Papers

Tennant claims that his Core logic $\mathbb{C}$ is paraconsistent. It means that the sequent of the First Lewis Paradox, i.e. $\lnot A, A \vdash B$ is declared false, and its corresponding antisequent, called `Claim~1', i.e. $\lnot A, A \nvdash B$ true, as in minimal logic $\mathbf{M}$. This paper proves that Claim~1 entails a contradiction in $\mathbb{C}$, so that, to preserve consistency, the Core logician must reject the claim that his system is paraconsistent. The proof is purely logical, in four steps within a five-rule fragment $\mathcal{F}$ of $\mathbb{C}$ and its refutation system in the sense of Lukasiewicz and Goranko; the Appendix certifies every step in Coq – with no axiom assumed and every commitment displayed as a named hypothesis – and the same certification is replayed independently in Lean~4.

An operator $A$ on a complex Hilbert space $\Hh$ is called antilinear if $A(x+y)=Ax+Ay$ and $A(\lambda x)=\ov{\lambda} Ax$ for $x,y\in \cD(A)$ and $\lambda\in \dC$. We investigate some classes of densely defined antilinear unbounded operators, especially antilinear normal operators. We give various characterizations of antilinear normal operators and study a class of such operators in detail. Our main result is a structure theorem for unbounded antilinear normal operators.

We introduce an auxiliary gradient-flow framework for variational problems with generalized Newtonian structure governed by an N-function. The key idea is to replace the nonlinear constitutive dependence on the gradient, or symmetric gradient, by an auxiliary scalar variable representing its squared magnitude. This shifts the nonlinearity from the state equation to the auxiliary variable, yielding a sequence of uniformly elliptic weighted linear problems. At the continuous level, we construct an auxiliary energy on a metric space adapted to the growth of the underlying N-function. In this topology, we prove lower semicontinuity, geodesic $\lambda$-convexity, and exponential convergence of the associated minimizing-movement scheme. At the finite element level, we derive a metric gradient flow through an explicit Riesz map, prove global well-posedness of the resulting semi-discrete ODE, and establish convergence to the finite element solution of the Euler–Lagrange equations of the generalized Newtonian energy. For the $p$-Laplacian and $p$-Stokes models, this gives a rigorous convergence result for $4/3\le p\le 4$, $p\ne2$, with asymptotic rate estimates beyond this range. We also propose practical time discretizations, including an operator-splitting scheme that gives the \kac iteration as a special case, and an adaptive pseudo-transient method that can be implemented using scalable linear solvers. Numerical experiments for power-law, Carreau–Yasuda, regularized Bingham, and optimal-design models demonstrate robustness, mesh-independent iteration counts in the tested regimes, and performance that matches or outperforms Newton's method.

To improve the accuracy of Monte Carlo estimation of expectations, a set of zero-mean feature functions, known as control variates, can be used. They can be used as feature functions for linear regression of the target function, and we can obtain an unbiased and variance-reduced estimate using its residual. One known way to construct such functions is a method using an equality called Stein's identity, but these functions are not sufficient for the case where the target distribution is multimodal. We propose a different approach to constructing these zero-mean functions based on distribution approximation and the density ratio. We demonstrate that combining the functions constructed by these two strategies can effectively reduce the estimation variance for a bimodal distribution.

We construct non-projective complete log canonical algebraic surfaces whose canonical divisor is semi-ample over any algebraically closed field other than the algebraic closure of a finite field. We provide a unified framework to construct such surfaces for any given non-negative Kodaira dimension, namely, zero, one, or two. Furthermore, we give a relatively elementary proof, within the framework of the minimal model theory, of the essentially known fact that any complete log canonical surface with Kodaira dimension minus infinity is projective. This fact confirms that our construction covers all possible Kodaira dimensions. As a byproduct, the proof shows that such a surface of Kodaira dimension minus infinity has at most one non-rational singular point.

In this paper we consider an implicit semi-discrete approximation of a degenerate reaction-cross-diffusion system. Due to the symmetry in the parabolic part, this system is known to preserve segregation of densities – initially non-overlapping densities belonging to different species remain segregated for all times, which leads to internal layers between different species. We show that time-discrete approximations exist and converge to a weak solution, as the timestep goes to zero.

