arXiv subjects
Publications and source records attributed to Siqi He.
This article equips the universal moduli spaces of non-degenerate $\mathbb{Z}/2\mathbb{Z}$ harmonic spinors with the structure of a tame Fr\'echet manifold and proves that the projection map to the parameter space is uniformly Fredholm. This recovers earlier work of Donaldson, Parker, and Takahashi in a unified framework.
We prove a 2-valued analogue of Calabi's intrinsic harmonicity theorem and use it to introduce the Calabi surgery method, a surgery theory for $\mathbb{Z}/2$ harmonic $1$-forms. Once the ambient metric is allowed to vary, the construction of new $\mathbb{Z}/2$ harmonic forms can be reduced to cutting and pasting closed 2-valued 1-forms, matching local harmonic models, and controlling the transitivity of the resulting singular foliation. For these constructions, the Nash--Moser-type analytic deformation problem that arises in singular gluing is replaced by local model matching and a global dynamical condition on the foliation. The resulting procedure gives a flexible way to construct and modify $\mathbb{Z}/2$ harmonic 1-forms under weak regularity assumptions. As applications, we obtain connected-sum and local replacement theorems, blow up isolated ordinary zeros by prescribed Euclidean models, split smooth $\vec{k}$-nondegenerate branching components, and desingularize graphic singular sets in dimensions 3 and 4 with suitable resolution models.
We prove a rigidity result for certain critical Z/2 eigensections of the Laplacian on S^2 associated to a flat real line bundle determined by a branch-point configuration. More precisely, we show that every minimal non-degenerate critical eigensection is deformation rigid: any sufficiently small deformation of the configuration that still admits a critical eigensection must come from an SO(3)-rotation. This generalizes the rigidity phenomenon previously discovered in symmetric examples of Taubes-Wu.
We study metric perturbations and deformation theory for degenerate Z/2-harmonic 1-forms. For a natural class of degenerate examples, we prove that after a suitable perturbation of the ambient Riemannian metric, the form can be deformed to a nearby non-degenerate Z/2-harmonic 1-form. Our argument combines analysis of the leading coefficients in the local expansion under metric perturbations with a quantitative Nash-Moser implicit function theorem.
Since viable $f(R)$ gravity models must reconcile early-universe inflation with late-time acceleration, we specifically study the dynamical behavior of such a theory during the matter-dominated to dark-energy-dominated transition epoch. By using $y_{H}(z)$ versus $z$ and the Hubble parameter, we solved the field equations. After appropriately choosing appropriate parameter values , we plotted a series of images. We mentioned that their current values are similar to latest observations data and $\Lambda$CDM-model values. Furthermore, we plotted the fitting of the distance modulus about this model using SN Ia observation data. Therefore we find that the $f(R)$ gravity model is consistent with the SN Ia data, meanwhile, explains the late-stage acceleration of the Universe. Finally, we used various diagnostic tools including $( r, s)$, $( r, q)$, $w_{D}-w'_{D}$ plane, growth rate analysis, statefinder hierarchy and $Om(z)$-diagnostic to evaluate the observational viability of our model, we perform a systematic comparison with the standard $\Lambda$CDM. We found that evolutionary images can be clearly distinguished this model from the $\Lambda$CDM.
Let $F$ be a non-Archimedean valued field, $\Sigma$ a closed Riemann surface of genus at least two, and $\Gamma$ its fundamental group. Building on the theory of equivariant harmonic maps into $\mathbb{R}$-trees, we study the non-Archimedean Hitchin map from the $\mathrm{SL}_2(F)$-character variety $\mathcal{X}_F(\Gamma)$, equipped with the non-Archimedean topology, to the space of holomorphic quadratic differentials on $\Sigma$. We prove that this map is continuous and that its image is contained in the space of Jenkins--Strebel differentials. Moreover, we establish a dynamical characterization of unbounded representations, showing that the induced action of $\Gamma$ on the Bruhat--Tits tree of $\mathrm{SL}_2(F)$ is never small.
