Papers

Hessian structure on a manifold is a geometric structure consisting of a pair $(\nabla,g)$ of an affine connection $\nabla$ and a Riemannian metric $g$ satisfying certain compatibility conditions. It is also known as a dually flat structure in information geometry. Hessian geometry is considered to be a real analogue of Kähler geometry; in particular, Hessian sectional curvature corresponds to holomorphic sectional curvature in Kähler geometry. The standard models of Hessian manifolds with constant Hessian sectional curvature were classified by Furuhata and Kurose (2013) in the nonpositive curvature case, whereas the positive curvature case is open. In this paper, we classify the standard models in the positive curvature case. For this end, we introduce Hessian structure on quotient manifolds of Kähler manifolds equipped with suitable Hamiltonian actions of abelian Lie groups and realizing the standard models as such quotient manifolds.

Let $S_g$ be a closed oriented surface of genus $g$. Ivanov conjectured that if $G$ a finitely generated subgroup of $\mathrm{Mod}(S_g)$ its finite-index Frattini subgroup $\Phi_f(G)$ is nilpotent. We prove this conjecture, using recent partial results of the first author about the Andersen-Masbaum-Ueno conjecture.

A storage code on a graph assigns a symbol to each vertex so that the symbol can be recovered from its neighbors. We prove that the binary full-parity storage codes on triangle-free coset graphs of primitive BCH codes have rate tending to one for every fixed error-correction capability of at least two. The conclusion also holds for an increasing error budget under an explicit growth condition. The proof bounds the binary rank of convolution matrices associated with polynomial maps. Low coordinate degree forces cancellation in the polynomial representing the image, and a polynomial rank bound converts this cancellation into a quantitative estimate for the storage rate. For the double-error-correcting family over $\mathbb{F}_{2^m}$, we prove that the storage deficiency is $\Theta(((1+\sqrt{5})/4)^m)$. This refinement follows from an exact Fibonacci formula for a polynomial coefficient rank and a matching exponential lower bound after finite-field evaluation.

Let $X$ be a nonsingular Hermitian variety of even dimension over a finite field, and let $\Lambda$ be the lattice spanned by the classes of its maximal totally isotropic linear subspaces defined over the base field. By work of Dummigan, Dummigan–Tiep and Shimada, these classes span the middle cohomology, and the intersection form on the primitive part is definite, as predicted by the standard conjecture of Hodge type. We give an elementary proof, which avoids the representation theory of finite unitary groups and uses only the dual polar graph attached to the Hermitian form. It shows that the intersection form on the primitive part is an explicit positive multiple of a Euclidean inner product. The advantage of the new proof is that it also controls $\Lambda$ $p$-adically. If $X$ is defined over $\F_{q^2}$ and has dimension $2m$, every $p$-adic elementary divisor of the intersection matrix of these subspaces divides $q^m$. Consequently $q^m$ times every algebraic cycle of codimension $m$ is numerically equivalent to an integral combination of them. As a by-product, the standard conjecture of Hodge type holds for varieties finitely covered by Hermitian varieties, and the $p$-adic bound descends along finite morphisms of degree prime to $p$.

We investigate how root allometry and below-ground root competition due to root overlap affect the conditions for pattern formation in non-local scalar vegetation models. We start from a spatially explicit two-component vegetation-water system in which plants acquire soil water through their laterally extended root systems, with root allometry represented by an interaction kernel whose spatial extent increases with the above-ground biomass density. In the limit of fast water dynamics, we eliminate the water variable, resulting in a non-local scalar model for the vegetation dynamics. This reduction reveals that accounting for root overlap naturally leads to a double integral structure in the scalar model, in contrast to the single convolution term commonly used to model non-local competition for resources. For the resulting scalar model, we identify two distinct mechanisms that can drive Turing instabilities. The first instability mechanism requires root growth to outpace mortality, while the second mechanism relies on the development of competitive pressure zones, in which vegetation growth is impeded by the cumulative competitive pressure exerted by neighbouring areas with higher biomass density. Root allometry promotes the occurrence of both instability mechanisms, while root overlap inhibits the emergence of competitive pressure zones. Hence, our results show that both root allometry and root overlap are important features to consider when studying vegetation pattern formation.

