Singular Spaces Research Articles

Transfer and the spectrum-level Siegel–Sullivan KO-orientation for singular spaces

Integrally oriented normally nonsingular maps between singular spaces have associated transfer homomorphisms on KO-homology at odd primes. We prove that such transfers preserve Siegel–Sullivan orientations, defined when the singular spaces are compact pseudomanifolds satisfying the Witt condition, for example pure-dimensional compact complex algebraic varieties. This holds for bundle transfers associated to block bundles with manifold fibers as well as for Gysin restrictions associated to normally nonsingular inclusions. Our method is based on constructing a lift of the Siegel–Sullivan orientation to a morphism of highly structured ring spectra which factors through L-theory.

Eguchi–Hanson metrics arising from Kähler–Einstein edge metrics

Calabi–Hirzebruch manifolds are higher-dimensional generalizations of both the football and Hirzebruch surfaces. We construct a family of Kähler-Einstein edge metrics singular along two disjoint divisors on the Calabi–Hirzebruch manifolds and study their Gromov–Hausdorff limits when either cone angle tends to its extreme value. As a very special case, we show that the celebrated Eguchi–Hanson metric arises in this way naturally as a Gromov–Hausdorff limit. We also completely describe all other (possibly rescaled) Gromov–Hausdorff limits which exhibit a wide range of behaviors, resolving in this setting a conjecture of Cheltsov–Rubinstein. This gives a new interpretation of both the Eguchi–Hanson space and Calabi’s Ricci flat spaces as limits of compact singular Einstein spaces.

Dynamic gain and frequency comb formation in exceptional-point lasers

Exceptional points (EPs)—singularities in the parameter space of non-Hermitian systems where two nearby eigenmodes coalesce—feature unique properties with applications such as sensitivity enhancement and chiral emission. Existing realizations of EP lasers operate with static populations in the gain medium. By analyzing the full-wave Maxwell–Bloch equations, here we show that in a laser operating sufficiently close to an EP, the nonlinear gain will spontaneously induce a multi-spectral multi-modal instability above a pump threshold, which initiates an oscillating population inversion and generates a frequency comb. The efficiency of comb generation is enhanced by both the spectral degeneracy and the spatial coalescence of modes near an EP. Such an “EP comb” has a widely tunable repetition rate, self-starts without external modulators or a continuous-wave pump, and can be realized with an ultra-compact footprint. We develop an exact solution of the Maxwell–Bloch equations with an oscillating inversion, describing all spatiotemporal properties of the EP comb as a limit cycle. We numerically illustrate this phenomenon in a 5-μm-long gain-loss coupled AlGaAs cavity and adjust the EP comb repetition rate from 20 to 27 GHz. This work provides a rigorous spatiotemporal description of the rich laser behaviors that arise from the interplay between the non-Hermiticity, nonlinearity, and dynamics of a gain medium.

The relationship between singular value decomposition orthogonal space vector and sound field reconstruction in equivalent source method

In acoustic holography, the equivalent source method represents a typical ill-posed inverse problem, characterized by the significant impact of measurement errors on the sought solution. To minimize the interference of noise with the sought solution, we analyze the relationship between orthogonal space vectors obtained through singular value decomposition and acoustic field reconstruction. The research findings indicate that the primary information of the acoustic field exhibits strong correlation with a small number of space vectors, while noise information demonstrates random correlation with all space vectors. Therefore, we retain the space vectors with higher correlation to the acoustic field while filtering out redundant ones, thereby stabilizing and accurately reconstructing the acoustic field. Numerical analysis of the acoustic field reconstruction demonstrates that compared to regularization methods such as Tikhonov and truncated singular value decomposition (TSVD), this approach achieves higher computational accuracy. Furthermore, the text combines practical examples to further explain the reasons behind the superior accuracy of this method.

End of the world brane networks for infinite distance limits in CY moduli space

Dynamical Cobordism provides a powerful method to probe infinite distance limits in moduli/field spaces parameterized by scalars constrained by generic potentials, employing configurations of codimension-1 end of the world (ETW) branes. These branes, characterized in terms of critical exponents, mark codimension-1 boundaries in the spacetime in correspondence of finite spacetime distance singularities at which the scalars diverge. Using these tools, we explore the network of infinite distance singularities in the complex structure moduli space of Calabi-Yau fourfolds compactifications in M-theory with a four-form flux turned on, which is described in terms of normal intersecting divisors classified by asymptotic Hodge theory. We provide spacetime realizations for these loci in terms of networks of intersecting codimension-1 ETW branes classified by specific critical exponents which encapsulate the relevant information of the asymptotic Hodge structure characterizing the corresponding divisors.