We study the asymptotic hyperk\"ahler geometry of the $\mathrm{SL}_2(\mathbb{C})$-Hitchin moduli space over the singular fibers of the Hitchin fibration. We extend the previously known exponential convergence results for solutions to the Hitchin equation to the class of locally fiducial Higgs bundles defined by a special local description at the singularities of the spectral curve. This condition is satisfied by the Higgs bundles contained in certain subintegrable systems introduced by Hitchin. We prove that the restriction of the hyperk\"ahler metric to the subintegrable system converges exponentially fast to the corresponding semi-flat metric along a ray $(\mathcal{E},t\varphi)$. This answers a question posed by Hitchin in \cite{Hitchin2021subintegrable_special_Kaehler}. More generally, we prove that for each stratum of quadratic differentials there is a closed subset of the corresponding Hitchin fibers, such that the restricted hyperk\"ahler metric converges to a generalized semi-flat metric.
Multimodal Large Language Models (MLLMs) have demonstrated significant potential to advance a broad range of domains. However, current benchmarks for evaluating MLLMs primarily emphasize general knowledge and vertical step-by-step reasoning typical of STEM disciplines, while overlooking the distinct needs and potential of the Humanities and Social Sciences (HSS). Tasks in the HSS domain require more horizontal, interdisciplinary thinking and a deep integration of knowledge across related fields, which presents unique challenges for MLLMs, particularly in linking abstract concepts with corresponding visual representations. Addressing this gap, we present HSSBench, a dedicated benchmark designed to assess the capabilities of MLLMs on HSS tasks in multiple languages, including the six official languages of the United Nations. We also introduce a novel data generation pipeline tailored for HSS scenarios, in which multiple domain experts and automated agents collaborate to generate and iteratively refine each sample. HSSBench contains over 13,000 meticulously designed samples, covering six key categories. We benchmark more than 20 mainstream MLLMs on HSSBench and demonstrate that it poses significant challenges even for state-of-the-art models. We hope that this benchmark will inspire further research into enhancing the cross-disciplinary reasoning abilities of MLLMs, especially their capacity to internalize and connect knowledge across fields.
This paper studies the relationship between an analytic compactification of the moduli space of flat $\mathrm{SL}_2(\mathbb{C})$ connections on a closed, oriented 3-manifold $M$ defined by Taubes, and the Morgan-Shalen compactification of the $\mathrm{SL}_2(\mathbb{C})$ character variety of the fundamental group of $M$. We exhibit an explicit correspondence between $\mathbb{Z}/2$ harmonic 1-forms, measured foliations, and equivariant harmonic maps to $\mathbb{R}$-trees, as initially proposed by Taubes. As an application, we prove that $\mathbb{Z}/2$ harmonic 1-forms exist on all Haken manifolds with respect to all Riemannian metrics. We also show that there exist manifolds that support singular $\mathbb{Z}/2$ harmonic 1-forms but have compact $\mathrm{SL}_2(\mathbb{C})$ character varieties, which resolves a folklore conjecture.
Given a pair of $\mathbb{Z}_2$-harmonic spinors (resp. 1-forms) on closed Riemannian 3-manifolds $(Y_1, g_1)$ and $(Y_2,g_2)$, we construct $\mathbb{Z}_2$-harmonic spinors (resp. 1-forms) on the connected sum $Y_1 \# Y_2$ and the torus sum $Y_1 \cup_{T^2} Y_2$ using a gluing argument. The main tool in the proof is a parameterized version of the Nash-Moser implicit function theorem established by Donaldson and the second author. We use these results to construct an abundance of new examples of $\mathbb Z_2$-harmonic spinors and 1-forms. In particular, we prove that for every closed 3-manifold $Y$, there exist infinitely many $\mathbb{Z}_2$-harmonic spinors with singular sets representing infinitely many distinct isotopy classes of embedded links, strengthening an existence theorem of Doan-Walpuski. Moreover, combining this with previous results, our construction implies that if $b_1(Y) > 0$, there exist infinitely many $\mathrm{spin}^c$ structures on $Y$ such that the moduli space of solutions to the two-spinor Seiberg-Witten equations is non-empty and non-compact.