Understanding loss landscapes is central to explaining neural-network training, yet their structure remains only partially understood even in simple models. We study the Gaussian population loss of shallow, bias-free ReLU networks in the teacher–student setting. This provides a simple model for studying essential aspects such as feature learning and overparameterization. For teacher networks with positive output weights and planar features, we show that including a learned linear skip removes all spurious local minima with non-negative student output weights once the student network is at least as wide as the teacher network. In contrast, without the skip, we construct a fixed teacher network with positive output weights and only three hidden neurons in input dimension two whose spurious local minima persist at every student width at least three. Thus, a learned linear skip can remove spurious minima that persist under arbitrary overparameterization. Furthermore, we show that a positive output weight student network always learns the subspace spanned by the teacher features: student features at local minima with non-negative student output weights lie in the span of the teacher features. For ReLU networks in two dimensions, even heavily overparameterized student networks have effective width controlled by the teacher width: every critical point with positive student output weights has at most twice as many distinct student feature directions as teacher neurons. Finally, we transfer the benignity result to empirical minima over parameter balls of any prescribed radius, with the required sampling accuracy depending on that radius.

For weak solutions to quasilinear degenerate parabolic equations of $p$-Laplace type, a central obstacle in applying the method of intrinsic scaling to prove their Hölder regularity is the derivation of forward-in-time propagation estimate for the spatial measure of level sets. In this paper, we revisit and overcome this difficulty for an inhomogeneous degenerate parabolic $p$-Laplace equation with bounded nonnegative time-independent forcing term and initial data. Our new proof is built upon the temporal Lipschitz estimate that we establish on any time interval separated from the initial time by a positive amount. The strong regularity gain furnished by this estimate allows us to derive the forward propagation of level-set measures directly, thereby bypassing the energy inequality that served as the starting point of the previous approach. Furthermore, we establish space-time $W^{1,\infty}_{x,t}$ estimates away from the initial time and construct counterexamples to demonstrate the sharpness of the positive waiting-time condition under merely bounded initial data.

In a recent paper [Kwaśnicki, Poisson kernels on the half-plane are bell-shaped, Bull. London Math. Soc. 58 (2026), no. e70303], the author proved that Poisson kernels for a class of elliptic operators in the half-plane are bell-shaped. By allowing one-sided barriers, we prove a converse result.

We study complete translating solitons for the mean curvature flow in arbitrary codimension under quadratic pinching of the second fundamental form. We prove a codimension reduction theorem at the Andrews-Baker threshold: if a complete $n$-dimensional connected translator isometrically immersed in $(\mathbb{R}^{n+p},\langle\cdot,\cdot\rangle)$ satisfies \[ |B|^2\leq \left(\frac{4}{3n}-\varepsilon_{0}\right)|H|^2 \] for some $\varepsilon_{0}>0$, then either it is an affine $n$-plane or $|H|>0$ everywhere and the translator is contained in an $(n+1)$-dimensional affine subspace. In particular, this improves in low dimensions the pinching range previously obtained from codimension reduction results for ancient mean curvature flows. Our proof is purely elliptic: viewing translators as weighted minimal submanifolds, we combine Simons-type identities for the drift Laplacian with refined gradient estimates and an Omori-Yau maximum principle. As a consequence, under the stronger pinching condition \[ |B|^2\leq (c_n-\varepsilon_{0})|H|^2,\qquad c_n=\min\left\{\frac{4}{3n},\frac{1}{n-2}\right\} \] for $n\geq3$, with $c_n=2/3$ when $n=2$, we obtain a rigidity theorem: the translator is either an affine plane or a bowl soliton contained in an $(n+1)$-dimensional affine subspace. No entropy assumption is required.

