Papers

We develop a graphical calculus for monoidal categories equipped with twisted pivotal structures, which are a generalization of pivotal structures originating from the study of orientation structures in the context of the Cobordism Hypothesis. This graphical calculus depends on a possibly singular foliation, and we use it to construct twisted string net modules for surfaces equipped with a Morse function or a Morse foliation. We prove that, despite the apparent dependence on this Morse function, the twisted string net modules assemble in an oriented categorified 2-TQFT. We study when the twisted string net module of the 2-sphere vanishes, relate it to the distinguished invertible object for finite tensor categories and exhibit examples of non-unimodular finite tensor categories with non-vanishing twisted string net module on the 2-sphere. This vanishing is expected to be the main obstruction for extending our categorified 2-TQFT to a non-compact 3-TQFT.

The numerical simulation of electromagnetic transients in fusion devices is essential for analyzing plasma stability and disruptive events. However, it remains computationally demanding due to the large-scale dense systems arising from integral formulations. This work proposes a model order reduction (MOR) strategy for transient electromagnetic problems based on integral formulations. Unlike operator-based compression techniques (such as $\mathcal{H}$-matrix approaches), the reduced space is constructed directly from the transient excitation. In contrast to classical snapshot- and transfer-function-based MOR approaches, the proposed formulation avoids repeated explicit inversions or factorizations of the dense integral operator during the MOR basis-construction stage. By combining wavelet-based temporal compression with source-driven Krylov projections, the method generates reduced models tailored to the dynamically reachable responses of the prescribed excitation families. Numerical validations on various plasma events and fusion-relevant scenarios demonstrate the robustness of the strategy, achieving substantial computational speedups while accurately preserving the transient electromagnetic response. Finally, the method is successfully applied to the null-field problem to efficiently generate training data for neural-network surrogates, contributing toward physics-consistent AI-enabled fusion modelling.

The existence of positive, pointwise decaying at infinity, weak solutions to a fractional $p$-Laplacian problem in the whole space and with singular reaction is established. Truncation arguments, variational methods, as well as suitable a priori estimates are exploited.

Let $X$ be a smooth proper variety over an algebraically closed field of characteristic zero, and let $\mathcal{A} \subset D^{b}_{\mathrm{coh}}(X)$ be an admissible subcategory. Let $Z \subset X$ be the union of set-theoretical supports of all objects in $\mathcal{A}$ and assume that $Z \neq X$. We show that for any morphism from $Z$ to an abelian variety each fiber has no isolated points; this implies, for example, that $Z$ cannot be isomorphic to an abelian variety. The key input is the fact that while not all line bundles on $Z$ lift to infinitesimal thickenings of $Z$, sufficiently many do: specifically, we show that for any infinitesimal thickening $Z \subset \widetilde{Z}$ the restriction morphism $\mathrm{Pic}^0(\widetilde{Z}) \to \mathrm{Pic}^0(Z)$ on the connected components of Picard schemes induces an isogeny between Albanese group schemes of those connected components.

Given a positive integer~$p$, we consider $W^{1,p}$-maps from a Euclidean domain of dimension $p+1$ into a closed Riemannian manifold $\NN$. The target manifold is required to satisfy a suitable topological condition, that is, the action of the fundamental group over the~$p$-th homotopy group must be trivial; however, we do not assume that $\NN$ is $(p-1)$-connected. Using tools from geometric measure theory — namely, chains with coefficients in the $p$-th homotopy group of~$\NN$— we associate to each map a new object that captures its topological point singularities. We provide a characterisation of strong and weak sequential closures of smooth maps in~$W^{1,p}$ in terms of their topological singularities.

