Curved Space Research Articles

Pluricanonical cycles and tropical covers

We extract a system of numerical invariants from logarithmic intersection theory on pluricanonical double ramification cycles, and show that these invariants exhibit a number of properties that are enjoyed by double Hurwitz numbers. Among their properties are (i) the numbers can be efficiently calculated by counts of tropical curves with a modified balancing condition, (ii) they are piecewise polynomial in the entries of the ramification vector, and (iii) they are matrix elements of operators on the Fock space. The numbers are extracted from the logarithmic double ramification cycle, which is a lift of the standard double ramification cycle to a blowup of the moduli space of curves. The blowup is determined by tropical geometry. We show that the traditional double Hurwitz numbers are intersections of the refined cycle with the cohomology class of a piecewise polynomial function on the tropical moduli space of curves. This perspective then admits a natural, combinatorially motivated, generalization to the pluricanonical setting. Tropical correspondence results for the new invariants lead immediately to the structural results for these numbers.

MADE: Multicurvature Adaptive Embedding for Temporal Knowledge Graph Completion.

Temporal knowledge graphs (TKGs) are receiving increased attention due to their time-dependent properties and the evolving nature of knowledge over time. TKGs typically contain complex geometric structures, such as hierarchical, ring, and chain structures, which can often be mixed together. However, embedding TKGs into Euclidean space, as is typically done with TKG completion (TKGC) models, presents a challenge when dealing with high-dimensional nonlinear data and complex geometric structures. To address this issue, we propose a novel TKGC model called multicurvature adaptive embedding (MADE). MADE models TKGs in multicurvature spaces, including flat Euclidean space (zero curvature), hyperbolic space (negative curvature), and hyperspherical space (positive curvature), to handle multiple geometric structures. We assign different weights to different curvature spaces in a data-driven manner to strengthen the ideal curvature spaces for modeling and weaken the inappropriate ones. Additionally, we introduce the quadruplet distributor (QD) to assist the information interaction in each geometric space. Ultimately, we develop an innovative temporal regularization to enhance the smoothness of timestamp embeddings by strengthening the correlation of neighboring timestamps. Experimental results show that MADE outperforms the existing state-of-the-art TKGC models.

On a geometric version of the Grothendieck conjecture for configuration spaces of hyperbolic curves

On a geometric version of the Grothendieck conjecture for configuration spaces of hyperbolic curves

Control point modifications that preserve the Pythagorean–hodograph nature of planar quintic curves

Although planar Pythagorean–hodograph (PH) curves are compatible with the standard Bernstein–Bézier representations, freely modifying the control points will compromise their PH nature. The present study focuses on identifying control point displacements that ensure a given planar PH curve remains a PH curve. In particular, for planar quintic PH curves r(t), t∈[0,1] it is shown that finitely-many simultaneous displacements of two control points yield modified quintic PH curves, identified as the solutions of quadratic and cubic equations. As a more practical approach, modification of PH quintics in canonical form with r(0)=0 and r(1)=1 by the displacement of a single interior control point is considered, with the remaining interior control points being used to minimize a measure of deviation from the original PH quintic. As illustrated by several examples, this approach provides an efficient and intuitive means of effecting reasonable shape modifications within the space of planar quintic PH curves.

On ELSV-type formulae and relations between Ω-integrals via deformations of spectral curves

The general relation between Chekhov–Eynard–Orantin topological recursion and the intersection theory on the moduli space of curves, the deformation techniques in topological recursion, and the polynomiality properties with respect to deformation parameters can be combined to derive vanishing relations involving intersection indices of tautological classes. We apply this strategy to three different families of spectral curves and show they give vanishing relations for integrals involving Ω-classes. The first class of vanishing relations are genus-independent and generalises the vanishings found by Johnson–Pandharipande–Tseng [34] and by the authors jointly with Do and Moskovsky [7]. The two other classes of vanishing relations are of a different nature and depend on the genus.

The strong maximal rank conjecture and moduli spaces of curves

The strong maximal rank conjecture and moduli spaces of curves

Metallic Structures on Product Manifolds and Ricci-Chen Inequalities

In this study, we discuss metallic structures on product manifolds and derive the Chen-Ricci inequalities for remarkable submanifolds determined by the behaviour of their tangent bundles with regard to the action of the metallic structure in a locally decomposable metallic Riemannian manifold whose components are spaces of constant curvature. Moreover, the equality cases are considered in order to characterize these submanifolds.

