Papers

For a locally finitely generated Grothendieck category, we characterise the Gorenstein injective covering property by several equivalent conditions, including closure under coproducts. To complete this characterisation, we construct a countable commutative Noetherian Cohen–Macaulay local ring of dimension one for which this property fails, answering negatively Iacob's question Question 1 whether character modules of Gorenstein injective modules over left Noetherian rings are Gorenstein flat. Using this characterisation, we prove that for Noetherian schemes the covering property is Zariski-local. Moreover, assuming that $\mathcal{GI}(\mathrm{QCoh}(X))$ is a covering class, we establish compatible adjunctions and recollements for homotopy categories of injectives and totally acyclic injective complexes under open restriction. However, this property is not stalk-local: we construct a commutative Noetherian ring $S$ such that $\mathcal{GI}(S_{\mathfrak{p}})$ forms a covering class for all prime ideals $\mathfrak{p}$, but $\mathcal{GI}(S)$ do not form a covering class.

We extend Uhlenbeck's classical results on the generic properties of Laplacian eigenfunctions. In particular, we find a residual set of $C^k$ domains in $\mathbb{R}^n$, $k > n$, such that the nodal set of any Dirichlet eigenfunction intersects the boundary orthogonally along a $C^{k-1}$ embedded submanifold, whenever the intersection is non-empty. This is achieved by analyzing the boundary asymptotics of Dirichlet eigenfunctions via homogeneous harmonic polynomials and comparing their respective nodal sets. Along the way, we give a complete proof that generic Euclidean domains have simple Dirichlet eigenfunctions with zero as a regular value in the interior and zero as a regular value of the boundary normal derivative. In the special case of planar domains, a different argument provides one-parameter families of perturbed domains with the desired properties.

We prove the Labastida–Mariño–Ooguri–Vafa conjecture for oriented links with arbitrary integer framings. Consequently, the framed reformulated invariants have finite integral LMOV expansions for every integer framing vector.

A closed subspace $E$ of a Banach space $F$ is \emph{$L$-embedded in $F$} if there is a projection $P\colon F\to E$ such that $\|x\|=\|Px\|+\|x-Px\|$ for all $x\in F$. We show that if $X$ is a proper, geodesic metric space with locally finite $\H^1$-measure, then its Lipschitz-free space, denoted by $\F(X)$, is $L$-embedded in its bidual. We also extend an argument by Cúth, Kalenda, and Kaplický, to show that if $E$ is a finite dimensional normed space and $C\ge 1$ is such that every subspace of $E$ is $C$-complemented in $E$, then $\cal F(E)$ is $C$-complemented in its bidual. In particular, for every two-dimensional normed space $E$, $\cal F(E)$ is $1$-complemented in its bidual.

We consider a natural extension of the classical saturation problem to oriented graphs, defined as the minimum number of edges in an oriented graph $H$ on $n$ vertices that avoids a fixed oriented graph $G$, but adding any missing edge to $H$ creates a copy of $G$. Our main result shows that saturation functions of oriented graphs attain at least four distinct orders of magnitude. In particular, if $G$ is acyclic, then its saturation function is bounded if and only if $G$ contains an isolated edge; otherwise, it is linear. Certain tournaments together with an isolated edge give further infinite examples of bounded and linear saturation functions. Two additional orders of magnitude are $\Theta(n \sqrt{\log n})$ and $\Theta(n \log n)$. For these two, the lower bounds are obtained by adapting a recently developed approach for the saturation problem in edge-ordered graphs, while the matching upper bounds are constructive. Finally, we study the semisaturation problem, in which we require only that adding any missing edge to $H$ creates a copy of $G$. We completely characterize the possible orders of magnitude: the semisaturation function is either bounded, linear, or $\Theta(n \log n)$.