Some Aspects of Differential Topology of Subcartesian Spaces

In this paper, we investigate the differential topological properties of a large class of singular spaces: subcarteisan space. First, a minor further result on the partition of unity for differential spaces is derived. Second, the tubular neighborhood theorem for subcartesian spaces with constant structural dimensions is established. Third, the concept of Morse functions on smooth manifolds is generalized to differential spaces. For subcartesian space with constant structural dimension, a class of examples of Morse functions is provided. With the assumption that the subcartesian space can be embedded as a bounded subset of an Euclidean space, it is proved that any smooth bounded function on this space can be approximated by Morse functions. The infinitesimal stability of Morse functions on subcartesian spaces is studied. Classical results on Morse functions on smooth manifolds can be treated directly as corollaries of our results here.

Obstacle avoidance planning for industrial robots based on singular manifold splitting configuration space

SummaryObstacle avoidance planning is the primary element in ensuring safe robot applications such as welding, assembly, and drilling. The states in the configuration space (C‐space) provide the pose information of any part of the manipulator and are preferentially considered in motion planning. However, it is difficult to express the environmental information directly in the high dimensional C‐space, limiting the application of C‐space obstacle avoidance planning. This paper proposes a singular manifold splitting C‐space method and designs a compatible obstacle avoidance strategy. The specific method is as follows: first, according to the specific structure of industrial robots, arm‐wrist separation obstacle avoidance planning is proposed to fix the robot as a 3R manipulator to reduce the dimension of C‐space. Next, the C‐space is segmented according to the singular manifolds, and the unique domain is delineated to complete the streamlining of the volume of the C‐space. Then, with the help of the point cloud, the obstacles are enveloped and mapped to the unique domain to construct the pseudo‐obstacle map. Industrial robots' obstacle avoidance planning is completed based on the pseudo‐obstacle map combined with an improved Rapidly‐Exploring Random Trees (RRT) algorithm. This method dramatically improves the efficiency of obstacle avoidance planning in the C‐space and avoids the effect of singularities on industrial robots. Finally, the effectiveness of the method is verified by physical experiments.

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.

Polarization-Controlled Exciton-Polaritons in WS2 Strongly Coupled with Low-Symmetry Photonic Crystal Nanostructures.

Atomically thin transition metal dichalcogenides (TMDs) with ambient stable exciton resonances have emerged as an ideal material platform for exciton-polaritons. In particular, the strong coupling between excitons in TMDs and optical resonances in anisotropic photonic nanostructures can form exciton-polaritons with polarization selectivity, which offers a new degree of freedom for the manipulation of the light-matter interaction. In this work, we present the experimental demonstration of polarization-controlled exciton-polaritons in tungsten disulfide (WS2) strongly coupled with polarization singularities in the momentum space of low-symmetry photonic crystal (PhC) nanostructures. The utilization of polarization singularities can not only effectively modulate the polarization states of exciton-polaritons in the momentum space but also facilitate or suppress their far field coupling capabilities by tuning the in-plane momentum. Our results provide new strategies for creating polarization-selective exciton-polaritons.

On the Degrees of Freedom Count on Singular Phase Space Submanifolds

On the Degrees of Freedom Count on Singular Phase Space Submanifolds

Beyond large complex structure: quantized periods and boundary data for one-modulus singularities

We study periods in an integral basis near all possible singularities in one-dimensional complex structure moduli spaces of Calabi-Yau threefolds. Near large complex structure points these asymptotic periods are well understood in terms of the topological data of the mirror Calabi-Yau manifold. The aim of this work is to characterize the period data near other boundaries in moduli space such as conifold and K-points. Using results from Hodge theory, we provide the general form of these periods in a quantized three-cycle basis. Based on these periods we compute the prepotential and related physical couplings of the underlying supergravity theory. Moreover, we elucidate the meaning of the model-dependent coefficients that appear in these expressions: these can be identified with certain topological and arithmetic numbers associated to the singular geometry at the moduli space boundary. We illustrate our findings by studying a wide set of examples.