Baxter's classical theorem for orthogonal polynomials on the unit circle establishes that the linear Fourier coefficients of a measure are in $\ell^1$ if and only if the nonlinear coefficients, i.e., the so-called Verblunsky coefficients, are in $\ell^1$. However, this equivalence is purely qualitative. We explore possible formulations of such quantitative theorems with norm and Lipschitz estimates for the $SU(2)$-valued nonlinear Fourier transform, for which the analog of Baxter's theorem has been recently obtained. In particular, the highlight of this paper is a number-theoretic construction for the NLFT which proves some negative results in this direction. We also prove a positive result and discuss the limitations of Baxter's method.

We prove two deterministic results for families of distances arising in the study of canonical processes. The first derives an admissible partition scheme from a growth condition. The second gives a representation in terms of parameterized separation trees and compares it with the corresponding majorizing-measure quantities. The main point is that the proofs do not depend on the distribution of the underlying process: once the initial distance and the family of distances are given, no random variables, independence, tail functions, or moment estimates are used. For canonical processes with regular log-concave tails, the assumptions of the abstract results follow from the usual regularity conditions. One direction of the separation-tree estimate also applies to Bernoulli processes without these additional assumptions, and we prove the reverse estimate for bounded convex unconditional index sets. We also give a version of the growth argument for points which, for finite index sets, leads to a recursive construction of admissible partitions and parameterized separation trees.

The $k$-Schur functions (resp., $K$-$k$-Schur functions) are the symmetric function representatives of Schubert classes in the homology (resp., $K$-homology) of the affine Grassmannian of type $A$. We prove that every $K$-$k$-Schur function is a nonnegative integer linear combination in the basis of $k$-Schur functions. This settles a conjecture by Lam, Schilling and Shimozono. The proof relies on realizations of $k$-Schur and $K$-$k$-Schur functions as Catalan and Katalan functions, respectively.

We present two particle methods for point set registration in bounded domains based on the Fokker-Planck equation. The first method relies on a moving least squares discretization of the S–formulation, in which moving grid points (particles) are advected by the drift with the target distribution, diffusion is resolved on a dynamically evolving particle cloud. This setting naturally leads to strong compression and expansion of the particle cloud. Obstacles are handled by enforcing reflective boundary conditions through a novel ghost point method. The second method is a Monte Carlo solver for the associated Langevin stochastic differential equation. It relies on a local approximation of the logarithmic gradient of the evolving particle density to extract macroscopic osmotic paths from individual stochastic trajectories. Owing to its inherent parallelism, this method exhibits excellent scalability and is well suited for high-dimensional registration problems. We illustrate the main features and performance of both approaches through extensive numerical experiments.

Let $K$ be any complete non-Archimedean field, not necessarily spherically complete. We prove that every polystable representation of a finite quiver over $K$ admits a split harmonic norm in the sense of Haiden–Katzarkov–Kontsevich–Pandit (HKKP), answering a question from their preprint Towards Categorical Kähler Geometry. Moret-Bailly's closed-image theorem gives a uniform separation from representations with destabilizing subobjects. A finite estimate on norm weights then yields a linear coercivity bound for stable representations and hence a minimum of the extended HKKP potential on the complete CAT(0) space of all norms. A direct first-variation computation identifies split minimizers with HKKP harmonic norms. Strict supporting hinges reduce split attainment to semistable reduction for quivers with arbitrary real radius labels. Small changes of the lifted splitting make each reversed extension nonsplit; a finite slope–rank descent proves termination. Consequently, every minimum in the completed norm space is attained by a split norm with the same arrow singular profiles, without restrictions on the value group or the quiver. Nonsplit minimizers illustrate split attainment in one and two radius modifications, and a two-parameter Berkovich analytic family separates polystability, semistability, and harmonicity of a chosen split norm.

We construct finite-dimensional Lie-group summaries of based rotational paths that retain the exact terminal attitude and selected iterated angular-velocity integrals. These summaries, called rotational trajectory synopses, compose exactly under path concatenation. Building on the classical works of Magnus and Chen, we establish endpoint-retaining body, spatial, and hybrid synopsis groups at every finite order and give a uniform matrix realization for the evaluated Magnus hierarchy. We distinguish this hierarchy from the full Chen signature from order three onward and prove \chg{that the exact endpoint is independent of every finite truncation of the body, spatial, or combined angular-velocity signature: the synopsis maps are surjective at every finite order}. An underactuated satellite example connects the retained coordinates to geometric phase in spacecraft attitude dynamics.