We establish a Weyl law for the codimension-one anisotropic volume spectrum on compact Riemannian manifolds. We then use this law to obtain equidistribution in the full Grassmannian bundle while keeping the ambient metric fixed. More precisely, on every closed connected smooth Riemannian manifold of dimension three through six, we construct smooth, embedded hypersurfaces stationary for anisotropies smoothly converging to area. The cumulative sum of their area measures, when lifted to the Grassmannian bundle, converges to the uniform measure.

We establish a Bogomolov–Gieseker-type inequality for tilt-semistable objects on smooth projective varieties of dimension $\geq 3$, in the spirit of the Bayer–Macrì–Toda conjecture. A key ingredient is an analysis of the action of the Frobenius morphism on tilt-semistable objects. As an application, we prove the Castelnuovo Bound Conjecture, up to an explicit constant term. Combining our results with work of Feyzbakhsh and Thomas, we further show that on a Calabi–Yau threefold, arbitrary rank Donaldson–Thomas invariants of semistable sheaves are determined by rank one invariants, and hence are symplectic invariants.

In this paper, we introduce a new class of data-driven ambiguity sets for distributionally robust optimization (DRO) that aim to enhance the local inference accuracy of an unknown distribution. Building on optimal transport-based discrepancy measures, we depart from the conventional paradigm of monolithic ambiguity balls, which rely on a single global transport budget to capture distributional uncertainty. Instead, we introduce local-discrepancy ambiguity sets, where the support of the distribution is partitioned into predefined regions and transport cost constraints are imposed locally to each region. This added flexibility allows for a more refined characterization of plausible distributions by preventing highly imbalanced transport across regions. As a result, the proposed approach can yield less conservative and more robust decisions, particularly in settings where decisions depend on the local properties of the sample space. We further derive tractable reformulations of the resulting DRO problems and demonstrate through simulations that our method can outperform traditional Wasserstein ambiguity sets in terms of decision quality.

The Klein-Gordon equation is a second order differential equation that appears in a variety of applications in mathematics, physics and engineering. This equation is often solved using numerical methods, since a general analytical solution is unattainable. However, one can find very useful analytical solutions for some special cases. In cosmology, the Klein-Gordon equation of a field can explain the physics of inflation and has important consequences on understanding the origin and evolution of the universe. An analytical solution to the Klein-Gordon equation in cosmology has been obtained in the small-field excursion approximation, and it turns out that the solution can be written in terms of the confluent hypergeometric limit function. In this work, we derive this result again while showing the details of the calculations and then generalize to the case of two coupled Klein-Gordon equations. We show that the solution of this coupled system can be written in terms of Kummer's hypergeometric limit function while considering minimal assumptions. We carry out a full numerical calculation of the coupled system and compare it to the results from the analytical expressions.

For $f(X)=X^r(X^s+a)\in\mathbb F_{p^m}[X]$, with $m,r,s\geqslant1$ and $a\in\mathbb F_{p^m}^\times$, put $g=\gcd(s,p^m-1)$. We prove that, for each fixed $g$, only finitely many prime characteristics $p$ admit such permutation binomials, even as $m,r,s,a$ vary. If $6\nmid g$, every such $p$ divides Wendt's determinant $W_g=\operatorname{Res}_X(X^g-1,(X+1)^g-1)$, which is nonzero. When $6\mid g$, this determinant vanishes; inverse power sums and iterated resultants yield an explicit nonzero integer divisible by every such $p>2g+1$. We determine the exact admissible characteristics for $1\leqslant g\leqslant10$, obtain a further divisibility condition over prime fields, and show that in prime fields $g=11$ occurs precisely when $p=23$.

