Papers

In their work on Brill-Noether theory, Eisenbud and Harris established the geometry of the universal parameter space of linear series over curves, proving that for even genus $g$ and degree $d = g/2 + 1$, the projection to the moduli space of curves is a finite cover of degree equal to the Catalan number $C_{g/2} = \frac{1}{g/2+1}\binom{g}{g/2}$. In this paper, we construct the tropical counterpart of this universal family: a polyhedral cone complex $\mathcal{G}_{g \to 0, d}^{\mathrm{trop}}$ parametrizing degree-$d$ tropical morphisms from genus-$g$ metric graphs to metric trees. For even $g$ and $d = g/2 + 1$, we prove that the forgetful projection $\Pi \colon \mathcal{G}_{g \to 0, d}^{\mathrm{trop}} \to \mathcal{M}_{g}^{\mathrm{trop}}$ is a branched cover of degree $C_{g/2}$ equipped with natural determinantal multiplicities. We compute this degree by showing that on caterpillars of loops the morphisms are in bijection with ballot sequences, and we establish its global invariance across $\mathcal{M}_{g}^{\mathrm{trop}}$ via a tropical balancing condition across codimension-$1$ walls. Via deformation and path lifting, this yields an effective method to construct Catalan-many gonality-witnessing maps for any generic metric graph, establishing that the tree gonality of any genus-$g$ metric graph is at most $\lceil g/2 \rceil + 1$.

Large linear programs can exceed the memory of a single compute node. Although first-order methods replace sparse factorizations with GPU-suited matrix-vector products, other solver phases can reintroduce a single-node memory limit. We present SHARDLP, a distributed GPU LP solver that keeps the matrix and primal-dual state partitioned from sharded input through solution output. On the Google PDLP benchmark, SHARDLP reaches the published criterion on nine of eleven instances, compared with eight in the published CPU PDLP study. On the largest benchmark, eight H200 GPUs solve a 1.185-billion-variable, 6.338-billion-nonzero LP in 9.9 minutes; the published CPU experiment reports 21.06 hours on different hardware. Beyond this benchmark, separately checked multi-node solves reach up to 13.604 billion variables and 40.807 billion nonzeros, while validated executions span up to 76 GPUs across 29 compute nodes. For column-partitioned solves, support-aware communication skips GPUs that store no coefficients for a row; on an LP with 2.76 billion nonzeros, it cuts modelled communication by 92.97% and improves solver time by 1.27x-1.52x

Fang and Huang proved that every cancellative pair $(\mathcal{A},\mathcal{B})$ of families of subsets of $[n]$ satisfies $|\mathcal{A}|\cdot|\mathcal{B}|\leq(\frac{9}{4})^n$. Their proof uses entropy. We give a Fourier-analytic proof based on Lifshitz's one-sided noise operator $\mathrm{T}^{1/4\to1/2}$. The main tool is a near-$L^1$ hypercontractive estimate for this operator.

We establish quantitative Parisi formulas and fluctuation bounds for the Sherrington-Kirkpatrick model at zero external field. In particular, using that the Parisi measure is supported on an interval, we show that for every fixed inverse temperature greater than one, the variance of the logarithmic partition function lies between orders $N^{4/15}$ and $N^{7/15}$. The same exponents hold for the ground-state variance. We also obtain upper and lower bounds on the finite-size bias of the mean free energy and ground-state energy, together with two-sided upper-tail estimates with exponent $6/5$ at both positive and zero temperature.

We prove the Trudinger inequality with exponent $\frac{s}{s-1}$ for $M^{1,s}$ Sobolev spaces on metric measure spaces under the sole measure growth assumption $\mu(B(x,r))\ge br^s$. This improves the previously known exponential integrability with power one. Our argument also yields sharp asymptotic bounds for the Sobolev constants as $p\uparrow s$. On doubling spaces, we further remove the connectedness assumption from the Trudinger inequality associated with a $(1,p)$-Poincaré inequality when $p<s$. An Ahlfors regular counterexample shows that this extension fails at the critical exponent $p=s$.

