arXiv subjects

Shing-Tung Yau

Publications and source records attributed to Shing-Tung Yau.

At least 73 records · Page 4Linked to original sources

arXiv2025

Computing singular simplicial homologies of digraphs and quivers

The dynamics of large complex systems are predominately modeled through pairwise interactions, the principle underlying structure being a network of the form of a digraph or quiver. Significant success has been obtained in applying the homology of the directed flag complex to study digraphs arising as networks within numerous scientific disciplines. This homology of directed cliques enjoys relative ease of computation when compared to other digraph homologies, making it preferable for use in applications concerning large networks. By extending the ideas of singular simplicial homology to quivers in categories of different morphism types, several new singular simplicial homology theories have recently been constructed. Computationally efficient homologies for quivers have in general not previously been seriously considered. In this paper we develop further the homotopy theory of quivers necessary to derive functors that realise isomorphisms between the singular simplicial quiver homologies and the homologies of certain spaces. The simplicial chains of these spaces arise in a conveniently compact form that is at least as convenient as the directed flag complex for computations. Moreover, our constructions are natural with respect to the isomorphisms on homology making them suitable for applications in conjunction with persistent homology for practical use. In particular, for each of the singular simplicial homologies considered, we provide efficient algorithms for the computation of their persistent homology.

math.AT↗

arXiv2025

Ollivier Ricci-flow on weighted graphs

We study the existence of solutions of Ricci flow equations of Ollivier-Lin-Lu-Yau curvature defined on weighted graphs. Our work is motivated by\cite{NLLG} in which the discrete time Ricci flow algorithm has been applied successfully as a discrete geometric approach in detecting complex networks. Our main result is the existence and uniqueness theorem for solutions to a continuous time normalized Ricci flow. We also display possible solutions to the Ricci flow on path graph and prove the Ricci flow on finite star graph with at least three leaves converges to constant-weighted star.

math.DG↗

arXiv2025

Yau-YauAL: A computer tool for solving nonlinear filtering problems

The Yau-Yau nonlinear filter has increasingly emerged as a powerful tool to study stochastic complex systems. To leverage it to a wider spectrum of application scenarios, we pack the Yau-Yau filtering ALgorithms (YauYauAL) into a package of computer software. Yau-YauAL was written in R, designed to simplify the implementation of the Yau-Yau filter for solving nonlinear filtering problems. Combining R's accessibility with C++ (via Rcpp) for computational efficiency, YauYauAL provides an intuitive Shiny-based interface that enables real-time parameter adjustment and result visualization. At its core, YauYauAL employs finite difference methods to numerically solve the Kolmogorov forward equation, ensuring a stable and accurate solution even for complex systems. YauYauAL's modular design and open-source framework further encourage customization and community-driven development. YauYauAL aims to bridge the gap between theoretical nonlinear filtering methods and practical applications, without requiring expertise in differential equation solving or programming, fostering its broader impact on various scientific fields, such as signal processing, finance, medicine, and biology among a long list.

stat.AP↗

arXiv2025

Minimal Path and Acyclic Models in the Path Complex

In this paper, firstly, we will study the structure of the path complex $(Ω_*(G;\Z),\partial)$ of a digraph $G$ via the $\Z$-generators of $Ω_*(G,\Z)$ under strongly regular condition, which is called the minimal path in \cite{HY}. In particular, we will study various examples of the minimal $3$-paths. Secondly, we will show that the supporting sub-digraph of minimal path has acyclic path homologies. Thirdly, we will consider the applications of such an acyclic model.

math.AT↗

arXiv2025

The Cellular Homology of Digraphs

In \cite{TY}, we investigate the pair $(P, \Supp(P))$ of minimal path $P$ and its supporting sub-digraph $\Supp(P)$ in the path complex of a digraph $G$ under the strongly regular condition. In this paper, first, we consider the special minimal path $P$ specified by the admissible condition (Definition \ref{admpair}), which means that $(P,\Supp(P))$ admits a singular cubical realization. Based on such a subset, we systematically introduce the definitions of cellular chain complex associated to $G$ and prove the well-definedness. Then we study several properties of such cellular homologies. Finally, we present several intriguing examples as well as some important observations.

math.CO↗

arXiv2025

A new flow solving the LYZ equation in Kähler geometry

We introduced a new flow to the LYZ equation on a compact Kähler manifold. We first show the existence of the longtime solution of the flow. We then show that under the Collins-Jacob-Yau's condition on the subsolution, the longtime solution converges to the solution of the LYZ equation, which was solved by Collins-Jacob-Yau [5] by the continuity method. Moreover, as an application of the flow, we show that on a compact Kähler surface, if there exists a semi-subsolution of the LYZ equation, then our flow converges smoothly to a singular solution to the LYZ equation away from a finite number of curves of negative self-intersection. Such a solution can be viewed as a boundary point of the moduli space of the LYZ solutions for a given Kähler metric.