We introduce and analyze a two-level overlapping additive Schwarz preconditioner for discontinuous Galerkin discretizations of virtual element spaces on general polygonal meshes in two dimensions, for a second-order elliptic problem whose coefficient is constant on each subdomain and may have large jumps across the interface. The discretization is a weighted symmetric interior penalty formulation: the averages carry the coefficient and the penalty is scaled by its harmonic mean across each edge, which keeps the method consistent and makes the constants of the analysis independent of the coefficient. The coarse space is obtained from a virtual, continuous partition of unity with a single degree of freedom per subdomain vertex, and is therefore applicable to the irregular subdomains produced by a graph partitioner; the discrete harmonic extension used to build it may be replaced by a cheaper projection onto polynomials of low degree. We prove that the condition number of the preconditioned system is bounded by $C(1+\log(H/h))(1+H/\delta)$, where $H$, $h$ and $\delta$ denote the subdomain diameter, the mesh size and the overlap, and where $C$ is independent of the number of subdomains and of the contrast of the coefficient, provided the latter is quasi-monotone. This is the same bound known for the conforming case and the analysis allows for hanging nodes. We also show, by an explicit example, that the quasi-monotonicity assumption cannot be dispensed with. Numerical experiments with structured triangular, hexagonal and Voronoi meshes, and with subdomains obtained both by uniform partitioning and by METIS, confirm the theory.

Rate splitting multiple access (RSMA) has emerged as a promising non-orthogonal transmission mechanism to improve spectral- and energy efficiency, and user fairness in multi-antenna broadband channels. To further enhance its performance, in this paper we investigate a novel cooperative RSMA (C-RSMA) transmission framework assisted by reconfigurable intelligent surface (RIS) and user relaying. More specifically, we first formulate the common- and private-rate maximization problem satisfying the transmit power constraint, by jointly optimizing the transmit beamformers at the BS and the reflective beamformer at the RIS. The above formulated problem is an intractable and NP-hard problem as the optimization variables are coupled and the strong residual self-interference (SI) is involved. To conquer the challenge mentioned above, we propose a low-complexity alternative optimization approach to jointly optimize the transmit and reflective beamforming. Simulation results show that our approach is capable of enabling superior sum-rate performance than the RIS-assisted space division multiple access (SDMA), half-duplex RSMA, and SDMA without using RIS schemes.

Let $\tau(G)$ denote the maximum number of pairwise edge-disjoint spanning trees contained in a graph $G$. Zhang and Fan [J. Graph Theory, 2026] posed the problem of determining the corresponding tight spectral-radius condition for every $s$-edge-connected graph with \[ k+1\leq s\leq2k-1, \] together with a characterization of the associated extremal graphs. In this paper, we completely resolve this problem.

Let $X$ be a smooth algebraic variety over $\mathbb{R}$. We study the separation of the connected components of $X(\mathbb{R})$ by equivariant cohomology, unramified cohomology, and signatures of unramified quadratic forms. Our results highlight how the geometry of $X$ influences these questions. We apply them to cubic hypersurfaces, complete intersections of two quadrics, abelian varieties, and Enriques surfaces. We also obtain new bounds on the exponent of the cokernel of higher signature morphisms in $\mathbf{I}$-cohomology and investigate the optimality of these bounds for smooth projective varieties.

In this paper, we study the last-iterate convergence of clipped stochastic gradient descent under heavy-tailed noise. We assume that the stochastic gradients are unbiased and that the noise has a bounded central $p$-th moment, where $1<p\le2$. For smooth convex objectives, a linearly decreasing learning rate gives a high-probability error bound of order $O(LR^2\log(1/\delta)/T+\sigma R(\log(1/\delta)/T)^{(p-1)/p})$. We further establish last-iterate guarantees for nonsmooth composite objectives and smooth composite objectives with proximal updates. In addition, when a causal adversary replaces at most $q$ gradients, the same clipped update achieves a bound with $\log(1/\delta)$ replaced by $\log(1/\delta)+q$. The corresponding lower bounds for the clean and corrupted oracle models are also provided.