Synchronisation by noise for McKean–Vlasov stochastic differential equations is investigated. A transfer principle is introduced by which synchronisation by noise and diagonal mixing can be transferred from an associated limiting frozen-diffusion SDE to a genuinely law-dependent McKean–Vlasov SDE. The usefulness of this principle is demonstrated through an application to ensemble Kalman sampling, thereby providing its first application in a sampling context. Synchronisation for SDEs with multiplicative noise is then revisited from the perspective of sampling. Existing general frameworks providing sufficient conditions for synchronisation by noise are refined and extended to a control-oriented setting on noncompact state spaces, and coefficient-level conditions are derived by which these criteria can be verified. Motivated by extrapolation schemes in sampling, the freedom in the construction of couplings for a fixed sampler is emphasised, together with the advantage of choosing couplings that exhibit favourable synchronisation properties.

This is the first part of a program to construct hyperholomorphic families of boundary conditions for the Kapustin–Witten B-twist of the Langlands QFT, otherwise known as (BBB)-branes. We define the category of quasi-algebraic sheaves over the Deligne moduli stack, which serves as an analog of the twistor space of Hitchin's moduli stack. This allow us to construct a representation of a simplified version of the Moore–Tachikawa category which is motivated by the relative Langlands program in the sense of Ben-Zvi–Sakellaridis–Venkatesh.

We provide a pathwise proof of global-in-time well-posedness for the generalised KPZ equation in the full subcritical regime by an adaptation of the strategy recently applied to the generalised Parabolic Anderson Model in [ES26]. Since this strategy relies crucially on the assumption that control of the supremum norm is sufficient to continue the solution, the main additional ingredient required is a treatment of the initial layer for gKPZ with merely $L^\infty$ initial data, rather than the more usual setting of $C^\theta$ initial data with $\theta > 0$. Our approach is based on an expansion around a deterministic profile followed by the introduction of an integrating factor in order to remove the terms which have critical scaling at time $0$. For pedagogical purposes, we first demonstrate the proof techniques in the case of the standard $(1+1)$-dimensional KPZ equation before turning to the more computationally involved case of generalised KPZ.

Jones described how the Ising and Potts models from statistical mechanics give rise, with appropriate choices of Boltzmann weights, to invariants of an oriented link. The Boltzmann weights that Jones proposed, however, work only with a correction factor that he does not mention. We fill in the missing details in two different ways. We also show that all of these invariants, which we call the Ising and Potts invariants, are given by evaluating the Jones polynomial at $t$ equal to one of these Boltzmann weights.

We study the estimation of infinite-dimensional optimal transport maps from noisy paired observations. The population map pushes a Gaussian reference measure forward to a target probability measure on a function space and takes the Cameron–Martin gradient form $T=I+\nabla_{\mathcal H}\phi$. Our estimator uses cylindrical gradient sieves based on finitely many Cameron–Martin coordinates, thereby reducing the problem to finite-dimensional empirical risk minimization. A local nonasymptotic oracle inequality separates cylindrical approximation and statistical estimation errors while accounting for the conditioning of the parametrization. The approximation analysis relies on regularity conditions governing coordinate decay and dependence on omitted input coordinates. For diagonal Gaussian and nonlinear block-interaction classes, we derive matching upper and lower bounds that establish the minimax rate $N^{-s/(2s+1)}$ in expected norm, where $s$ measures weighted coordinate regularity. We further analyze a continuous two-groups model with Gaussian–Laplace mixtures and derive a prediction-risk bound for the resulting transport-map estimator.

Let $M$ be a complete open $5$-manifold with nonnegative Ricci curvature. If its universal cover $\tilde M$ has Euclidean volume growth, then $\pi_1(M)$ is finitely generated and virtually $\mathbb Z^k$ for some $0\le k\le4$. This confirms a conjecture of Pan and Rong \cite{PR18} in dimension five; see also \cite{BNS25,BrNaSe25}. In dimension six, under the same volume growth assumption, we also obtain a quantitative virtual abelianness result without assuming finite generation of the fundamental group.