Role of Multistate Models to Predict Patency, Limb Salvage, and Survival: New Concepts to Analyse Data in Peripheral Arterial Disease

In peripheral arterial disease, patency, limb salvage, and survival rates are mostly reported using Kaplan-Meier analyses. When comparing different revascularisation techniques, these methods have limitations in analysing complex patient flows over time. This study aimed to present, illustrate, and discuss new concepts based on multistate models of analysing outcome parameters in peripheral arterial disease. Previously published data from a single centre, randomised controlled trial (RCT) with 218 cases that underwent either vein bypass surgery (bypass group, n = 109) or nitinol stent angioplasty (stent group, n = 109) of long femoropopliteal lesions were re-analysed using non-homogeneous Markov models. A step by step description of the concepts of states, state space, definitions, and illustration of transition probability curves as well as the benefits of multistate models is given. The RCT was registered at ISRCTN.com (ISRCTN18315574). Transition probability curves over time showed similar patterns in the bypass and stent groups. Significant differences in the transition probabilities were found for transitions from primary patency as well as secondary patency to end of patency. The transition probability for patients with preserved primary patency at 24 months who moved to end of patency at 48 months was 19.9% in the stent group vs. 6.4% in the bypass group (p < .001). The proposed method can answer important questions, such as: Did patients after femoropopliteal stenting with preserved primary patency at two years lose their patency more quickly within the following years compared with bypass surgery? and Did stent patients after a re-intervention to maintain patency at one year lose their patency more quickly compared with bypass surgery within the following years? Completely new research questions can now be raised and answered to optimise treatment and follow up strategies; this might lead to better identification of subgroups at higher risk of clinical deterioration following revascularisation procedures.

Lipschitz regularity of sub-elliptic harmonic maps into CAT(0) space

Abstract We prove the local Lipschitz continuity of sub-elliptic harmonic maps between certain singular spaces, more specifically from the 𝑛-dimensional Heisenberg group into CAT ⁡ ( 0 ) \operatorname{CAT}(0) spaces. Our main theorem establishes that these maps have the desired Lipschitz regularity, extending the Hölder regularity in this setting proven in [Y. Gui, J. Jost and X. Li-Jost, Subelliptic harmonic maps with values in metric spaces of nonpositive curvature, Commun. Math. Res. 38 (2022), 4, 516–534] and obtaining same regularity as in [H.-C. Zhang and X.-P. Zhu, Lipschitz continuity of harmonic maps between Alexandrov spaces, Invent. Math. 211 (2018), 3, 863–934] for certain sub-Riemannian geometries; see also [N. Gigli, On the regularity of harmonic maps from RCD ⁢ ( K , N ) \mathrm{RCD}(K,N) to CAT ⁢ ( 0 ) \mathrm{CAT}(0) spaces and related results, preprint (2022), https://arxiv.org/abs/2204.04317; and A. Mondino and D. Semola, Lipschitz continuity and Bochner–Eells–Sampson inequality for harmonic maps from RCD ⁡ ( k , n ) \operatorname{RCD}(k,n) spaces to CAT ⁡ ( 0 ) \operatorname{CAT}(0) spaces, preprint (2022), https://arxiv.org/abs/2202.01590] for the generalisation to RCD spaces. The present result paves the way for a general regularity theory of sub-elliptic harmonic maps, providing a versatile approach applicable beyond the Heisenberg group.

Divisors and curves on logarithmic mapping spaces

We determine the rational class and Picard groups of the moduli space of stable logarithmic maps in genus zero, with target projective space relative a hyperplane. For the class group we exhibit an explicit basis consisting of boundary divisors. For the Picard group we exhibit a spanning set indexed by piecewise-linear functions on the tropicalisation. In both cases a complete set of boundary relations is obtained by pulling back the WDVV relations from the space of stable curves. Our proofs hinge on a controlled technique for manufacturing test curves in logarithmic mapping spaces, opening up the topology of these spaces to further study.

GIT quotient of holomorphic foliations on ℂℙ2 of degree 2 and quartic plane curves

Abstract We study the quotient variety of the space of foliations on ℂ ⁢ ℙ 2 {\mathbb{CP}^{2}} of degree 2 up to change of coordinates. We find the intersection Betti numbers of this variety. As a corollary, we have that these intersection Betti numbers coincide with the intersection Betti numbers of the quotient variety of quartic plane curves. Finally, we give an explicit isomorphism between the space of foliations of degree 2 with different singular points, without invariant lines and the space of smooth quartic plane curves.