Distributed uncertainty-management systems often combine local probabilistic models along aggregation trees chosen by communication, privacy, or scheduling constraints. The final density should depend on the weighted sources, not on the particular order in which intermediate nodes combine them. We study this requirement as an algebraic compositionality problem for binary fusion of weighted probability densities. The central question is when a local fusion rule can be executed hierarchically while remaining order-invariant. We establish a compositional boundary for local segment-valued fusion rules. Within the class of continuous binary rules with additive output weights and weight-only coefficients, order-invariant hierarchical execution characterizes normalized weighted linear pooling; norm-induced segment balancing realizes the corresponding coefficient. Smooth endpoint-to-candidate $f$-divergence balancing has a different local geometry: its quadratic expansion induces square-root effective weights, showing why pairwise solvability alone is insufficient for schedule-independent fusion. We show that this obstruction is local to endpoint-to-candidate binary balancing, whereas global divergence barycenters retain additive-weight local limits. Finally, Gaussian mixtures show how the same issue appears in finite model classes: exact fusion is compositional, whereas stepwise compression is compositional only under a congruence condition on unnormalized component measures. These results distinguish exact schedule-independent fusion from global aggregation objectives and local approximation heuristics.

We study the topological zeta function of a loopless matroid $\mathsf{M}$ and its Möbius transform. We provide a novel and manifest description (a “Master Theorem”) for both functions and all of their coefficients, which can be used to give transparent solutions to several open questions and conjectures on topological zeta functions of matroids, even in greater generality than what was anticipated. As applications we solve conjectures of van der Veer (2019), Kutler (2023), and Mengesha, Miranda, and Sun (2026).

Sustaining cooperation in resource-constrained populations requires allocation mechanisms that balance individual incentives, resource sustainability, and distributional fairness. This paper proposes a network common-pool resource game in which individuals are embedded in complex networks, participate in multiple overlapping local resource pools, and face endogenous resource constraints during strategy evolution. Within this framework, we first examine two representative allocation mechanisms, equal allocation and proportional allocation. The results show that equal allocation produces fair but inefficient outcomes by weakening contribution incentives, whereas proportional allocation can temporarily promote cooperation but amplifies accumulated advantages and leads to severe inequality. To overcome these limitations, we develop a graph neural network-based reinforcement learning framework in which a learned social planner allocates local pool resources without directly controlling individual strategies. Simulation results under four representative network topologies show that the learned planner sustains higher cooperation levels and average accumulated resources, and reduces inequality compared with the baselines. Furthermore, we interpret the learned policy and distill it into two simpler mechanisms: a resource-dependent mixture mechanism for regular networks and a degree-conditioned mixture mechanism for heterogeneous networks. These mechanisms reveal that effective allocation should adapt to both local resource states and structural positions, providing an interpretable route from reinforcement learning policy search to mechanism design in networked resource-sharing systems.

We show that every derivation of a simple and rational vertex operator algebra of CFT type is an inner derivation.

This paper considers geometric problems of the following type: given a point set $P \subset \mathbb R^2$, one seeks a large subset avoiding a prescribed geometric configuration. Our main result states that, for any $P \subset \mathbb R^2$, there exists a subset $P' \subset P$ with $|P'| \gg |P|^{1/3}$ such that all of the distances determined by $P'$ are distinct. This improves a result of Charalambides. We make heavy use of a result of Li and Postle concerning the independence number of hypergraphs which satisfy some edge distribution conditions, as well as tools from incidence geometry.

We compare the classical phase-plane analysis of a dynamic two-dimensional IS–LM model with a sign-based trend approach. Applying both methods to the same specification, we examine which qualitative properties of the system can be established solely from sign relations and how these methods can complement each other. In summary, these two approaches capture complementary perspectives on the qualitative behaviour of the considered model, combining the detailed local classification of equilibrium dynamics provided by phase-plane analysis with the broader admissible qualitative structures captured by trend-based analysis.