Time Functions on Lorentzian Length Spaces

AbstractIn general relativity, time functions are crucial objects whose existence and properties are intimately tied to the causal structure of a spacetime and also to the initial value formulation of the Einstein equations. In this work we establish all fundamental classical existence results on time functions in the setting of Lorentzian (pre-)length spaces (including causally plain continuous spacetimes, closed cone fields and even more singular spaces). More precisely, we characterize the existence of time functions by K-causality, show that a modified notion of Geroch’s volume functions are time functions if and only if the space is causally continuous, and lastly, characterize global hyperbolicity by the existence of Cauchy time functions, and Cauchy sets. Our results thus inevitably show that no manifold structure is needed in order to obtain suitable time functions.

Poles, shocks, and tygers: The time-reversible Burgers equation.

We construct a formally time-reversible, one-dimensional forced Burgers equationby imposing a global constraint of energy conservation, wherein the constant viscosity is modified to a fluctuating state-dependent dissipation coefficient. The system exhibits dynamical properties which bear strong similarities with those observed for the Burgers equationand can be understood using the dynamics of the poles, shocks, and truncation effects, such as tygers. A complex interplay of these give rise to interesting statistical regimes ranging from hydrodynamic behavior to a completely thermalized warm phase. The end of the hydrodynamic regime is associated with the appearance of a shock in the solution and a continuous transition leading to a truncation-dependent state. Beyond this, the truncation effects such as tygers and the appearance of secondary discontinuity at the resonance point in the solution strongly influence the statistical properties. These disappear at the second transition, at which the global quantities exhibit a jump and attain values that are consistent with the establishment of a quasiequilibrium state characterized by energy equipartition among the Fourier modes. Our comparative analysis shows that the macroscopic statistical properties of the formally time-reversible system and the Burgers equationare equivalent in all the regimes, irrespective of the truncation effects, and this equivalence is not just limited to the hydrodynamic regime, thereby further strengthening the Gallavotti's equivalence conjecture. The properties of the system are further examined by inspecting the complex space singularities in the velocity field of the Burgers equation. Furthermore, an effective theory is proposed to describe the discontinuous transition.

Disconnected gauge groups in the infrared

Gauging a discrete 0-form symmetry of a QFT is a procedure that changes the global form of the gauge group but not its perturbative dynamics. In this work, we study the Seiberg-Witten solution of theories resulting from the gauging of charge conjugation in 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 theories with SU(N) gauge group and fundamental hypermultiplets. The basic idea of our procedure is to identify the ℤ2 action at the level of the SW curve and perform the quotient, and it should also be applicable to non-lagrangian theories. We study dynamical aspects of these theories such as their moduli space singularities and the corresponding physics; in particular, we explore the complex structure singularity arising from the quotient procedure. We also discuss some implications of our work in regards to three problems: the geometric classification of 4d SCFTs, the study of non-invertible symmetries from the SW geometry, and the String Theory engineering of theories with disconnected gauge groups.

Variational autoencoder-based techniques for a streamlined cross-topology modeling and optimization workflow in electrical drives

The generation and optimization of simulation data for electrical machines remain challenging, largely due to the complexities of magneto-static finite element analysis. Traditional methodologies are not only resource-intensive, but also time-consuming. Deep learning models can be used to shortcut these calculations. However, challenges arise when considering the unique parameter sets specific to each machine topology. Building on two recent studies (Parekh et al. in IEEE Trans. Magn. 58(9):1–4, 2022; Parekh et al., Deep learning based meta-modeling for multi-objective technology optimization of electrical machines, 2023, arXiv:2306.09087), that utilized a variational autoencoder to cohesively map diverse topologies into a singular latent space for subsequent optimization, this paper proposes a refined architecture and optimization workflow. Our modifications aim to streamline and enhance the robustness of both the training and optimization processes, and compare the results with the variational autoencoder architecture proposed recently.