Computational and human perception are often considered separate approaches for studying sound changes over time; few works have touched on the intersection of both. To fill this research gap, we provide a pioneering review contrasting computational with human perception from the perspectives of methods and tasks. Overall, computational approaches rely on computer-driven models to perceive historical sound changes on etymological datasets, while human approaches use listener-driven models to perceive ongoing sound changes on recording corpora. Despite their differences, both approaches complement each other on phonetic and acoustic levels, showing the potential to achieve a more comprehensive perception of sound change. Moreover, we call for a comparative study on the datasets used by both approaches to investigate the influence of historical sound changes on ongoing changes. Lastly, we discuss the applications of sound change in computational linguistics, and point out that perceiving sound change alone is insufficient, as many processes of language change are complex, with entangled changes at syntactic, semantic, and phonetic levels.
In this article, we study the eigenvalues and eigenfunction problems for the Laplace operator on multivalued functions, defined on the complement of the 2n points on the round sphere. These eigenvalues and eigensections could also be viewed as functions on the configuration spaces of points, introduced and systematically studied by Taubes-Wu. Critical eigenfunctions, which serve as local singularity models for gauge theoretical problems, are of particular interest. Our study focuses on the existence and rigidity problems pertaining to these critical eigenfunctions. We prove that for generic configurations, the critical eigenfunctions do not exist. Furthermore, for each n>1, we construct infinitely many configurations that admit critical eigensections. Additionally, we show that the Taubes-Wu tetrahedral eigensections are deformation rigid and non-degenerate. Our main tools are algebraic identities developed by Taubes-Wu and finite group representation theory.
In this article, we explore Higgs bundles on a projective manifold $X$, focusing on their spectral bases, a concept introduced by T.Chen and B.Ngô. The spectral base is a specific closed subscheme within the space of symmetric differentials. We observe that if the spectral base vanishes, then any reductive representation $ρ: π_1(X) \to \text{GL}_r(\mathbb{C})$ is both rigid and integral. Additionally, we prove that for $X=Ω/Γ$, a quotient of a bounded symmetric domain $Ω$ of rank at least $2$ by a torsion-free cocompact irreducible lattice $Γ$, the spectral base indeed vanishes, which generalizes a result of B.Klingler.
We introduce the RHDLPP, a flux-limited multigroup radiation hydrodynamics numerical code designed for simulating laser-produced plasmas in diverse environments. The code bifurcates into two packages: RHDLPP-LTP for low-temperature plasmas generated by moderate-intensity nanosecond lasers, and RHDLPP-HTP for high-temperature, high-density plasmas formed by high-intensity laser pulses. The core radiation hydrodynamic equations are resolved in the Eulerian frame, employing an operator-split method. This method decomposes the solution into two substeps: first, the explicit resolution of the hyperbolic subsystems integrating radiation and fluid dynamics, and second, the implicit treatment of the parabolic part comprising stiff radiation diffusion, heat conduction, and energy exchange. Laser propagation and energy deposition are modeled through a hybrid approach, combining geometrical optics ray-tracing in sub-critical plasma regions with a one-dimensional solution of the Helmholtz wave equation in super-critical areas. The thermodynamic states are ascertained using an equation of state, based on either the real gas approximation or the quotidian equation of state (QEOS). Additionally, RHDLPP includes RHDLPP-SpeIma3D, a three-dimensional spectral simulation post-processing module, for generating both temporally-spatially resolved and time-integrated spectra and imaging, facilitating direct comparisons with experimental data. The paper showcases a series of verification tests to establish the code's accuracy and efficiency, followed by application cases, including simulations of laser-produced aluminum (Al) plasmas, pre-pulse-induced target deformation of tin (Sn) microdroplets relevant to extreme ultraviolet lithography light sources, and varied imaging and spectroscopic simulations.