We prove nonlinear stability of the Guderley imploding shock under radial perturbations for the three-dimensional compressible Euler equations in the adiabatic range $1<\gamma\leq\frac{5}{3}$. The perturbed solutions develop an asymptotically self-similar implosion in finite time. Our result allows perturbations with small, nonzero constant pressure in the quiescent region ahead of the shock. The proof combines weighted $L^\infty$ estimates along characteristics with a computer-assisted proof of stability inequalities for the Guderley profile.

We construct solutions to the two-dimensional stationary Prandtl system under a constant adverse pressure gradient exhibiting a family of distinct separation laws. More precisely, for every integer $\ell\ge1$, we construct smooth initial data for which separation occurs at a finite location $x=x^\ast$, with \[ \partial_y u(x,0) =\mathbf C_\ell(x^\ast-x)^{\ell/2}\bigl(1+o(1)\bigr), \qquad x\uparrow x^\ast, \] where $\mathbf C_\ell>0$ depends on the initial data. We determine the local asymptotic structure of the solution near separation through matched inner and outer expansions based on linearization around the ground state in von Mises variables. This description addresses Open Problem~5 posed by Oleinik and Samokhin~\cite{OleinikSamok-book-99}. We also prove that the square-root separation regime, corresponding to $\ell=1$, is dynamically stable under small admissible perturbations of the initial data in a suitable topology.

In this paper we explore and extend the state of the art for understanding the skill and utility of early warnings of critical transitions (tipping points) in forced nonlinear systems. We highlight some of the challenges and opportunities of transforming estimates of a dynamics-based early warning system, based on trends in stability or resilience into forecasts with high skill. We highlight the importance of (a) quantifying consequences of possible actions in response to warnings that may be false negatives or positives (b) considering finite time horizon predictions to give verifiable predictions (c) assumptions necessary for valid extrapolations of trends. We evaluate approaches to improving the skill and utility of the forecast.

We classify the set of finite simple undirected graphs whose corresponding graph operator system is C*-nuclear. We prove that the operator system associated to the graph is C*-nuclear if and only if the graph is 2-chordal, as defined by McCullough. In particular, we present the first known example of a non-chordal graph whose graph operator system is C*-nuclear, as well as the first known examples of graph operator systems which are not C*-nuclear.

We prove (under some mild assumptions) that the strong-strong twice epidifferentiability of a convex functional is equivalent to the weak-$\star$ twice epidifferentiability of its conjugate functional. Further, the corresponding subderivatives are (up to scaling) conjugates of each other. This result can be used to facilitate the computation of weak-$\star$ second subderivatives, which are crucial ingredients in no-gap second-order optimality conditions. We also show that weak-$\star$ second subderivatives are invariant under suitable changes of the underlying function space setting.

We study the construction of ternary semi-direct products in semi-abelian categories, following a definition of higher semi-direct product previously introduced by Carrasco and Cegarra for groups and Lie algebras. We show that these objects are determined by actions and equivariant morphisms satisfying a certain compatibility condition in the ambient category, and that actions by semi-direct products are a special case of this construction. We also show how their structure can be further simplified in algebraically coherent or locally algebraically cartesian closed categories. We also give a concrete description of the structure of ternary semi-direct products in concrete categories, and how our categorical interpretation of the necessary structure relates to the group and Lie-theoretic versions given by Carrasco and Cegarra.