The cycle space of a graph $G$, denoted $\mathcal{C}(G)$, is a vector space over ${\mathbb F}_2$, spanned by all incidence vectors of edge-sets of cycles of $G$. If $G$ has $n$ vertices, then $\mathcal{C}_n(G)$ denotes the subspace of $\mathcal{C}(G)$, spanned by the incidence vectors of Hamilton cycles of $G$. We consider several known sufficient conditions for Hamiltonicity and show that an appropriate and fairly mild strengthening of each such condition in fact ensures the stronger property $\mathcal{C}_n(G) = \mathcal{C}(G)$. In particular, we consider the classical Chvátal-Erdős criterion and prove that (under various additional restrictions) if $n$ is odd and $\kappa(G) \geq c \alpha(G)$, where $c$ is a sufficiently large absolute constant, then $\mathcal{C}_n(G) = \mathcal{C}(G)$. Moreover, considering the McDiarmid-Yolov criterion we prove that if $n$ is odd and $\delta(G) \geq \max \left\{2 \tilde{\alpha}(G) + 9, \tilde{\alpha}(G) + 18 \right\}$, where $\tilde{\alpha}(G)$ is the so-called bipartite independence number of $G$, then $\mathcal{C}_n(G) = \mathcal{C}(G)$. We also prove that if $n$ is odd and $G$ admits $16 \alpha(G) + 12$ pairwise disjoint connected dominating sets, $\mathcal{C}_n(G) = \mathcal{C}(G)$. Finally, we consider an effective Chvátal-Erdős type criterion for bipartite graphs and prove that if $G$ is a balanced bipartite graph on $2n$ vertices, satisfying $\alpha_{\emph{BIP}}(G) \leq 2 \delta(G) - 24$, then $\mathcal{C}_{2n}(G) = \mathcal{C}(G)$.

The classical Wallman compactification of a $T_1$-space and the Stone–Čech compactification of a completely regular space rely on choice principles. We show that, by representing a space by its powerset MT-algebra (McKinsey–Tarski algebra), both constructions admit choice-free compactifications. More generally, from any Wallman basis of a spatial $T_1$ MT-algebra we construct a compact $T_1$ MT-algebra which is a compactification of the original algebra. Taking the basis of all closed elements yields a choice-free Wallman compactification of every spatial $T_1$ MT-algebra, while taking the basis of zero-elements yields a choice-free Stone–Čech compactification of every spatial completely regular MT-algebra. Choice is only needed to show that the resulting compactifying algebras are spatial, and hence to recover the usual compactifying spaces. We also show that these constructions recover the corresponding compactifications of frames of opens.

The skew $(\mathfrak{gl}_{n}, \mathfrak{gl}_{r})$ Howe duality states that the exterior algebra $\Lambda(\mathbb{C}^{nr})$ admits a multiplicity-free decomposition under the natural actions of $\mathfrak{gl}_{n}\times \mathfrak{gl}_{r}$. In this paper, by using certain Lagrange interpolation polynomials of degree $r-1$, we extend the action of $\mathfrak{gl}_{n}$ on $\Lambda(\C^{nr})$ to its loop algebra $L(\mathfrak{gl}_{n})$. View $\Lambda(\C^{nr})$ as a module for the loop algebra $L(\mathfrak{sl}_{n})$ of $\mathfrak{sl}_{n}$ by taking restriction. We prove that every highest weight vector of $\mathfrak{gl}_{n}\times \mathfrak{gl}_{r}$ in $\Lambda(\C^{nr})$ generates a local Weyl module of $L(\mathfrak{sl}_{n})$. Furthermore, we obtain in this way an explicit realization of all local Weyl modules for $L(\mathfrak{sl}_{n})$.