Investigation of the complex dynamical structure of bifurcation and dark soliton solutions to fractional generalized double [formula omitted]-Gordon equation

In the current research work, the classical wave equation is combined with a nonlinear sinh source term in the sinh-Gordon equation. It has been used in several scientific domains such as differential geometry theory, integrable quantum field theory, kink dynamics, and statistical mechanics. It makes more comprehensible the dynamics of strings and multi-strings in the constant curvature space. The current study has three main objectives. Examine the governing model to get novel solutions by employing the modified Kudryashov technique. Then compare it with the numerical technique of modified variational iteration method (MVIM) to calculate the error terms. Furthermore, employing bifurcation theory to produce a dynamical system. Additionally, use the dynamical system’s sensitivity analysis to investigate the model’s sensitivity. At last, for the validation of acquired results, the cryptography technique of novel image encryption and decryption is used. The research is greatly enhanced by the presentation of thorough 2D and 3D phase portraits. The field of mathematics and other sciences will benefit from these discoveries.

Totally Geodesic Subvarieties of the Moduli Space of Curves and Linear Systems

Abstract We construct a linear system on a general curve in a totally geodesic subvariety of the moduli space of curves. As a consequence, one obtains rank bounds for totally geodesic subvarieties of dimension at least two. Furthermore, this leads to a classification of totally geodesic subvarieties of dimension at least two in strata with at most two zeros.

Projective structures and Hodge theory

AbstractEvery compact Riemann surface X admits a natural projective structure $$p_u$$ p u as a consequence of the uniformization theorem. In this work we describe the construction of another natural projective structure on X, namely the Hodge projective structure $$p_h$$ p h , related to the second fundamental form of the period map. We then describe how projective structures correspond to (1, 1)-differential forms on the moduli space of projective curves and, from this correspondence, we deduce that $$p_u$$ p u and $$p_h$$ p h are not the same structure.

Research on time-energy optimal trajectory planning of articulated heavy-duty robot

Articulated heavy-duty robots are more and more widely used in the construction environment. Aiming at the problems of low working efficiency and high energy consumption of this kind of robot, a multi-objective adaptive trajectory planning method combining path optimization and trajectory optimization is proposed to improve the working efficiency of the robot and reduce energy consumption. Firstly, based on kinematics and dynamic analysis considering friction, the energy consumption model of robot trajectory is constructed. Then, according to the posture of the key points in the known workspace, the optimal path points in the joint space are obtained by a path point solution method considering the joint load characteristics. On this basis, a multi-objective model considering running time and energy consumption is established to plan the interpolation trajectory of the joint space of the quintic B-spline curve. Finally, constrained by the velocity, acceleration and torque of the joint of the manipulator, the optimal solution is obtained by using the elite non-dominated sorting genetic algorithm (NSGA-II), and the optimal weight factor is selected by a multi-objective adaptive optimization method to obtain the optimal position-time series. The experimental results show that compared with the quintic polynomial trajectory optimization method, the energy consumption of the proposed method is reduced by 10.90% under the same working efficiency. The research results of this paper provide a new optimization idea for other target trajectory planning.

Deep hyperbolic convolutional model for knowledge graph embedding

Recent advancements in knowledge graph embedding have enabled the representation of entities and relations in continuous vector spaces. Performing link prediction on incomplete knowledge graphs using these embeddings has emerged as a challenging task, drawing considerable research interest. Knowledge graphs in the real world typically exhibit a natural hierarchical structure. Hyperbolic embedding methods have shown promising results in modeling hierarchical data, especially in low-dimensional representations. However, existing hyperbolic models for knowledge graph embedding are generally shallow, resulting in limited expressiveness. Additionally, some deep hyperbolic models exist but are often constrained to fixed curvature spaces, which may not optimally capture varying hierarchical structures. In this work, we propose DeER, a novel multi-layer hyperbolic model that leverages a trainable curvature space, marking an advancement in the depth and adaptability of hyperbolic modeling for knowledge graph embeddings. We specifically design a hyperbolic convolutional neural network to learn the heterogeneous interactions between entities and relations within this adaptable hyperbolic space. Additionally, we introduce a hyperbolic feedforward neural network to transform hyperbolic features across multiple layers, enhancing the model’s ability to capture complex hierarchical relationships. Experimental results on three benchmark datasets demonstrate that DeER significantly enhances expressiveness and hierarchical modeling performance when compared to both shallow and other deep baseline approaches.

