy-critical nonlinear Schrödinger equation with conservative linear Marcus noise. For arbitrary deterministic energy-space data, we construct a probabilistically strong solution, unique in the critical spacetime class, and prove convergence of finite-activity approximations. The spatial profiles need only be bounded and Lipschitz, while the centered pure-jump Lévy noise is assumed to have finite second moment. If the temporal noise coefficient is square integrable on the half-line, the solution scatters almost surely, with uniform mean-square tail convergence and no smallness restriction. The solution conserves mass and satisfies an exact energy balance. The proof combines deterministic energy-critical stability with Poisson Strichartz estimates; convergence of the interaction-representation martingale gives the critical spacetime control needed for continuation and scattering.

We study the Delay Constrained Routing (DCR) problem arising in IP computer networks to support high bandwidth applications. The goal is to route IP packets, from source to destination, subject to quality of service constraints – in particular, bounding the worst case delay – while minimizing the total allocated bandwidth in the network. DCR can be formulated as a Mixed-Integer Second-Order Cone Program, and solved using standard solvers; while generally efficient, this hinges on off-the-shelf tools and does not scale to large instances. We devise a solver-free bespoke approach that uses nested Benders-Lagrange, namely, a double decomposition approach that exploits the special structure of the problem. In particular, we use a nonconvex nonlinear formulation of the problem, that results in a nonconvex Benders value function whose optimization we tackle with a bespoke Spatial Branch-and-Bound approach using custom piecewise-convex cuts. We show that our approach provides very tight upper and lower bounds in competitive running times w.r.t. previous state-of-the-art approaches.

We introduce the tangloid algebra $\mathcal{T}_n$, a diagrammatic algebra arising from the extended tangloid category $\Ti$, and its subalgebra, the braidoid algebra $\mathcal{B}_n$. The construction begins with the unoriented tangloid category $UTC$, inspired by Turaev's theory of knotoids, which is obtained from the unoriented welded tangleoid category $UWTC$ by removing the welded relation while retaining the forbidden moves and imposing some additional relations. Within this framework, the braidoid category arises naturally as a subcategory of $UTC$. We then introduce the extended tangloid category $\Ti$ by adjoining a placeholder morphism $\emptyset$, which allows the generators to be extended to endomorphisms of a fixed $n$-box. The interaction of the placeholder morphism with the crossing morphisms gives rise to diagonal morphisms, $/$ and $\backslash$, which are used in the defining relations of $\Ti$. The corresponding extended braidoid category $\mathbf{Brd}_{\Ti}$ is realised as a subcategory of $\Ti$. For each $n\geq 0$, we then define the tangloid algebra $\mathcal{T}_n$ by linearising the endomorphism algebra $\operatorname{End}_{\Ti}(n)$, as a unital $\mathbb{C}$–algebra presented by diagrammatic generators and relations. We also define the braidoid subalgebra $\mathcal{B}_n$, which provides an algebraic framework for the theory of braidoids. We further introduce the reduced tangloid algebra $\delta\mathcal{T}_n$ by imposing two additional reduction relations for trivial knot and trivial knotoid components, and we construct a natural diagrammatic bilinear pairing on $\delta\mathcal{T}_n$ using diagrammatic reflection, composition, and closure. Our categorical and algebraic setting provides a theoretical framework for studying diagrammatic structures with intrinsic endpoints, namely knotoids, linkoids, and braidoids, and related diagram algebras.

We extend a six-functor formalism $D\colon\mathrm{Span}(C,E)\to\mathrm{Cat}$ to a lax symmetric monoidal functor of $(\infty,2)$-categories $\mathbf{Span}^2(C,E)^P_I\to\mathbf{Cat}$, where $P$ and $I$ are the classes of $D$-proper and $D$-étale morphisms, respectively. This proves a conjecture of Mann and generalizes a special case of a theorem of Cnossen, Lenz, and Linskens. To prove this result, we develop a theory of internal $\mathsf{E}$-monoidal categories and $\mathsf{E}$-operads, where $\mathsf{E}$ is a local class of morphisms in an $\infty$-topos. These notions generalize the internal symmetric monoidal categories and operads developed by Martini and Wolf.

We prove that any Fano orbifold admits a free Campana curve in the sense of Campana. Moreover, assuming that any klt log Fano pair admits a very free rational curve in its smooth locus, we prove that any Fano orbifold is Campana rationally connected. As an application, we prove the finiteness of the orbifold fundamental groups for Fano orbifolds.

We study spin-current fluctuations of the folded XXZ chain, which describes the strong-anisotropy limit of the XXZ chain. We obtain an exact expression for the moment-generating function of the time-integrated spin current at infinite temperature. A long-time asymptotic analysis establishes that the rescaled time-integrated current converges to a non-Gaussian distribution described by the M-Wright function. The derivation relies on a nonlocal map to the one-dimensional t-0 model and its exact spin-charge separation. By extending this approach, we also show that the dynamical correlation function of the spins summed over two adjacent sites exhibits a diffusive Gaussian profile at long times.