We prove that the maximum modulus of every nonzero second Neumann eigenfunction on a bounded convex planar domain $\Omega$ is attained only on the boundary, without boundary smoothness or eigenvalue simplicity. The key is a rigidity theorem for any positive Neumann eigenvalue $\mu$: an interior maximum-modulus point $p$ with $\sqrt{\mu}\,\operatorname{dist}(p,\partial\Omega)<\pi/\sqrt{3}$ forces the domain to be a rectangle and the eigenfunction to be a cosine of one coordinate. In particular, $\mu\ge4\mu_2(\Omega)$, and the factor $4$ is sharp. This result reveals how a local distance condition at a single interior maximum-modulus point can enforce global geometric rigidity, thereby linking extremum localization to the classification of domains and eigenfunctions.

In this paper we present partial progress towards two conjectures on the variation of maximal operators of convolution type $\mathrm{M}_{\varphi}$, associated to non-negative and radial decreasing kernels $\varphi$ of integral $1$. In dimension $d=1$ we prove that $\mathrm{var}(\mathrm{M}_{\varphi} f) \leq \mathrm{var}(f)$ when $f$ is a characteristic function. In higher dimensions we prove that $\mathrm{var}(\mathrm{M}_{\varphi} f) \leq C_d \mathrm{var}(f)$ when $f$ is a radial characteristic function, with a constant $C_d$ depending only on the dimension $d$ but not on the kernel $\varphi$. We further show that $C_d \sim \sqrt{d}\log d $ is the sharp dependence on $d$ for such an inequality. This last claim relies on a novel explicit asymptotic estimate for the Hardy-Littlewood maximal function of the unit ball in $\mathbb R^d$, a result of independent interest.

Let $F:M^n\to\mathbb{S}^{2n+1}$ be a minimal Legendrian immersion of a connected closed manifold, where $n\ge3$, and let $B$ be its second fundamental form. We prove that if $|B|^2\le(n-1)(n+2)/n$, then, up to a unitary transformation, $F$ parametrizes a totally geodesic sphere or is a finite covering of the image of the minimal Calabi product of a point and a totally geodesic Legendrian sphere. This proves Conjecture 1 of Luo and Sun [Ann. Inst. H. Poincaré C Anal. Non Linéaire 40 (2023), p. 548].

We consider the image of the comparison homomorphism from bounded cohomology to ordinary cohomology. Its dimension is called the bounded Betti number. In our earlier work we proved that the total bounded Betti number of a closed Riemannian $n$-manifold satisfying $\Ric\geq -(n-1)$ and $\Diam\leq D$ is bounded by a constant depending only on $n$ and $D$. The original proof used the amenability of the images of sufficiently small local fundamental groups. Since that work was written, Kapovitch and Wilking proved Gromov's generalized Margulis lemma under a lower Ricci curvature bound. Their theorem gives the stronger conclusion that these local groups are uniformly virtually nilpotent. We explain how their result supplies a direct modern input for the bounded Betti number theorem. We also discuss the topological meaning of the bounded part of ordinary cohomology and its relation to the $\ell^1$-seminorm on homology.

We characterize several properties of higher-arity function classes, specifically finite $\VC_k$-dimension, slicewise finite $\VC$-dimension, and finite slicewise Littlestone dimension, in terms of generalizations of Rademacher complexity. These characterizations sets up simple new proofs of several results on finite $\VC_k$-dimension and slicewise finite $\VC$-dimension. These include the preservation of finite $\VC_k$-dimension under taking expectations and uniform law of large numbers statements for both properties. All of these results apply naturally to real-valued function classes as well as boolean-valued classes. We also prove a new adversarial uniform law of large numbers for classes with finite slicewise Littlestone dimension.