We study a $\lambda$-biased random walk $(X_n)_{n\ge0}$ on the deterministic infinite rooted tree $\mathcal{T}=\{(i,j): i\ge0,\,0\le j\le i\}$, whose backbone is $\{(i,0):i\ge0\}$ and, for each $i\ge1$, the segment $\{(i,j):1\le j\le i\}$ forms a trap attached to $(i,0)$. The trapping effect induces long sojourns, yielding asymptotics markedly different from simple random walks. The walk is recurrent for $\lambda\ge1$ and transient for $0<\lambda<1$. In the transient regime it is sub-ballistic: its distance from the root grows logarithmically, with \[ \liminf_{n\to\infty}\frac{|X_n|}{\log n}=\frac{1}{\log(1/\lambda)},\quad \limsup_{n\to\infty}\frac{|X_n|}{\log n}=\frac{2}{\log(1/\lambda)},\quad\text{a.s.}. \] A contrast between spatial and temporal regeneration emerges. Let $C(n)$ be the number of cutpoints among the first $n$ backbone vertices and $M(N)$ the number of cut times up to time $N$. Then \[ \lim_{n\to\infty}\frac{C(n)}{n}= 1-\lambda,\qquad \lim_{N\to\infty}\frac{M(N)}{\log N}=\frac{1-\lambda}{\log(1/\lambda)},\quad\text{a.s.}, \] so cutpoints have positive linear density while cut times grow only logarithmically.

The paper deals with Ulrich wildness of decomposable threefold scrolls $X$ over Hirzebruch surfaces $\mathbb{F}_a$, for any $a \geqslant 0$. Our Main Theorem enstablishes that for $a=0$, the moduli space of rank-$r$ Ulrich bundles, for any $r \geqslant 2$ and of given Chern classes, contains a generically smooth, unirational component $\mathcal{M}(r)$ of computed dimension whose general point corresponds to a slope-stable Ulrich bundle; in particular $X$ turns out to be Ulrich wild. When $a \geqslant 1$ and in presence of modular obstructions, $X$ is nevertheless shown to be Ulrich wild too.

In the breakthrough paper \cite{Mu-jems}, it is constructed the first example of a simply connected compact $5$-manifold (aka.\ Smale-Barden manifold) which admits a K-contact structure but does not carry a Sasakian structure. In this work we clarify some aspects of the construction of \cite{Mu-jems}, determining explicitly the number $N$ of symplectic surfaces needed to have an isotropy locus that produce a $5$-manifold that is K-contact but not Sasakian. Also, in order to analyse the geography problem of determining which Smale-Barden manifolds admit K-contact but not Sasakian structures, we refine and generalize the constructions of symplectic surfaces in a symplectic $4$-manifold with transversal intersections giving rise to such manifolds.

Demazure weaves are combinatorial representations of maps between Bott–Samelson cells and have been used to construct cluster structures on braid varieties. We show the compatibility of these maps and the resulting cluster structures with the standard Poisson structure on the Bott–Samelson variety. Adding frayed strands to Demazure weaves, we further construct Poisson compatible cluster structures on other affine charts of the Bott–Samelson variety in a manner that transition functions across charts become rational quasi-cluster. The mutation sequences we construct for these quasi-cluster morphisms are closely related to those of Ménard for open Richardson seeds.

We prove the finiteness of the set of $j$-invariants of Drinfeld modules of rank 2 over $\mathbb{F}_q[T]$ which are CM and $S$-units, for $S$ the infinite set of primes with even degrees. The proof is based on the study of ordinary reduction and supersingular reduction of Drinfeld modules, and on the splitting behaviour of primes dividing the difference of two Drinfeld singular moduli. We also provide an algorithm to compute a polynomial with coefficients in $\mathbb{F}_q[T]$ and roots the $j$-invariants having CM by a given order, and use it to compute some explicit examples, providing for instance counterexamples to a conjecture of Dorman. For a maximal order $\mathcal{O}$, we prove by a universality argument that our algorithm computes the Hilbert modular polynomial $H_\mathcal{O}$.

We show that singular instanton Floer homology with the Chern–Simons filtration can be used to produce exotic pairs of slice disks. We moreover construct a strongly invertible $\mathbb{Z}$-slice knot for which any symmetric pair of $\mathbb{Z}$-disks are exotic, and remain exotic after stabilizing by $n\smash{\mathbb{CP}}^2$ or $n\smash{\overline{\mathbb{CP}}}^2$ (or by standard $n\smash{\mathbb{RP}}^2$ or $-n\smash{\mathbb{RP}}^2$) for any $n$. Our methods apply more generally to stabilization by any simply connected definite manifold, or by any number of exotic embedded projective planes of the same sign. We also provide an example of a strongly invertible knot which is $\mathbb{Z}$-slice and equivariantly slice, but not equivariantly $\mathbb{Z}$-slice. Along the way, we partially compute various symmetry actions on the singular instanton Floer complexes of two-bridge knots via an explicit analysis of their traceless $\mathit{SU}(2)$-character varieties.