Greedy box splitting reads gradient signs at the centre and retains one smaller box. Its summable travel budget makes an incorrect exclusion irreversible. We determine when this geometry is safe. Halving is minimax optimal in the sign-vector oracle model on a coordinatewise sign-consistent class that need not be convex. A dimensionless sign defect yields a sharp scalar envelope, attained by one smooth convex potential; a three-dimensional trajectory ends farther from the target than its initial radius. For quadratics, universal convergence with aspect $w$ and contraction beta holds exactly when the weighted absolute row defect is at most $(2 \beta - 1)$. Perron weights minimise this defect. Adaptive half-sizes avoid initial enlargement and, under persistent error, converge to an exact capped fixed-point limit. For binary comparisons, an independent probe fraction $q$ permits universal localisation on common-target monotone sections exactly when $(q \leq 2 \beta - 1)$; coupled probes give the sharp threshold $2/3$. Matching information bounds distinguish comparisons from sign-vector queries. Error bands give finite-time guarantees and sharp residual floors. Extensions and reproducible experiments identify the assumptions and costs of freezing, smoothing, voting, multistart and random-matrix certificates.

We prove that the Ehrhart $h^*$-polynomial of a rank-two matroid with exactly three parallel classes is real-rooted whenever its smallest parallel class has size at most three. This bound is sharp: the rank-two matroids with parallel-class sizes $(4,561,600)$ and $(4,a,a+29)$, for all sufficiently large integers $a$, have $h^*$-polynomials that are not real-rooted. These counterexamples disprove Ferroni's real-rootedness conjecture. Their duals are cycle matroids of theta graphs and have the same $h^*$-polynomials. Nevertheless, every matroid of rank two or corank two has a positive $h^*$-coefficient sequence that is ultra log-concave of order equal to the polynomial's degree. In particular, the unimodality conjecture holds in both cases.

We supply the details of the subcritical dry Ten Martini statement announced in [6, Remark 1.3]: for every irrational frequency, every subcritical Type I energy with resonant rotation number is an endpoint of an open spectral gap. The proof combines the all-frequency Puig argument developed there with quantitative almost reducibility. A different proof was recently given by Li [7].

Erdős, Odlyzko and Sárközy conjectured that every reduced residue class modulo $q$ contains a product of two primes not exceeding $q$; their conjecture remains open even under the GRH. We prove that its analogue holds, unconditionally and in the wider range $q^{\Theta}$ for any fixed $\Theta > 3/4$, in Granville's random model of the primes. Almost surely, every reduced class modulo every sufficiently large modulus is representable, and the number of representations is asymptotic to its expected order. The choice of the model is decisive; the corresponding statement in the random prime model of Banks, Ford and Tao fails with positive probability, an elementary local defect that Granville's discard step removes.

The construction of absolutely irreducible $KG$-modules for a field $K$ and a finite group $G$ is a basic problem in representation theory. For the symmetric group $\operatorname{S}_n$, Gordon James gave an explicit solution using subgroup data associated to a $\mu$-tableau $t$ for partitions $\mu\vdash n$. Here, we want to investigate in which way James's method can be extended beyond symmetric groups. We identify suitable subgroup data from a finite group and use it to construct a module such that an associated factor module is absolutely irreducible or zero.

We prove that (n + 1)-dimensional de Sitter spacetime is locally the unique $(\Lambda > 0)$-vacuum spacetime whose Killing horizons intersect the conformal boundary $\mathscr{I}$. The intersection occurs at the isolated essential zeros of the extension of the Killing vector field to $\mathscr{I}$. We then reconstruct all de Sitter Killing vector fields from their boundary value and obtain a characterization of those admitting a Killing horizon in terms of a conformal invariant $\sigma$: $\sigma<0$ characterizes non-degenerate bifurcate horizons, whereas $\sigma = 0$ yields degenerate non-bifurcate horizons. Finally, we derive an intrinsic, conformally invariant formula for the surface gravity of the horizons in de Sitter, $\kappa^2 = |\sigma|$.

Fix a target accuracy $\varepsilon$, a gradient-noise level $s$, and a horizon $N$. We wish to design algorithms which minimize the probability of observing a suboptimality gap which exceeds the target accuracy, i.e., $\mathcal{E}_N = -\log \sup_{f,P} \mathbb{P}_P(f(x_A) - f_\star \ge \varepsilon)$, with the noise law $P$ known only to be sub-Gaussian. We single out a uniformly averaged schedule which is harmonic (of the form $h_k = R^2/(\varepsilon (N+m-k))$) and prove, via an optimized exponential supermartingale argument, that it attains the optimal exponent $\mathcal{E}_N^\star = \varepsilon^2 N (1+o(1))/(2R^2 s^2)$. Optimality is certified by a matching impossibility result: under Gaussian noise, a gradient-masking change of measure caps the exponent of every algorithm at the same leading order. In a small-noise limit, our setting degenerates into the adversarial-error model of Gösgens and van Parys (2025) for subgradient methods.