Closed bosonic lattice systems do not generally admit operator norm Lieb–Robinson bounds uniformly in the initial state. The reason is that the information propagation velocity can grow with the local boson occupancy, which can be macroscopically large. Here, we show that local dissipation restores an operator-norm Lieb–Robinson bound. We consider the dissipative Bose–Hubbard model described by a Lindbladian operator with on-site $\ell$-photon loss, $\ell>2$ and, for our main result, we establish an almost-ballistic Lieb-Robinson bound. Dissipation rapidly depletes highly occupied sites, thus regularizing the state on the Sobolev-type scale of local particle moments. The resulting moment bounds diverge as time approaches zero but remain integrable near zero at sufficiently low orders. We extend these ideas to treat cat-code dissipation and, for initial states in the code space, we prove a local adiabatic approximation uniform in the total volume. Further consequences include local channel approximation, a thermodynamic limit, and efficient digital quantum simulation. The point is that these applications are now available uniformly in the input state, as for quantum spin systems, but in contrast to closed Bose–Hubbard systems.

Motivated by hyperconnected urban parcel networks for same-day and time-sensitive delivery, we study the fixed-line routing problem that arises when evaluating a candidate set of recurring inter-hub vehicle services. Vehicles follow predefined cyclic sequences of service points, or lines, and parcels may transfer between lines. Such transfers expand path choice but require loading and unloading operations that make vehicles dwell at service points. Because handled volumes are routing-dependent, dwell times become endogenous, linking routing decisions to line cycle times, effective service frequencies, and transfer waiting times. We introduce the Routing in Line Networks with Handling Times (RLN-HT) problem and formulate it as a bilinear generalized capacity-flow assignment model. We characterize optimal dwell times and the resulting frequencies as functions of handling workloads, enabling a reformulation solely in terms of commodity flows. We exploit its convex substructure to derive two second-order conic relaxations that preserve the original feasible region and therefore yield feasible incumbent solutions along with valid lower bounds. We also develop a residual-frontier framework that characterizes the tradeoff between the convex relaxation objective and the remaining nonconvex residual, yielding a general mechanism for constructing lower-bound certificates. Computational experiments show that ignoring handling times can yield poor or infeasible routing, whereas accounting for endogenous dwell times can improve average parcel travel times by up to 26.5%. In instances with up to 257 nodes and 25 lines, the relaxations solve quickly, yield near-optimal routing, and obtain small optimality gaps, whereas off-the-shelf spatial branch-and-bound struggles to produce useful bounds.

We establish complexity lower bounds for stochastic first-order algorithms in nonconvex–concave minimax optimization, allowing algorithms to use variance reduction. Our main contribution is a lower bound for a zero-respecting algorithm class that permits variance reduction, extending beyond the algorithmic restrictions imposed by some existing lower bounds. We consider objectives with an $L$-Lipschitz continuous joint gradient, a compact convex dual domain of Euclidean radius at most $D_Y$, and a primal value function, defined by maximizing the objective over the dual variable, with initial suboptimality at most $\Delta$. The target accuracy $\varepsilon$ is measured by the gradient norm of the Moreau envelope of the constrained primal value function with parameter $1/(2L)$. Under an unbiased stochastic first-order oracle with variance at most $\sigma^2$ and mean-square smoothness, we prove the lower bound $\Omega\!\left(L^2D_Y\Delta\varepsilon^{-3}+L^3D_Y^2\Delta\sigma^2\varepsilon^{-6}\right)$. This result quantifies the dependence on accuracy, dual-domain radius, and oracle noise even when variance reduction is allowed. We also establish complementary lower bounds for nonconvex–strongly-concave minimax optimization. With dual strong-concavity parameter $\mu>0$ and condition number $\kappa:=L/\mu$, we obtain $\Omega\!\left(L\Delta\sqrt{\kappa}\,\varepsilon^{-2}+L\Delta\kappa\sigma^2\varepsilon^{-4}\right)$ under the bounded-variance oracle model. Under the additional mean-square smoothness condition with constant $\bar L$, we obtain $\Omega\!\left(L\Delta\sqrt{\kappa}\,\varepsilon^{-2}+\Delta\bar L\sigma\kappa^{3/2}\varepsilon^{-3}\right)$. Together, these results identify complexity barriers across the concave and strongly concave regimes, with the main nonconvex–concave bound remaining valid for algorithms that use variance reduction.