We study one-dimensional spinless bosons with semirelativistic spinless-Salpeter dispersion and attractive pairwise contact interactions. Because the dispersion becomes linear at large momentum, the contact interaction is marginal by power counting and produces a logarithmic ultraviolet divergence. We construct the renormalized many-body theory using an enlarged Fock space and a Schur-complement representation of the resolvent, eliminating the bare coupling in favor of the physical zero-total-momentum two-body bound-state energy. The resulting cutoff-independent resolvent defines a self-adjoint Hamiltonian in each fixed particle-number sector. We treat the two-body problem explicitly and show, in the norm-resolvent sense, that the nonrelativistic limit reproduces the attractive Lieb–Liniger Hamiltonian. We also formulate a mean-field approximation directly within the renormalized theory. In the massless and deeply bound large-particle-number regimes, it predicts an exponentially increasing binding scale whose exponent is determined by a one-dimensional variational problem.

We study word maps with constants on quasisimple algebraic groups over a local field $L$. We prove that, for such a group $G=\mathbf{G}(L)$ and a word $w\in (G\ast\mathbf{F}_r)\setminus G$, either $w$ has a so-called Tomanov-small critical constant or the minimal dimension of the word image $w(G^r)\subseteq G$ of such a word is bounded from below by a function $c(G)>1$. We compute the optimal value of $c(G)$ for most quasisimple linear algebraic groups.

Modern science and engineering increasingly rely on time-varying data, yet the mathematical tools used to model temporal phenomena are often developed within separate disciplines, obscuring common principles and limiting the transfer of ideas across fields. This chapter presents the theory of narratives, an abstract framework for time-varying objects of any mathematical kind that supports both theoretical investigations and applications. To illustrate this perspective, the chapter develops three vignettes, each illustrating a different research direction. The first addresses a general concern: What information loss can occur when switching between different representations of temporal data? The second concerns structural and algorithmic approaches: How can we systematically decompose time-varying data into simple pieces and obtain invariants describing its structural complexity? The third is an application to control theory: How can we model multi-agent systems with switching communication topologies? More important than any individual vignette, the central message of this invitation is that a suitable abstract perspective can organize and guide research across remarkably diverse mathematical and scientific domains.

As an extension of the Point Estimate Method (PEM) to evaluate probabilistic moments of quantities of interest (QoI) in general $n$-dimensional spaces, the Quadratic Point Estimate Method (QPEM) has been recently developed. This new method is defined to fully represent up to fifth-order input moments in the Gaussian space, providing general analytical expressions for sample locations and weights, without requiring any numerical optimization. The QPEM can significantly improve the estimation accuracy of the output QoI moments, in relation to PEM-based methods whose numbers of sigma points grow linearly with the problem dimension, while at the same time having an affordable and competitive computational cost up to a considerable number of dimensions. The QPEM is further enhanced in this work by enabling copula integration into the framework, which enables effective modeling of the joint input probability density function by estimating marginals and the dependence structure of the involved random variables. The validity and efficient performance of the copula-based QPEM are showcased against numerous other sampling methods in various examples considering two practical scenarios: (i) when the joint dependence structure can be inferred from data, and (ii) when only marginal distributions and correlation matrices are known.

Let $g_1H_1,\ldots,g_nH_n$ be left cosets of subgroups of a group $G$ whose union $U$ is a proper subset of $G$. We prove that $G$ is the union of at most $2^n$ left translates of $A:=G\setminus U$. For finite $G$ this yields $|A|\ge|G|/2^n$, which settles a conjecture of Tărnăuceanu and the author. Equality $|A|=|G|/2^n$ can only hold when $A$ is a left coset of $K:=H_1\cap\ldots\cap H_n$. Moreover, every subgroup occurring in an irredundant cover of an arbitrary group by $n$ cosets has index at most $2^{n-1}$. This improves a lemma of Neumann.

We classify complete toric threefolds $X$ such that the automorphism group $\text{Aut}(X)$ acts on $X$ with an open orbit whose complement does not contain a divisor. The latter condition means that for any ray $\rho$ of the fan $\Sigma_X$ there is a Demazure root of $\Sigma_X$ associated with $\rho$. We also find among these varieties those $X$ for which the group $\text{Aut}(X)$ is transitive on the smooth locus $X^{\text{reg}}$. The classifications are based on Gale-dual interpretations of these properties.