math.DG↗

arXiv2025

Homotopy Groups and Puppe Sequence of Digraphs

We introduce homotopy groups of digraphs that admit an intuitive description of grid structures, which is a variation of the GLMY homotopy groups introduced by Grigor'yan, Lin, Muranov and Yau in 2014. This direct approach enables a descriptive interpretation of GLMY theory in applications such as network science. Furthermore, we prove that there exists a long exact sequence of homotopy groups of digraphs associated to any based digraph map, that is, there exists a digraph version of the Puppe sequence.

math.AT↗

arXiv2025

Calculating Higher Digraph Homotopy Groups

We give the first tractable and systematic examples of nontrivial higher digraph homotopy groups. To do this we define relative digraph homotopy groups and show these satisfy a long exact sequence analogous to the relative homotopy groups of spaces. We then define digraph suspension and Hurewicz homomorphisms and show they commute with each other. The existence of nontrivial digraph homotopy groups then reduces to the existence of corresponding groups in the degree 1 path homology of digraphs.

math.AT↗

arXiv2025

Geometric Inequality for Axisymmetric Black Holes With Angular Momentum

In an effort to understand the Penrose inequality for black holes with angular momentum, an axisymmetric, vacuum, asymptotically Euclidean initial data set subject to certain quasi-stationary conditions is considered for a case study. A new geometric definition of angular velocity of a rotating black hole is defined in terms of the momentum constraint, without any reference to a stationary Killing vector field. The momentum constraint is then shown to be equivalent to the dynamics of a two-dimensional steady compressible fluid flow governed by a quasi-conformal mapping. In terms of spinors, a generalised first law for rotating black holes (possibly with multi-connected horizon located along the symmetry axis) is then proven and may be regarded as a Penrose-type inequality for black holes with angular momentum.

gr-qc↗

arXiv2024

C-R-T Fractionalization, Fermions, and Mod 8 Periodicity

Charge conjugation (C), mirror reflection (R), time reversal (T), and fermion parity $(-1)^{\rm F}$ are basic discrete spacetime and internal symmetries of the Dirac fermions. In this article, we determine the group, called the C-R-T fractionalization, which is a group extension of $\mathbb{Z}_2^{\rm C}\times\mathbb{Z}_2^{\rm R}\times\mathbb{Z}_2^{\rm T}$ by the fermion parity $\mathbb{Z}_2^{\rm F}$, and its extension class in all spacetime dimensions $d$, for a single-particle fermion theory. For Dirac fermions, with the canonical CRT symmetry $\mathbb{Z}_2^{\rm CRT}$, the C-R-T fractionalization has two possibilities that only depend on spacetime dimensions $d$ modulo 8, which are order-16 nonabelian groups, including the famous Pauli group. For Majorana fermions, we determine the R-T fractionalization in all spacetime dimensions $d=0,1,2,3,4\mod8$, an order-8 abelian or nonabelian group. For Weyl fermions, we determine the C or T fractionalization in all even spacetime dimensions $d$, which is an order-4 abelian group. We only have an order-2 $\mathbb{Z}_2^{\rm F}$ group for Majorana-Weyl fermions. We determine the maximal number of linearly independent Dirac and Majorana mass terms and construct them explicitly. We also discuss how the conventional Dirac and Majorana mass terms break the symmetries C, R, or T. We study the domain wall dimensional reduction of the fermions and their C-R-T fractionalization: from $d$-dim Dirac to $(d-1)$-dim Dirac or Weyl and from $d$-dim Majorana to $(d-1)$-dim Majorana or Majorana-Weyl.

hep-th↗

arXiv2024

RC-positivity, Schwarz's lemma and comparison theorems

It is well-known that the classical Schwarz lemma yields an explicit comparison of two Hermitian metrics with uniform constant negative curvature bounds through holomorphic maps between complex manifolds. In this paper, we establish Schwarz lemmas for holomorphic bundle maps between abstract Hermitian holomorphic vector bundles with various positive curvature bounds. As applications, we prove Schwarz lemmas for holomorphic maps between complex manifolds whose curvature tensors are described by the notion ``RC-positivity''. In particular, new diameter and volume comparison theorems are obtained by using Schwarz lemmas.

math.DG↗

arXiv2024

Primitive path homology

In this paper we introduce a primitive path homology theory on the category of simple digraphs. On the subcategory of asymmetric digraphs, this theory coincides with the path homology theory which was introduced by Grigor'yan, Lin, Muranov, and Yau, but these theories are different in general case. We study properties of the primitive path homology and describe relations between the primitive path homology and the path homology. Let $a,b$ two different vertices of a digraph. Our approach gives a possibility to construct primitive homology theories of paths which have a given tail vertex $a$ or (and) a given head vertex $b$. We study these theories and describe also relationships between them and the path homology theory.

math.AT↗

arXiv2024

Estimating unknown parameters in differential equations with a reinforcement learning based PSO method