Between April 1 and May 15, 2026, a group of 49 mathematicians compiled a dataset of research-level mathematics questions with known answers. Most of the work was done during the three-day workshop Benchmarks in Leipzig with 35 participants at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany. We present the resulting collection of 100~questions. We evaluated these questions in three stages: a single attempt by five state-of-the-art LLMs and their predecessors, followed by a 20-runs-per-model evaluation with three of these models, and finally a 3-run attempt with two heavy-thinking models. After Stage 1, 41 questions remained completely unsolved; after Stage 2, this count dropped to 16; and we concluded Stage 3 with only 2 unsolved questions. This demonstrates that the mathematical reasoning capabilities of LLMs are becoming impressive. In September 2026, we added a fourth stage in which the next generation of models attempted all 100 questions once more, after which only 1 question remains unsolved.

We develop the Dirac formalism for infinite-dimensional quadratic $\mathbb{Z}$-graded color Lie algebras with finite-dimensional components. Cubic Dirac operators are defined in completions of the quantum Weil algebra determined by the $\mathbb{Z}$-grading. The same grading fixes the normal-ordering convention. Normal ordering introduces a cohomological obstruction to the construction, measured by a color analogue of the Kac-Peterson class. When this class is trivial, we construct cubic and relative cubic Dirac operators satisfying the expected invariance properties and Parthasarathy-type square formulas. We further extend the Chern-Weil homomorphism to completed $\mathfrak{g}$-differential algebras and use it to identify the classical precursor of the cubic Dirac operator with the Chern-Simons element associated with the invariant quadratic polynomial determined by the quadratic structure. As applications, we consider symmetrizable Kac-Moody superalgebras. In this setting, the Kac-Peterson class is trivial, with primitive given by the Weyl vector, which yields the linear correction defining the cubic Dirac operator. We then use the relative Dirac operator to extract representation-theoretic information from highest weight supermodules. As an explicit example, for the affine Kac-Moody superalgebra associated with $\mathfrak{osp}(1\vert 2n)$, we compute the kernel of $\operatorname{D}_{\mathfrak{g},\mathfrak{g}_{\bar{0}}}$ on integrable highest weight supermodules. Finally, for unitarizable highest weight supermodules, we explain why the usual Dirac inequality is not available in the affine setting.

The purpose of this paper is to establish a local theory of the non-abelian Noether–Lefschetz locus. Given a family of projective manifolds over a complex variety $S$, the isomonodromic deformation of the initial $\mathbb C$-PVHS defines a holomorphic family of flat bundles and defines a real analytic family of Higgs bundles by the non-abelian Hodge correspondence. The non-abelian Noether–Lefschetz locus exactly consists of those points in $S$ on which the isomonodromic deformed Higgs bundle underlies a graded structure. Esnault-Kerz ask whether the non-abelian Noether–Lefschetz locus is precisely the maximal complex analytic subvariety on which the real analytic isomonodromic deformation of Higgs bundles becomes holomorphic. Our main result gives an affirmative answer to this question. The proof is based on the deformation equation of the harmonic metric solved by the non-abelian Hodge correspondence, and we use it to study higher order deformation class of the isomonodromic deformation of a graded Higgs bundle, which is expressed in terms of the differential graded Lie algebra of the joint real analytic deformation. We introduce a sequence of obstruction classes measuring the failure of holomorphicity and show that their vanishing forces the graded structure to lift to arbitrary finite order. This yields a local characterization of the non-abelian Noether–Lefschetz locus in terms of the holomorphicity of the isomonodromic deformation of Higgs bundles.

We show that for any uncountable cardinal $\kappa$, there is a coloring $c: [\kappa]^2\to \omega$ such that $c''A \otimes B = \omega$ for any $A, B\subseteq \kappa$ of order type $\omega_1$ that are stationary in their common supremum. In particular, the stationary version of Erdős-Rado theorem and the higher dimensional Friedman's property are both inconsistent. We demonstrate that the theorem is optimal in various ways.