LVMB manifolds are a class of non-Kähler compact complex manifolds with a remarkably rich geometry: in many cases they admit a holomorphic bundle structure over a compact toric manifold. In fact, such a bundle is determined by an algebro-combinatoric datum encapsulating a simplicial fan. This is reflected in a close relationship between the geometry of an LVMB manifold and that of its toric base space. Throughout this paper, we restrict to this subclass of LVMB manifolds. We provide a formula for the characteristic class of the bundle in terms of the original LVMB datum. Subsequently, this expression is employed to address the existence of Hermitian metrics satisfying “special” conditions. We consider balanced and SKT metrics, showing that the former are obstructed in most cases. Moreover, within the LVMB class, we construct a new example of a manifold admitting a balanced metric. Finally, restricting ourselves to the subclass of LVM manifolds, we characterize those admitting an SKT metric.

In this Part 2 of our article we give a detailed discussion of the compatibility between the analytic Gysin maps we have defined in Part 1 and the topological Gysin maps defined by the second author. A significant role is played by a bordism-like description of K-homology due to Jakob which is closely related to the geometric K-homology theory of Baum and Douglas. We give a self-contained proof of the equivalence of the former with the analytic K-homology theory of Kasparov. As an intermediate step towards proving our main result we use Thom's transversality theorem to describe Gysin maps compatibly with Jakob's definition of K-homology.

In 1988, William Duke showed that CM points of fundamental discriminant $D$ are equidistributed in the complex upper half-plane $\mathcal H$ as $D \to -\infty$. He also showed a similar result for RM curves (a positive discriminant analog of CM points). In this paper, we investigate analogous problems concerning the distribution of CM points and RM curves along fixed geodesics in $\mathcal H$, and around fixed points in $\mathcal H$. Specifically, we show that CM points and RM curves are equidistributed along every fixed rational geodesic in $\mathcal H$, and around every fixed CM point in $\mathcal H$. To prove these results, we solve the aggregate Linnik problem for arbitrary binary quadratic forms.

This paper studies the stability of low-rank implicit regularization in deep matrix factorization, a tractable model for understanding how gradient-based training can favor low-complexity structure. We first revisit the noiseless setting and derive sufficient spectral conditions under which gradient descent exhibits a nonempty low-rank interval. These conditions clarify how the target spectrum, initialization, and step size jointly determine when a low-rank phase is observable along the optimization trajectory. We then analyze the perturbed problem, where the target matrix is subject to an additive perturbation. By studying the perturbed gradient descent dynamics at the eigenvalue level, we prove convergence guarantees and quantify how the perturbation size affects iteration complexity and eigenvalue recovery. Finally, we establish stability of the low-rank phase under perturbation: the effective rank of the iterates remains close to that of the rank-L approximation of the noiseless target over a perturbed low-rank interval, with explicit dependence on the perturbation size. Numerical illustrations support the theoretical predictions and illustrate the role of spectral structure in determining when this stability is observed.

Analyzing in detail the analytic continuation of the Riemann zeta function we are able to generate several new identities which may be useful for application in physics and mathematics.

We introduce the twisted homology jump loci of a space $X$: the jump loci for homology with coefficients in rank-one local systems, twisted by a fixed finite-dimensional representation $\sigma$ of $\pi_1(X)$. These loci refine the classical characteristic varieties, and their defining equations in degree one are the twisted Alexander polynomials of knot theory. Our main theorem is that their tropicalizations bound from above the Bieri–Neumann–Strebel–Renz (BNSR) $\Sigma$-invariants of $X$. Twisting gains real ground. The resulting bound is strictly stronger than the untwisted tropical bounds obtained from the usual characteristic varieties: for a one-relator group whose $\Sigma^1$ was computed by Brown, the untwisted bound excludes only two directions in $H^1(G;\mathbb{R})$, whereas the twisted bound determines $\Sigma^1(G)$ exactly. For a compact orientable $3$-manifold $M$ with toroidal or empty boundary, the twisted bound is sharp: the union of the twisted tropical varieties over all finite-image integral representations of $\pi_1(M)$ computes $\Sigma^1(\pi_1(M))$, and hence recovers the fibered faces of the Thurston norm ball. Sharpness genuinely requires twisting: a non-fibered class enters the tropical variety through the vanishing of a twisted Alexander polynomial along it, and the untwisted polynomial need not vanish. For a compact Kähler manifold $X$, we prove that the first twisted Alexander polynomial $\Delta^{\sigma}(X)$ is either $0$ or $1$, for every representation $\sigma$ over every field, and that $\Sigma^1(\pi_1(X))$ is controlled by the hyperbolic orbifold fibrations of $X$ for every $\sigma$. The obstruction to Kählerianity that comes out of this is strictly finer than its untwisted counterpart: we exhibit groups $G$ with $\Delta(G)=1$ but $\Delta^{\sigma}(G)\ne 1$, which the twisted test excludes from being Kähler and the classical one does not.