Counting meromorphic differentials on CP1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}\\mathbb {P}}^1$$\\end{document}

We give explicit formulas for the number of meromorphic differentials on CP1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{C}\\mathbb{P}^1$$\\end{document} with two zeros and any number of residueless poles and for the number of meromorphic differentials on CP1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{C}\\mathbb{P}^1$$\\end{document} with one zero, two poles with unconstrained residue and any number of residueless poles, in terms of the orders of their zeros and poles. These are the only two finite families of differentials on CP1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{C}\\mathbb{P}^1$$\\end{document} with vanishing residue conditions at a subset of poles, up to the action of PGL(2,C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PGL}(2,\\mathbb {C})$$\\end{document}. The first family of numbers is related to triple Hurwitz numbers by simple integration and we show its connection with the representation theory of SL2(C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{SL}_2(\\mathbb {C})$$\\end{document} and the equations of the dispersionless KP hierarchy. The second family has a very simple generating series, and we recover it through surprisingly involved computations using intersection theory of moduli spaces of curves and differentials.

Exponential Decay Estimates and Smoothness of the Moduli Space of Pseudoholomorphic Curves

The compacified moduli space of bordered stable maps carries a Kuranishi structure with boundary. Smoothness of Kuranishi structure along the boundary requires smoothness of coordinate changes along the boundary. The proof of smoothness is written by the authors in [AMS/IP Stud. Adv. Math., 46.1, 2009, xii+396 pp.], [AMS/IP Stud. Adv. math., 46.2, 2009, pp. i–xii and 397–805], and [Surv. Differ. Geom., vol. 22, pp. 133–190, 2018] based on some uniform exponential decay estimates of the stable maps with respect to a parameter T T , the length of the gluing cylindrical neck, near the boundary of the moduli space. In this paper we establish this exponential decay estimates in a precise manner by carefully examining the dependence on the parameter T T of the gluing construction in the setting of bordered Riemann surfaces with boundary punctures. We also show that the smoothness of the collar follows from the aforementioned exponential estimates by taking s = 1 / T s = 1/T as the radial coordinate of the collar.

Wall crossing for K‐moduli spaces of plane curves

AbstractWe construct proper good moduli spaces parametrizing K‐polystable ‐Gorenstein smoothable log Fano pairs , where is a Fano variety and is a rational multiple of the anticanonical divisor. We then establish a wall‐crossing framework of these K‐moduli spaces as varies. The main application in this paper is the case of plane curves of degree as boundary divisors of . In this case, we show that when the coefficient is small, the K‐moduli space of these pairs is isomorphic to the GIT moduli space. We then show that the first wall crossing of these K‐moduli spaces are weighted blow‐ups of Kirwan type. We also describe all wall crossings for degree 4,5,6 and relate the final K‐moduli spaces to Hacking's compactification and the moduli of K3 surfaces.

Inverse localization and global approximation for some Schrödinger operators on hyperbolic spaces

We consider the question of whether the high-energy eigenfunctions of certain Schrödinger operators on the d-dimensional hyperbolic space of constant curvature −κ2 are flexible enough to approximate an arbitrary solution of the Helmholtz equation Δh + h = 0 on Rd, over the natural length scale O(λ−1/2) determined by the eigenvalue λ ≫ 1. This problem is motivated by the fact that, by the asymptotics of the local Weyl law, approximate Laplace eigenfunctions do have this approximation property on any compact Riemannian manifold. In this paper we are specifically interested in the Coulomb and harmonic oscillator operators on the hyperbolic spaces Hd(κ). As the dimension of the space of bound states of these operators tends to infinity as κ ↘ 0, one can hope to approximate solutions to the Helmholtz equation by eigenfunctions for some κ &amp;gt; 0 that is not fixed a priori. Our main result shows that this is indeed the case, under suitable hypotheses. We also prove a global approximation theorem with decay for the Helmholtz equation on manifolds that are isometric to the hyperbolic space outside a compact set, and consider an application to the study of the heat equation on Hd(κ). Although global approximation and inverse approximation results are heuristically related in that both theorems explore flexibility properties of solutions to elliptic equations on hyperbolic spaces, we will see that the underlying ideas behind these theorems are very different.