Transition from Quasi‐Unidirectional to Unidirectional Guided Resonances in Leaky‐Mode Photonic Lattices

AbstractUnidirectional light emission from planar photonic structures is highly advantageous for a wide range of optoelectronic applications. Recently, it has been demonstrated that unidirectional guided resonances (UGRs) can be realized by utilizing topological polarization singularities in momentum space. However, the practical application of these topological unidirectional emitters has been limited due to their intricate geometric configurations, requiring special efforts with high‐cost fabrication processes. In this study, it is demonstrated that unidirectional light emission can be achieved in conventional 1D zero‐contrast gratings (ZCGs), which can be easily fabricated using current nanofabrication technologies. In ZCGs, the interband coupling between even‐like and odd‐like waveguide modes leads to the formation of quasi‐UGRs, characterized by significantly higher decay rates in either the upward or downward direction compared to the opposite direction. It is demonstrated that these quasi‐UGRs evolve into genuine UGRs with an gradual increase in grating thickness. Moreover, the emission direction of UGRs can be selectively steered either upward or downward by adjusting the lattice parameters. In addition to quasi‐UGRs and UGRs, the study also reveals additional topological phenomena in ZCGs, including exceptional points and quasi‐BICs.

"Thank You for Sharing this Fantastic Performance"

YouTube is one of the most important music media in the world today. Musical practice is reflected there in a variety of forms, from sound recordings to music videos to tutorials. The media environment that YouTube creates as a platform inscribes itself – to some degree – in the actions of those who use it. This is also the case with the YouTube drummer phenomenon, which this article examines in more detail in terms of the interplay between persona construction, music-making, media presentation, and transmedia content concatenation. The thesis is pursued that the practice of YouTube drummers creates a singular interference space of conventions, modes of action, and social roles, which in some places has novelty value with regard to music-related persona constructions and their implementation in platform-based audiovisual media, and which transcends media boundaries. The specificity of the persona construction essentially results from the mediating position of the drum-playing actors: they are not only performers who, like social media influencers, push into the light of publicity, but at the same time musical experts and instructors, quasi-curatorial music communicators who select, prepare, and present music of the past, and intermediaries of the corporeality of popular music. The analyses presented are based on exploratory investigations of the YouTube drummer field itself and a case study.

Space-time entropy, space of singularities and gravity origin: A case study

A new definition of entropy is introduced using a model that simulates an expanding space-time compatible with the fundamental principle of cosmology. The entropy is obtained by mean of a state function that measures the variation of the space-time normal curvature, from a highly compressed space to a lower compressed space. The defined entropy leads to work out a new understanding of the earliest conditions that last for a period estimated to 380,000 years after the Big Bang. It leads to understand via a short period of inflation the process that generates the uniform distribution of matter and energy at the surface of the last scattering. It involves gravitational singularities in a process of gradual decompression propitious to the incubation of matter recombination, and it allows to trace back gravity origin.

Spatial Information Lasing Enabled by Full-k-Space Bound States in the Continuum.

Optical amplification and massive information transfer in modern physics depend on stimulated radiation. However, regardless of traditional macroscopic lasers or emerging micro- and nanolasers, the information modulations are generally outside the lasing cavities. On the other hand, bound states in the continuum (BICs) with inherently enormous Q factors are limited to zero-dimensional singularities in momentum space. Here, we propose the concept of spatial information lasing, whose lasing information entropy can be correspondingly controlled by near-field Bragg coupling of guided modes. This concept is verified in gain-loss metamaterials supporting full-k-space BICs with both flexible manipulations and strong confinement of light fields. The counterintuitive high-dimensional BICs exist in a continuous energy band, which provide a versatile platform to precisely control each lasing Fourier component and, thus, can directly convey rich spatial information on the compact size. Single-mode operation achieved in our scheme ensures consistent and stable lasing information. Our findings can be expanded to different wave systems and open new scenarios in informational coherent amplification and high-Q physical frameworks for both classical and quantum applications.

On the ∂—-Equation with L2 Estimates on Singular Complex Spaces

Abstract In this paper, we present the unsolvability of $\overline \partial $-equation with weighted $L^{2}$ estimates involved curvature terms on any singular normal complex space in general. Moreover, in the non-normal case, we also give a complete description on $L^{2}$-solvability of the $\overline \partial $-equation with weighted $L^{2}$ estimates for plane curve singularities and their variants in the higher dimension.