Using the results of Nikulin and Vinberg on the groups of isometries generated by reflections, we construct a subvariety called the Nikulin-Vinberg locus in the moduli space of polarized hyperkahler manifolds. It is obtained as a finite union of components of higher Noether-Lefschetz loci which parameterize manifolds with certain special Neron-Severi lattices. The Nikulin-Vinberg locus is the closure of the set of hyperkahler manifolds with Picard number $\geq 3$ which have finite groups of birational automorphisms. Using this construction and a refinement of an argument by Oguiso, we show that any non-trivial family of projective deformations of a hyperkahler manifold with $b_2(M)\geq 6$ has a dense set of fibers which have an infinite group of birational automorphisms.

We prove two conjectures on the spaces generated by multiple $q$-zeta values. More precisely, we show that the spaces $Z_q^{\mathrm{o}}$ and $Z_{q,1}^{\mathrm{o}}$ already generate the larger spaces $Z_q$ and $Z_{q,1}$, respectively. Our result is stronger than the equality of $\mathbb{Q}$-vector spaces: for every generator in the larger spaces, we construct an explicit expression with integer coefficients in terms of the smaller generating families. We first establish these formulas at the finite level, where suitable finite $q$-analogues admit recursive descriptions through generating series, and then pass to the infinite limit.

We prove that every concatenation of $10$ or more binary squares contains an overlap. The bound $10$ is best possible. In contrast, over a ternary alphabet, there are infinitely long overlap-free words that consist of a concatenation of squares.

Positive systems describing networks with inherently non-negative states and inputs arise naturally in routing, logistics, and compartmental modelling. We consider problems modelled as positive linear systems in incidence form with linear cost. The addition of capacity constraints on states (storage) and inputs (flows between nodes) significantly increases the problem complexity. Leveraging the analytic structure of the unconstrained problem, an explicit suboptimal admissible controller is constructed. This yields graph-computable performance bounds and a minimum stabilising horizon length for a model predictive controller without terminal conditions. A convex program enables efficient computation of the optimal bound and horizon. These results highlight how system structure enables explicit MPC guarantees that are typically not available.

We study Einstein metrics on complex projective spaces that are invariant under cohomogeneity one actions of compact connected Lie groups, under the assumption that the singular orbits are totally geodesic. These actions were classified by Takagi into five models. For each of them, we write the Einstein equation for diagonal invariant metrics and determine the corresponding smoothness conditions at the singular orbits. Our main result is the nonexistence of smooth globally defined diagonal invariant Einstein metrics with totally geodesic singular orbits in four of the five models and a necessary condition for global existence in the remaining one.