Nonlinear wave equations exhibit rich interplay among dispersion, nonlinearity, coupling, dissipation, and external forcing, producing diverse coherent and complex wave structures. Although exact travelling-wave solutions provide analytical benchmarks, their construction alone does not reveal underlying phase-space, bifurcations, stability, or responses to perturbations. This review presents a reduction-based methodological framework connecting nonlinear partial differential equations(PDEs) to reduced dynamical systems, invariant phase-space structures, exact waveform reconstruction, stability analysis, and verification in PDEs. Its applicability and limitations are examined across Schrödinger-type, coupled, nonparaxial, dissipative, magnetic, shallow-water, and fractional wave equations. Particular emphasis is placed on correspondence between analytical waveforms and invariant orbits. Equilibria, periodic trajectories, homoclinic orbits, and heteroclinic connections geometrically represent constant, periodic, localized, and front-like structures, respectively. This review distinguishes existence and stability of invariant structures from onset of chaos, emphasizing complementary diagnostics rather than reliance on phase-portraits or finite-time indicators alone. Major methodological gaps include incomplete parameter-space characterization, weak correspondence between reduced models and full-PDE dynamics, ambiguities in generalized and fractional formulations, limited robustness analysis, inadequate numerical reproducibility, and insufficient links to experimental observables. The framework prioritizes physical admissibility, stability, robustness, and predictive relevance over generation of additional formal solutions. It supports reliable nonlinear wave prediction, stability assessment, control, and system design in fluid, optical, plasma, and other nonlinear physical systems.

We consider a minimal pressure-based model of heterogeneous cell populations consisting of proliferative and non-proliferative cells with different mobilities. The model is formulated as a system of reaction–cross–diffusion equations describing the spatio-temporal dynamics of the cell densities. The model is known to admit one-dimensional travelling wave solutions with strictly segregated components: non-proliferative cells occupy a finite region at the leading edge, while proliferative cells remain at the rear. However, the speed, parameter dependence, and stability of these waves remain poorly understood. In this work, we derive an almost explicit variational bound on the wave speed by reformulating the problem as a free-boundary problem for a generalised porous–Fisher equation. The estimates we obtain apply to general pressure laws and growth kinetics, agree closely with the results of numerical simulations, and become sharp in the incompressible limit, where we formally recover a fully explicit characterisation of the wave speed. We then analyse the stability of the waves to show that segregated waves are stable only when non-proliferative cells are more mobile than proliferative cells. Finally, motivated by numerical observations of finger-like protrusions, we investigate the stability of incompressible segregated circular waves through asymptotic shape-perturbation analysis. This yields explicit expressions for the pressure, interface velocity, and growth rates of angular modes, thereby making evident the destabilisation mechanisms that may lead to the emergence of fingering instability. Interestingly, we find that, in contrast with the one-dimensional case, the stability of such circular waves is not determined solely by the relative value of the mobility coefficients, and thus instabilities may arise irrespective of which cell type has the larger mobility.

We give several characterizations of an inhomogeneous basis of the ring of symmetric functions whose evaluations are the character values of the irreducible representations of the rook monoid (symmetric inverse semigroup). Using Schur–Weyl duality, we show that the transition coefficients of this basis with the power symmetric basis are the characters of the propagating partition algebra (dual symmetric inverse monoid algebra). In addition, the structure coefficients of this basis are equal to the coefficients in the smash (or Heisenberg) product of Schur functions.