We establish the existence of abelian envelopes for interpolation categories of wreath product groups $G\wr S_n$, for a fixed finite group $G$ with the symmetric groups $S_n$, for $n\ge0$. Our approach consists of showing directly via essentially combinatorial methods that certain generalized restriction functors admit adjoints.

We introduce exotic Kloosterman sums and exotic inverted Kloosterman sums attached to non-commutative finite-dimensional semisimple algebras over a finite field $\mathbf{F}_q$, and prove their reduction formulae to exotic Kloosterman and exotic inverted Kloosterman sums over commutative étale $\mathbf{F}_q$ algebras. We then obtain square-root estimates for these sums; for inverted sums an explicit correction term may appear.

This paper establishes the global well-posedness for the 3D anisotropic MHD system with partial dissipation: $\Delta_\mathrm{h}u$ for velocity and $\partial_1^2b$ for magnetic field, near background field $(0,1,0)$. Crucially, only horizontal components $(u^\mathrm{h}_0,b^\mathrm{h}_0)$ need to be small in $H^2(\R^3)$, while $(u^3_0,b^3_0)$ can be arbitrarily large. Our analysis develops novel techniques including component-decoupled energies and iterative control of dangerous nonlinearities using the background field structure. This establishes the global result for anisotropic MHD equations allowing large vertical data, breaking the full-smallness requirement of previous works.

Graph Neural Networks (GNNs) have demonstrated remarkable performance across a range of applications involving graph-structured data, particularly in high-stakes domains. However, the opaque nature of their decision-making processes limits their trustworthiness and broader adoption. Existing post-hoc explanation methods aim to improve explainability by identifying subgraphs that influence GNN predictions and adopt mixup strategies to alleviate the out-of-distribution (OOD) issue caused by using subgraphs for prediction. Yet, these approaches typically rely on soft masks, which are inherently unable to fully eliminate label-irrelevant information, allowing redundant structures to leak into the mixup process and hindering the resolution of the OOD problem, thereby degrading explanation fidelity. In this work, we propose HPME, a Hard-Perturbation Mixup Explanation framework grounded in a generalized Graph Information Bottleneck, which leverages graph pooling to extract discrete explanatory subgraphs and to yield an information-capacity bound to thoroughly compress label-irrelevant components. Furthermore, we introduce a novel mixup strategy built upon structure-level replacement, generating in-distribution explanations to effectively mitigate the distribution shift. Extensive experiments on diverse tasks demonstrate that HPME achieves state-of-the-art performance in generating robust and interpretable explanations across both synthetic and real-world datasets.

In this paper, we establish in Theorem 1.2 the reciprocity of wavefront sets for irreducible admissible representations $\pi$ of classical groups $G$ over any local field $F$ of characteristic zero if $\pi$ has a generic local $L$-parameter. Over archimedean local fields, based on the progress made in our previous work (arXiv:2207.04700), we prove in Theorem 1.5 that for an irreducible Casselman–Wallach representation $\pi$ with a generic local $L$-parameter, the Wavefront Set Conjecture (arXiv:2207.04700, Conjecture 1.2) and its refinement (Conjecture 1.1) hold for the arithmetic wavefront set ${\mathrm{WF}}_{\mathrm{ari}}(\pi)$ as defined by the associated enhanced local $L$-parameter of $\pi$ and the wavefront set ${\mathrm{WF}}_{\mathrm{tr}}(\pi)$ defined by the Harish–Chandra distribution character $\Theta_\pi$ of $\pi$. Hence the microlocal structure of $\Theta_\pi$ is completely determined by the arithmetic information carried by the enhanced local $L$-parameter of $\pi$. The relations with the algebraic wavefront set ${\mathrm{WF}}_{\mathrm{wm}}(\pi)$ defined by the degenerate Whittaker models are extensively discussed by means of the composition law (Theorem 3.4) over all local fields of characteristic zero. Under Conjecture 1.3, the Wavefront Set Conjecture is fully established over archimedean local fields. As a consequence, we prove a refinement of Vogan's maximal-orbit principle (Theorem 1.7).