For graphs $G, F_1$ and $F_2$, we say that $G$ is $(F_1,F_2)$-free if neither $F_1$ nor $F_2$ is an induced subgraph of $G$. We say that $G$ is $k$-vertex-critical if the chromatic number of $G$ is $k$, but every proper induced subgraph of $G$ has chromatic number at most $k-1$. The $\textit{chair}$ graph is a $5$-vertex graph obtained by adding a pendant vertex to one of the two central vertices of a path on $4$ vertices. The $\textit{cricket}$ graph is a $5$-vertex graph obtained by adding two pendant vertices to a common vertex of a triangle. The path on $5$ vertices is denoted by $P_5$. We prove that for every $k \geq 1$, there are only finitely many $(P_5,\text{chair})$-free $k$-vertex-critical graphs. We also prove that the same conclusion holds if $\text{chair}$ is replaced by $\text{cricket}$. We further characterize all $5$-vertex-critical $(P_5,\text{chair})$-free graphs, all $5$-vertex-critical $(P_5,\text{cricket})$-free graphs and all $6$-vertex-critical $(P_5,\text{cricket})$-free graphs. Our proofs rely on bounding the size of antichains and developing Ramsey-theoretic ideas. For any fixed integer $k \geq 1$, our results imply the existence of a polynomial time algorithm to decide whether a $(P_5,\text{chair})$-free (or $(P_5,\text{cricket})$-free) graph is $(k-1)$-colourable such that this algorithm can also present a negative constant-size certificate in case the graph is not $(k-1)$-colourable.

We study vector-valued incidence maps obtained from ordinary graph incidence maps by linear observation of the free vertex space. Let $\mathbb{F}$ be a field, $D = (X, E, s, t)$ a finite directed multigraph, $U$ an $\mathbb{F}$-vector space, and $\phi : X \to U$ a vertex labeling with $\mathbb{F}$-linear extension $\hat{\phi} : \mathbb{F}^X \to U$. The vector-valued incidence map $\partial_\phi : \mathbb{F}^E \to U$, $\partial_\phi(\mathbf{1}_e) = \phi(t(e)) - \phi(s(e))$, factors as $\partial_\phi = \hat{\phi} \circ B_D$, where $B_D$ is the classical incidence map of $D$. We prove the formula $\dim_{\mathbb{F}} \mathrm{Ker}(\partial_\phi) = |E| - |X| + c(D) + \delta_\phi,$ where $c(D)$ is the number of weakly connected components of $D$ and $\delta_\phi := \dim_{\mathbb{F}}(\mathrm{Im}(B_D) \cap \mathrm{Ker}(\hat{\phi}))$ is the defect invariant. We apply this framework to directed tensor-labeled hypergraphs $\mathcal{H} = (Q_0, Q_1, \beta)$, in which each hyperedge carries a pair of boundary tensors $(A_e, B_e)$ in the tensor algebra $T(\mathbb{F}^{Q_0})$, and prove that $\delta(\mathcal{H}) = 0$ over any field for each of the six standard constructions, including symmetric encodings that degenerate in positive characteristic. Over $\mathbb{F} = \mathbb{R}$, the edge Gram operator $\mathcal{L} = \partial_\beta^* \partial_\beta$ has rank $|V_{\mathrm{macro}}| - c_{\mathrm{macro}} - \delta(\mathcal{H})$, and its degree-truncated operators form a Loewner-monotone filtration whose rank increments equal the decrements of the defect filtration. We further realize the cycle space of every oriented hypergraph (in the sense of Reff–Rusnak) as $\mathrm{Ker}(\partial_\beta)$ within this framework, and exhibit a four-edge inclusion–exclusion example with $\delta(\mathcal{H}) = 1$.