In this article, we study the Hitchin morphism over a smooth projective variety $X$. The Hitchin morphism is a map from the moduli space of Higgs bundles to the Hitchin base, which in general not surjective when the dimension of X is greater than one. Chen-Ngô introduced the spectral base, which is a closed subvariety of the Hitchin base. They conjectured that the Hitchin morphism is surjective to the spectral base and also proved that the surjectivity is equivalent to the existence of finite Cohen-Macaulayfications of the spectral varieties. For rank two Higgs bundles over a projective manifold $X$, we explicitly construct a finite Cohen-Macaulayfication of the spectral variety as a double branched covering of $X$, thereby confirming Chen-Ngô's conjecture in this case. Moreover, using this Cohen-Macaulayfication, we can construct the Hitchin section for rank two Higgs bundles, which allows us to study the rigidity problem of the character variety and also to explore a generalization of the Milnor-Wood type inequality.
Beam monitoring and evaluation are very important to boron neutron capture therapy (BNCT), and a variety of detectors have been developed for these applications. However, most of the detectors used in BNCT only have a small detection area, leading to the inconvenience of the full-scale 2-D measurement of the beam. Based on micromegas technology, we designed a neutron detector with large detection area and high counting rate. This detector has a detection area of 288 mm multiples 288 mm and can measure thermal, epithermal, and fast neutrons with different detector settings. The BNCT experiments demonstrated that this detector has a very good 2-D imaging performance for the thermal, epithermal, fast neutron and gamma components, a highest counting rate of 94 kHz/channel, and a good linearity response to the beam power. Additionally, the flux fraction of each component can be calculated based on the measurement results. The Am-Be neutron source experiment indicates that this detector has a spatial resolution of approximately 1.4 mm, meeting the requirements of applications in BNCT. It is evident that this micromegas-based neutron detector with a large area and high counting rate capability has great development prospects in BNCT beam monitoring and evaluation applications.
Following the work of Mazzeo-Swoboda-Weiss-Witt and Mochizuki, there is a map $\overline{\Xi}$ between the algebraic compactification of the Dolbeault moduli space of $\mathsf{SL}(2,\mathbb{C})$ Higgs bundles on a smooth projective curve coming from the $\mathbb{C}^\ast$ action, and the analytic compactification of Hitchin's moduli space of solutions to the $\mathsf{SU}(2)$ self-duality equations on a Riemann surface obtained by adding solutions to the decoupled equations, known as ``limiting configurations''. This map extends the classical Kobayashi-Hitchin correspondence. The main result of this paper is that $\overline{\Xi}$ fails to be continuous at the boundary over a certain subset of the discriminant locus of the Hitchin fibration. This suggests the possibility of a third, refined compactification which dominates both.
The expression of one gene might be regulated by its corresponding protein, which is called autoregulation. Although gene regulation is a central topic in biology, autoregulation is much less studied. In general, it is extremely difficult to determine the existence of autoregulation with direct biochemical approaches. Nevertheless, some papers have observed that certain types of autoregulations are linked to noise levels in gene expression. We generalize these results by two propositions on discrete-state continuous-time Markov chains. These two propositions form a simple but robust method to infer the existence of autoregulation in certain scenarios from gene expression data. This method only depends on the Fano factor, namely the ratio of variance and mean of the gene expression level. Compared to other methods for inferring autoregulation, our method only requires non-interventional one-time data, and does not need to estimate parameters. Besides, our method has few restrictions on the model. We apply this method to four groups of experimental data and find some genes that might have autoregulation. Some inferred autoregulations have been verified by experiments or other theoretical works.