Differential equations offer a foundational yet powerful framework for modeling interactions within complex dynamic systems and are widely applied across numerous scientific fields. One common challenge in this area is estimating the unknown parameters of these dynamic relationships. However, traditional numerical optimization methods rely on the selection of initial parameter values, making them prone to local optima. Meanwhile, deep learning and Bayesian methods require training models on specific differential equations, resulting in poor versatility. This paper reformulates the parameter estimation problem of differential equations as an optimization problem by introducing the concept of particles from the particle swarm optimization algorithm. Building on reinforcement learning-based particle swarm optimization (RLLPSO), this paper proposes a novel method, DERLPSO, for estimating unknown parameters of differential equations. We compared its performance on three typical ordinary differential equations with the state-of-the-art methods, including the RLLPSO algorithm, traditional numerical methods, deep learning approaches, and Bayesian methods. The experimental results demonstrate that our DERLPSO consistently outperforms other methods in terms of performance, achieving an average Mean Square Error of 1.13e-05, which reduces the error by approximately 4 orders of magnitude compared to other methods. Apart from ordinary differential equations, our DERLPSO also show great promise for estimating unknown parameters of partial differential equations. The DERLPSO method proposed in this paper has high accuracy, is independent of initial parameter values, and possesses strong versatility and stability. This work provides new insights into unknown parameter estimation for differential equations.

cs.LG↗

arXiv2024

Rigid Schubert classes in partial flag varieties

A Schubert class is called rigid if it can only be represented by Schubert varieties. The rigid Schubert classes have been classified in Grassmannians and orthogonal Grassmannians. In this paper, we study the rigidity problem in partial flag varieties (type A) and orthogonal partial flag varieties (type B and type D). In particular, we give numerical conditions that ensure a Schubert class is rigid.

math.AG↗

arXiv2024

Multi-rigidity of Schubert classes in partial flag varieties

In this paper, we study the multi-rigidity problem in rational homogeneous spaces. A Schubert class is called multi-rigid if every multiple of it can only be represented by a union of Schubert varieties. We prove the multi-rigidity of Schubert classes in rational homogeneous spaces. In particular, we characterize the multi-rigid Schubert classes in partial flag varieties of type A, B and D. Moreover, for a general rational homogeneous space $G/P$, we deduce the rigidity and multi-rigidity from the corresponding generalized Grassmannians (correspond to maximal parabolics). When $G$ is semi-simple, we also deduce the rigidity and multi-rigidity from the simple cases.

math.AG↗

arXiv2024

Futaki Invariants and Reflexive Polygons

Futaki invariants of the classical moduli space of 4d N=1 supersymmetric gauge theories determine whether they have a conformal fixed point in the IR. We systematically compute the Futaki invariants for a large family of 4d N=1 supersymmetric gauge theories coming from D3-branes probing Calabi-Yau 3-fold singularities whose bases are Gorenstein Fano surfaces. In particular, we focus on the toric case where the Fano surfaces are given by the 16 reflexive convex polygons and the moduli spaces are given by the corresponding toric Calabi-Yau 3-folds. We study the distribution of and conjecture new bounds on the Futaki invariants with respect to various topological and geometric quantities. These include the minimum volume of the Sasaki-Einstein base manifolds as well as the Chern and Euler numbers of the toric Fano surfaces. Even though the moduli spaces for the family of theories studied are known to be K-stable, our work sheds new light on how the topological and geometric quantities restrict the Futaki invariants for a plethora of moduli spaces.

hep-th↗

arXiv2024

Strong field behavior of Wang-Yau Quasi-local energy

We look at the strong field behavior of the Wang-Yau quasi-local energy. In particular, we examine the limit of the Wang-Yau quasi-local energy as the defining spacelike $2$-surface $Σ$ approaches an apparent horizon from outside. Assuming that coordinate functions of the isometric embedding are bounded in $W^{2,1}$ and mean curvature vector of the image surface remains spacelike, we find that the limit falls in two exclusive cases: 1) If the horizon cannot be isometrically embedded into $R^3$, the Wang-Yau quasi-local energy blows up as $Σ$ approaches the horizon while the optimal embedding equation is not solvable for $Σ$ near the horizon; 2) If the horizon can be isometrically embedded into $R^3$, the optimal embedding equation is solvable up to the horizon with the unique solution at the horizon corresponding to isometric embedding into $R^3$ and the Wang-Yau quasi-local mass admits a finite limit at the horizon. We discuss the implications of our results in the conclusion section.

gr-qc↗

arXiv2024

Weak Stability Conditions as Limits of Bridgeland Stability Conditions

In this paper, we give a definition of weak stability condition on a triangulated category. The difference between our definition and existing definitions is that we allow objects in the kernel to have non-maximal phases. We then construct four types of weak stability conditions that naturally occur on Weierstrass ellitpic surfaces as limites of Bridgeland stability conditions.

math.AG↗