Motivated by the Engel and Pierce expansions, we introduce a signed Engel expansion. We expand each $x\in(0,1)\setminus\mathbb{Q}$ uniquely as $$x=\frac{\epsilon_{1}(x)}{d_{1}(x)}+\frac{\epsilon_{2}(x)}{d_{1}(x)d_{2}(x)}+\cdots+\frac{\epsilon_{n}(x)}{d_{1}(x)d_{2}(x)\cdots d_{n}(x)}+\cdots,$$ where $\epsilon_{1}(x)\coloneqq1$ and $\epsilon_{n}(x)\in\left\{1,-1\right\}$ for $n\geq2$. The digit sequence $\left\{d_{n}(x)\right\}_{n\geq1}$ satisfying $d_{n+1}(x)\geq d_{n}(x)+2$ when $\epsilon_{n+1}(x)=-\epsilon_{n}(x)$ forms a non-decreasing sequence of even positive integers tending to infinity. On the one hand, we obtain the law of large numbers, the central limit theorem and the law of the iterated logarithm regarding $d_{n}(x)$ and $\Delta_{n}(x)\coloneqq d_{n}(x)-d_{n-1}(x)\ (n\geq2)\ (\Delta_{1}(x)\coloneqq d_{1}(x))$. On the other hand, we prove a Borel–Bernstein theorem on the zero-one law on the Lebesgue measure of the set $$\left\{x\in(0,1)\colon R_{n}(x)\geq\phi(n)\ \textnormal{ for infinity many } n\right\},$$ where $R_{n}(x)\coloneqq\frac{d_{n}(x)}{d_{n-1}(x)}\ (n\geq2)\ (R_{1}(x)\coloneqq d_{1}(x))$ and $\phi$ is an arbitrary positive function defined on the set of positive integers.

This paper introduces the Myerson interaction index (MII), an extension of the Shapley interaction index to cooperative games with communication structures restricted by graphs. We establish a formal framework for interaction indices on graphs and provide an axiomatic characterization of the MII based on component efficiency, fairness, and veto partnership consistency. Furthermore, we analyse network-induced interaction to distinguish between effects arising from the graph topology and those stemming from the game's intrinsic synergies. This framework offers a robust tool for analysing coalitional behaviour in structured environments, with potential applications in social network analysis and explainable artificial intelligence.

In this note, we formulate the notions of the direct and inverse problems and contact points for the Feynman diagrams. For the electron-photon interaction case, the solutions of these direct and inverse problems are presented. The interrelations between Feynman diagrams and the classical colored graphs are discussed.

Let \(G\) and \(H\) be graphs, and let \(G\times H\) denote their direct product. For a graph \(G\), let \(\operatorname{im}(G)\) be the largest integer \(t\) such that \(G\) contains a \(K_t\)-immersion. Collins, Heenehan, and McDonald conjectured that if \(\operatorname{im}(G)=t\) and \(\operatorname{im}(H)=r\), then \[\operatorname{im}(G\times H)\ge (t-1)(r-1)+1.\] We disprove this conjecture by constructing an infinite family of connected bipartite counterexamples.

We give examples of proper minimal immersions in Euclidean space with very rapid area growth. The first is a proper embedding into $\bf{R}^4$ that yields a stable minimal surface, while the second is a proper immersion into $\bf{R}^3$. These results are motivated by [CM1] that shows that proper minimal submanifolds confined in space satisfy strong structural constraints.

We consider interpolation-based derivative-free optimization in settings where only some derivatives are available. Such situations arise in scientific computing applications involving simulations, adjoint-enabled components, legacy software, or partially differentiable models. We introduce a Birkhoff interpolation framework that permits arbitrary patterns of derivative availability and enables the construction of local polynomial models using mixtures of function values and partial derivative information. In contrast to Hermite interpolation approaches, the proposed framework does not require all available derivatives to be queried at every interpolation point. We develop conditions under which the resulting interpolation systems are poised and establish corresponding model-accuracy bounds for fully quadratic interpolation models. We develop a trust-region framework that maintains poised interpolation sets while selectively incorporating derivative information. The method generalizes an established class of interpolation-based derivative-free optimization algorithms and naturally bridges derivative-free and derivative-based settings. We evaluate our approach on a collection of CUTEst test problems with synthetically generated derivative-availability patterns, as well as a stellarator design problem in which only some derivatives are available.

For any positive integer $n$, the author previously constructed several minimal simplicial $n$-complexes which necessarily contain a non-splittable two-component link, consisting of an $(n-1)$-sphere and an $n$-sphere, in any embedding into $\mathbb{R}^{2n}$. In this paper, we present an additional simplicial $n$-complex with the same property.