Finite-horizon optimal control becomes difficult when a dynamical model can be simulated reliably, but its derivatives are unavailable, discontinuous, or unreliable. Hamiltonian Primal-Dual Policy Optimization (HPDPO) is introduced as a derivative-free, model-based reinforcement learning framework for this setting. The control is represented as a time-indexed finite-horizon policy. Candidate actions are evaluated by deterministic suffix rollouts of the supplied transition model, while a regularized Hamiltonian dual formulation supplies structured future-cost information through block-coordinate multiplier updates. No differentiation of the state dynamics or learned critic is required. The method is validated on nonsmooth problems with exact solutions, an analytical and learned ReLU biomarker cascade, an amyloid Neural ODE trained from longitudinal ADNI data, and three multidimensional-control problems with up to five simultaneous controls. Across these examples, HPDPO closely reproduces exact or reference objective values, trajectories, and control structures. The results show that forward simulation and Hamiltonian dual structure can be combined to solve deterministic finite-horizon control problems involving nonsmooth, learned, or black-box dynamics.

Sampling from high-dimensional, heavy-tailed distributions is a fundamental challenge in computational statistics, as many standard Markov chain Monte Carlo (MCMC) methods mix poorly in such settings. Recently, Stereographic MCMC [Yang et al., 2024] and the Sub-Cauchy Projection Sampler [Grazzi et al., 2026] have been shown to perform successfully on such tasks. However, establishing their non-asymptotic convergence properties remains an important open problem. In this work, we fill this gap by establishing non-asymptotic upper bounds on the mixing time of the stereographic projection and sub-Cauchy projection samplers. Our results demonstrate that, under certain conditions on the target and initial distributions, the mixing time is polynomial in dimension for a broad class of distributions, including light- and heavy-tailed cases. Motivated by the theoretical analysis, we further establish a new weighted isoperimetric inequality that extends the classical version for log-concave distributions to the heavy-tailed setting with optimal dimension dependence. The proof techniques provide new insights into the geometric properties of heavy-tailed distributions that govern rapid mixing in high dimensions.

The modular data of the doubles of the Haagerup–Izumi categories is a grafting of two finite metric groups (conjectured by Evans–Gannon, proved for untwisted categories by Gannon–Schopieray–Yadav). Gang and Kim found non-unitary modular data of this shape in the topological twist of rank-$0$ $S$-fold theories, from the Bethe vacua of an abelianised gauge theory. The aim is to understand rigorously the significance of the handle-gluing and fibering operators, and to construct the Bethe vacua. For every hyperbolic once-punctured-torus monodromy $\varphi$, a word in $L$ and $R$, the two groups are $A_\mp=\coker(\varphi\mp1)$, with the torsion linking forms of the mapping tori of $\varphi$ and $-\varphi$. Over all words of trace $m^2+2$ these forms reproduce the Evans–Gannon table of Q-systems, one printed entry being corrected, and the distinct modular data in the list of Gannon–Schopieray–Yadav. The Bethe data are given in closed form by the word: the level matrix, the positive vacuum, $c_{\eff}=\#L$, and the handle-gluing values $2^{\#L}\lambda^{\pm1}|A_-||A_+|$ at the identity classes, $\lambda$ the stretch factor. Each kernel vector of $\varphi\mp1$ modulo $|A_\mp|$ at which an explicit trigonometric formula does not vanish gives a solution of the Bethe equations in closed form. Under the half-index dictionary, with the unit in the class of $\sigma V^*$, $c\equiv\#L-6\#R\pmod{24}$.

We construct non-holomorphic Siegel modular forms whose coefficients encode linking numbers in arithmetic hyperbolic manifolds of simplest type. Our proofs rely on an improved understanding of the transgression forms appearing in Kudla–Millson theory, and along the way we establish some structural properties including in particular their expected current equations.