For a third-order symmetric tensor, the Comon rank gap is the difference between its CP rank and its symmetric rank. We study direct sums of Lovitz's rational $27$-dimensional tensor, whose two ranks are $55$ and $56$. Two copies have ranks $110$ and $112$, and three copies have ranks $165$ and $168$, over both the real and the complex fields. Thus direct sums yield concise cubic tensors on spaces of dimensions $54$ and $81$ with Comon rank gaps two and three.

Given a symmetric simple random walk $(X_n)_{n \ge 0}$, the family of all symmetric simple random walks $(Y_n)_{n \ge 0}$ adapted to the filtration of $(X_n)_{n \ge 0}$ was studied in Collevecchio et al. (2022). In particular, the authors established necessary and sufficient conditions under which the suitably normalized two-dimensional process $((X_n, Y_n))_{n \ge 0}$ converges weakly to a two-dimensional Brownian motion. When this occurs, we say that the random walk $(Y_n)_n$ is normal (with respect to $(X_n)_n$). In this paper, we investigate whether a "randomly selected" $(Y_n)_{n \ge 0}$ is normal. We consider a very general randomization procedure and look at both the quenched and annealed settings.

Jin, Milojević, Tomon and Zhang established a powerful recursive estimate relating the positive eigenvalues of a graph whose least eigenvalue is small in absolute value. We establish a refinement of this result, yielding improved estimates across spectral graph theory and discrepancy theory, and for Chowla's cosine problem. Recent results of Janzer, Tomon and Yip allow us to directly transfer least eigenvalue estimates to the corresponding surplus estimates. Using these tools, we fully resolve a conjecture of Räty, Sudakov and Tomon. We show that, for an $n$-vertex graph $G$ with least eigenvalue $\lambda_n$ and surplus $\operatorname{sp}(G)$, if $G$ is $\epsilon$-far from all disjoint unions of cliques, then $|\lambda_n|\geq \Omega_\epsilon(n^{1/4})$ and $\operatorname{sp}(G)\geq \Omega_\epsilon(n^{5/4})$. Furthermore, we show that when $G$ is $n^{-o(1)}$-far from all disjoint unions of cliques, we have $|\lambda_n|\geq n^{1/4 - o(1)}$ and $\operatorname{sp}(G)\geq n^{5/4 - o(1)}$. Finally, we show that the surplus of a $K_t$-free graph with $m$ edges is at least $m^{0.614 - o(1)}$ as $m$ tends to infinity.

This paper is concerned with establishing estimates on averaged Green's functions for a uniformly elliptic divergence form partial difference equation with random coefficients on the $d$ dimensional integer lattice $\mathbb{Z}^d$. It has previously been shown that the averaged Green's function is point-wise well approximated by the homogenized Green's function at large length scales. Corresponding results also hold for (fractional) derivatives of the averaged Green's function up to but not including the second derivative. Second derivative results have been established when the elliptic equation generates a positive semi-group, as in the diagonal case. Here the second derivative result is shown more generally by using an insight of Bourgain.

The honeycomb and Kelvin problems ask how space can be partitioned into cells of equal volume with the least surface area. We resolve this question in four dimensions for convex bodies that tile space by translations, showing that the regular $24$-cell uniquely minimizes surface area at fixed volume. We develop a framework combining an exact deficit identity, valid in every dimension, with a descent argument. As a consequence, the regular $24$-cell also minimizes surface area among such tiles containing a fixed ball.

We prove asymptotic bias bounds for stochastic gradient Langevin dynamics in Wasserstein distance of order two. We assume that the negative log-density is strongly convex with a Lipschitz gradient, and that the stochastic gradient estimator is unbiased with an error satisfying a mean-square Lipschitz condition. The bounds are of order $h$ under a fourth moment assumption on the stochastic gradient error and of order $h^{1/2}$ under only a second moment assumption, where $h$ is the stepsize. A spiked-noise example shows that a second moment assumption alone is insufficient for a bound of order $h$ that is uniform over noise distributions with a fixed variance.