In chapter 9 of his book "The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal", Woodin shows how to force the Strong Chang Conjecture over models of determinacy using $\mathbb{P}_{\mathrm{max}}$. We show here how a modification of the proof implies that such extensions actually verify the stronger cofinal version of the conjecture. This stronger version has important consequences on the semi-properness of small forcing, allowing us to prove the consistency of the theory "ZFC + Namba forcing is semiproper + $\Theta^{UB}=\omega_3$". We then use the constructions of this proof to also show that Woodin $(\ast)_{UB}$ axiom implies the conjecture.

We study inference in stochastic block models (SBMs) through the lens of optimal transport (OT). We first establish that maximum likelihood variational inference (MLVI) can be interpreted as a semi-relaxed Gromov-Wasserstein (srGW) projection with entropic regularization. While this formulation yields accurate clustering, the entropic regularization prevents transport plans to be sparse, hindering intrinsic model selection. Consequently, we investigate unregularized srGW estimators, and prove that they consistently recover both the SBM connectivity matrix and latent cluster assignments in the asymptotic regime. However, this asymptotic property does not translate into reliable model selection in finite samples, and calls for additional mechanisms to promote sparsity in the inferred cluster proportions. We empirically show that such a regularized formulation yields estimators that simultaneously recover model parameters and select the number of clusters in a single optimization problem, thereby avoiding costly grid search or heuristic model selection procedures.

We reduce the eight-dimensional weak form of the bilinear Boltzmann collision operator to a five-dimensional kinematic core by rigidly rotating the laboratory frame to align with the colliding pair and integrating over the $\mathrm{SO}(3)$ rotation group. This reduction yields an exact Wigner–Eckart factorization within a spectral Galerkin framework of associated Laguerre polynomials and spherical harmonics. The decomposition decouples the angular geometry from the scattering physics. The former, represented by Clebsch–Gordan coefficients, is evaluated exactly, while the latter is evaluated to machine precision by a spectrally convergent singular quadrature strategy. By explicitly zeroing specific entries, the macroscopic collision invariants are embedded without approximation. Cache-optimized contractions deliver up to a 37-fold single-core speedup and a 1000-fold memory reduction over standard dense Cartesian formulations. The approach is validated against analytical solutions for Maxwell molecules and infinite-order Chapman–Enskog viscosity coefficients for hard spheres.

We introduce and study dual Chow functions associated to kernels in incidence algebras of weakly ranked posets. Given a kernel, its dual Chow function is defined as the Chow function associated to the sign-twisted reverse kernel. For kernels satisfying a natural skew-symmetry condition, such as the Eulerian kernel of an Eulerian poset or the kernel given by R-polynomials on Bruhat intervals, this construction recovers the ordinary Chow function. In contrast, when this skew-symmetry fails, the dual Chow function gives a genuinely different invariant. The main example considered in this paper is the dual Chow function associated to the characteristic function. We develop the basic theory of these dual Chow functions, with particular emphasis on posets arising from matroids. We prove chain formulas, unimodality and gamma-positivity results, formulas under standard poset operations, and deletion formulas for matroids. Along the way, we also obtain a general deletion formula for the ab-index of matroids, which leads to new formulas for extended ab-indices and, in turn, specializes to several deletion formulas appearing in the literature.

Under the duality between the two-dimensional XY model and an integer-valued height function, the BKT transition is expected to correspond to the disappearance of a corner in the surface tension; in the delocalised phase, its zero-slope curvature should determine the Gaussian free field prefactor. We prove that the XY mass equals the right derivative at zero slope of the dual height function's free energy, with a uniform quadratic error bound near the origin. Thus the massive phase is exactly the corner regime, while in the BKT phase the surface tension is quadratically bounded at zero slope. The proof combines Kadanoff–Ceva duality, Ginibre's inequality, and a pushing lemma (an estimate of Russo–Seymour–Welsh type) for the cable height function.