Let $\Omega_{n}=\{-1,1\}^n$ be the $n$-dimensional hypercube equipped with the normalized uniform measure, let $\nabla$ be the Walsh gradient and let $\Delta$ be the Walsh Laplacian. For every $1<p\leq 2$ we prove the following estimate \[ \|\nabla f\|_{L_p(\Omega_n;\ell_2^n)} \leq c_{{\rm abs}}(p-1)^{-2}\|\Delta^{1/p}f\|_{L_p(\Omega_n)}. \] The exponent $\frac1p$ is optimal, thus this settles the open problem on the sharp fractional Riesz estimate by Efraim and Lust-Piquard \cite{E-LP2008} which was subsequently highlighted by Ivanisvili and Volberg \cite{I-V2022}. We also establish the higher-order counterpart. As applications of our results, we obtain simpler proofs of the optimal short-time estimate for $\nabla e^{-t\Delta}$, and the Bernstein-Markov type inequality for $d$-bounded degree functions.

We establish a linear quantum query lower bound for high-accuracy convex optimization over an explicit family of $n$-dimensional ellipsoids. We focus on linear optimization with an explicitly given objective, where the feasible set is accessed through a membership oracle. We show that any algorithm that, for every unit linear objective, returns an exactly feasible point with additive objective error $\Theta(n^{-1/2})$ requires $\Omega(n)$ membership queries, where the ellipsoids are contained in the unit ball. The same lower bound can be shown to hold if the returned point is only required to be approximately feasible, within $\Theta(n^{-3/2})$ distance from the feasible set. This resolves an open question posed by Chakrabarti, Childs, Li, and Wu~(Quantum, 2020) and by van Apeldoorn, Gilyén, Gribling, and de Wolf~(Quantum, 2020). Coupled with the upper bounds in these papers, the query complexity of high-accuracy convex optimization is characterized tightly up to logarithmic factors. The proof is built around a lower bound for determinant computation that is derived via a novel polynomial method based on Fourier-rank. In the continuous matrix phase-query model, computing the determinant of a real $n\times n$ matrix requires at least $n/2$ matrix-vector product queries. The construction also yields an $\Omega(n)$ phase-query lower bound for estimating the minimum eigenvalue of a real symmetric $n\times n$ matrix to additive accuracy $\Theta(n^{-2})$. These results extend the determinant and minimum-eigenvalue lower bounds of Childs, Hung, and Li~(ICALP 2021) from finite fields to the real-valued setting. Based on the same constructions, we also prove an $\Omega(\min\{\sqrt\kappa,n\})$ gradient-query lower bound for constant-accuracy optimization of $1$-strongly convex and $\kappa$-smooth functions, which is optimal up to a logarithmic factor.

Chain-of-thought reasoning provides a structured computation between a model's input and final answer. Yet it is often evaluated through endpoint accuracy, which ignores the path taken to reach that answer. An emerging line of work addresses this limitation using entropy profiles, which track how uncertainty evolves over the reasoning process but do not reveal which competing hypotheses account for that uncertainty. We introduce answer-distribution trajectories, a stochastic-dynamics-inspired representation that tracks the model's full predictive distribution over answers as reasoning unfolds. As a strictly finer representation than endpoint and entropy summaries, answer-distribution trajectories enable us to characterize a trace through a dynamical reasoning profile spanning exploration, revision, motion, and commitment, and to distinguish different dynamical mechanisms of reasoning success and failure. Across sixteen open-weight language models and four reasoning benchmarks, we show that traces with the same endpoint and similar entropy profiles can exhibit substantially different reasoning dynamics. We further find substantial variation in these dynamics both within and across models and tasks, with different objectives favoring different dynamical profiles. Additionally, we show that training and inference choices systematically reshape these profiles. Our results suggest that answer-distribution trajectories provide a rich framework for analysing and evaluating the dynamics of LLM reasoning.

We establish a necessary and sufficient weight condition so that a modular inequality holds for a generalized integral operator $T_\phi$ on non-negative non-increasing functions in weighted variable Lebesgue spaces $L_w^{p(x)}.$ We give a condition under which the mentioned modular inequality leads to a norm inequality. Lastly, examples are given to demonstrate our main result, to obtain modular inequalities for some well known operators.