Two-level overlapping Schwarz preconditioners with Generalized Dryja-Smith-Widlund (GDSW) and Reduced GDSW (RGDSW) coarse spaces for high-order finite element methods (FEM) and virtual element methods (VEM) are investigated and applied to the Poisson problem in 2D. The GDSW coarse space is a well-established robust choice for preconditioners in domain decomposition, while the RGDSW coarse space offers a competitive alternative by reducing the coarse problem dimension while maintaining robustness. On the theoretical side, a condition number bound for both preconditioners is proved for conforming $\mathbb{P}_k$ and $\mathbb{Q}_k$ finite element spaces with explicit dependence on the polynomial degree $k$. Through numerical experiments on structured triangular and quadrilateral meshes for FEM and on polygonal meshes for VEM, it is demonstrated that both GDSW and RGDSW maintain scalability properties across polynomial orders up to $k=9$ for selected degrees. The RGDSW approach achieves a reduction in the coarse problem dimension by a factor of approximately three compared to GDSW while maintaining competitive iteration counts.

This work provides a framework to treat optimal control problems on separable Banach spaces in the viscosity sense. First, it extends the notion of $B$-viscosity solutions to the Banach setting and relates the extension to the notions of viscosity solutions used in metric space settings. As the main results, the paper proves a strict comparison principle and characterizes the value function of an optimal control problem as the unique viscosity solution to an infinite-dimensional HJB equation.

Let $\TP_k$ denote the triangular pyramid with $k$ layers. For a graph family $\mathcal F$, let $\ex(n,\mathcal F)$ be the maximum number of edges in an $n$-vertex graph containing no member of $\mathcal F$ as a subgraph. For a graph $H$, let $\cJ(H)$ be the family of subgraph-minimal graphs among all such graphs $H-I$, where $I\subseteq V(H)$ is an independent set. For every integer $k\ge4$ and all sufficiently large $n$, we prove that \[ \ex(n,\TP_k)=\max_{0\le m\le n}\{m(n-m)+\ex(m,\cJ(\TP_k))\}+O_k(n)=\left\lfloor n^2/4\right\rfloor+\Theta_k(n^{4/3}). \] Moreover, for $k=4$, we characterize all extremal graphs and show that \[ \ex(n,\TP_4)=\max_{0\le m\le n}\bigl\{m(n-m)+\ex\big(m,\cJ(\TP_4)\big)\bigr\} \] for sufficiently large $n$.

Analogous to quasi-Frobenius rings, dual rings, and Johns rings, the classes of $p$-quasi-Frobenius rings, $p$-dual rings, $p$-Johns rings are introduced and used to discuss and settle principal analogues of the Faith-Menal, Faith, and FGF conjectures. A chain symmetry theorem establishes that for quasi-injective semi-projective modules with semiprime endomorphism ring, endonoetherianity and endoartinianity are equivalent and force the endomorphism ring to be semisimple artinian. Within this unified setting, it is shown that (1) two-sided $p$-Johns rings are $p$-quasi-Frobenius, giving a principal Faith-Menal analogue, with a counterexample showing that the one-sided hypothesis is insufficient; (2) semiprimary principal right ideal rings that are right principally injective and right Kasch are $p$-quasi-Frobenius-this affirmatively answers a question of Faith (Q2) in the principal two-sided ideal ring case and settles the principal Faith conjecture, with a counterexample showing the hypotheses to be indispensable; and (3) right $p$-Johns rings that are left CF and left perfect are $p$-quasi-Frobenius, yielding a principal FGF analogue, with a counterexample demonstrating that the hypotheses are essential. Wherever needed, examples are provided to show that various classes introduced are different from the existing classes in the literature.

We prove an effective version of the Chebotarev density theorem, conditional on the quasi-Riemann hypothesis for the Dedekind zeta function. Drawing on recent breakthroughs of OpenAI and Cureton–Thorner, we make unconditional the Ellenberg–Venkatesh bounds on $\ell$-torsion in class groups of number fields and extend an averaged result of Lemke Oliver–Smith to arbitrary families of degree-$d$-fields. Further applications are given to the least prime splitting and to small generators in number fields.

We study the irrationality of smooth Fano complete intersection fourfolds in weighted projective spaces, using an atom theoretic approach based solely on Gromov–Witten invariants. We prove symplectic irrationality, and hence irrationality for every smooth member, in a wide range of families for which irrationality was previously known only for very general members.