We study simultaneous color densities in packing colorings of integer distance graphs. For $D(1,6)$, we determine several exact density regions and prove that colors $1$ through $7$ have maximum combined density $211/252$. When this maximum is approached, the seven individual color frequencies are forced to converge to a specified vector. On an optimal low-color layer, some density vectors have nonperiodic realizations but no periodic realization; we determine how much accumulated density loss is necessary for switching between the relevant configurations. For sufficiently large additional color indices, a fixed finite graph describes the joint density region. In particular, we determine a seven-vertex region for every $i\equiv8\pmod{14}$ with $i\ge36$ and prove that $36$ is the first stable index in this residue class. The proofs combine finite-state integer certificates with explicit constructions and limit arguments. Applications give $17\le\chi_\rho(D(1,6))\le20$, $18\le\chi_\rho(D(1,8))\le22$, and $\chi_\rho(D(1,9))\le17$.

We prove the existence of arbitrarily high-dimensional families of minimal surfaces of any prescribed genus and conformal structure in even-dimensional round spheres. More precisely, let $n\geq2$ and let $\Sigma$ be any closed Riemann surface of genus $g$. We construct a sequence of degrees $d_\ell\to+\infty$ such that, for every $\ell$, there exists a complex manifold of complex dimension $2d_\ell+n^2(1-g)$ consisting of linearly full branched superminimal immersions of $\Sigma$ into $\mathbb S^{2n}$ of degree $d_\ell$. In particular, this yields parametrized families of linearly full branched minimal immersions whose dimensions tend to infinity.

In this paper we consider two-dimensional water waves, under the action of gravity and surface tension. We prove a controllability result for irrotational waves in a fluid domain with finite depth and general bottom topography. The result holds for an open and dense set of bottom topographies in $H^{s+1/2}(\mathbb{T})$ (where $s$ is sufficiently large) that do not touch the free surface. We point out that the bottom topographies that we allow for are not necessarily small perturbations of the flat bottom case: this leads to many technical difficulties, since the eigenvalues of the Dirichlet-Neumann operator at a still free surface with general bottom topography are not explicit, nor are they necessarily close (for low frequencies) to the eigenvalues of the corresponding operator for the flat bottom case. In turn, this leads to a more involved argument to prove Ingham-type estimates, which are needed to prove observability, and motivates the restriction mentioned above on the admissible bottom topographies. We also prove a controllability result for waves with constant vorticity in a fluid domain with flat bottom topography.

We show that the capacity of a quantum channel demarcates a phase transition: while reliable transmission below capacity is always possible, any attempt to transmit information above it fails catastrophically. Specifically, we prove exponential strong converse theorems for unassisted quantum and classical communication over arbitrary finite-dimensional memoryless quantum channels. At rates beyond the respective capacity, the entanglement-generation fidelity and the success probability for classical communication decay exponentially with the number of channel uses. This rules out transmission above capacity even when one tolerates arbitrarily large errors. Our proof follows the classical Arimoto strategy, augmented by a crucial new ingredient: integral representations of Rényi information measures that lead to asymptotic continuity bounds for Rényi capacities.

A family of paths strongly separates the edges of a graph if every two distinct edges are separated in both directions by paths in the family. Bonamy, Botler, Dross, Naia, and Skokan proved that every $n$-vertex graph admits such a family of at most $19n$ paths. We improve this bound to $10n-o(n)$.

We study the number \(N_I(L)\) of zeros of the Hardy–Szegő zero process in the horizontal window \([0,L]\times I\) as \(L\to\infty\), where \(I=[\alpha,\beta]\Subset(0,\infty)\) is fixed. We obtain sharp point-probability asymptotics at the lower endpoint of the density scale. In the fixed-count regime, for every fixed integer \(k\geq0\), we determine a full asymptotic formula for \(\mathbb P\{N_I(L)=k\}\), identifying its exponential rate, order-one correction, and \(k\)-dependent polynomial prefactor; the case \(k=0\) gives the hole probability. In the vanishing-density regime, we prove a uniform local asymptotic formula for \(b_L\leq n\leq\varepsilon_L L\), where \(b_L\to\infty\), \(\varepsilon_L\downarrow0\), and \(b_L\leq\varepsilon_L L\), identifying the large-deviation exponent, endpoint correction, and Gaussian prefactor.