Singular Spaces Research Articles

Hamiltonian facets of classical gauge theories on E-manifolds

Manifolds with boundary, with corners, b-manifolds and foliations model configuration spaces for particles moving under constraints and can be described as E-manifolds. E-manifolds were introduced in Nest and Tsygan (2001 Asian J. Math. 5 599–635) and investigated in depth in Miranda and Scott (2021 Rev. Mat. Iberoam. 37 1207–24). In this article we explore their physical facets by extending gauge theories to the E-category. Singularities in the configuration space of a classical particle can be described in several new scenarios unveiling their Hamiltonian aspects on an E-symplectic manifold. Following the scheme inaugurated in Weinstein (1978 Lett. Math. Phys. 2 417–20), we show the existence of a universal model for a particle interacting with an E-gauge field. In addition, we generalise the description of phase spaces in Yang–Mills theory as Poisson manifolds and their minimal coupling procedure, as shown in Montgomery (1986 PhD Thesis University of California, Berkeley), for base manifolds endowed with an E-structure. In particular, the reduction at coadjoint orbits and the shifting trick are extended to this framework. We show that Wong’s equations, which describe the interaction of a particle with a Yang–Mills field, become Hamiltonian in the E-setting. We formulate the electromagnetic gauge in a Minkowski space relating it to the proper time foliation and we see that our main theorem describes the minimal coupling in physical models such as the compactified black hole.

Long time existence of Yamabe flow on singular spaces with positive Yamabe constant

In this work we establish long-time existence of the normalized Yamabe flow with positive Yamabe constant on a class of manifolds that includes spaces with incomplete cone-edge singularities. We formulate our results axiomatically, so that our results extend to general stratified spaces as well, provided certain parabolic Schauder estimates hold. The central analytic tool is a parabolic Moser iteration, which yields uniform upper and lower bounds on both the solution and the scalar curvature.

Dynamic behaviors for the graphene nano/microelectromechanical system in a fractal space

The microelectromechanical devices have triggered rocketing interest in various advanced technologies, that is, sensors, material science, and energy harvesting, and now the devices tend to a nanoscale size, making the technology much more attractive in both academic and industrial communities. This paper takes a graphene nano/microelectromechanical system as an example to study its dynamical property, which is the foundation for the optimal design and reliable operation. As the system is inextricably complex with singularity and zero conditions in a fractal space, the iteration perturbation method is used to obtain the periodic solution of the system. The results show clearly the low-frequency property of the system and pull-in instability; furthermore, the fractal dimension affects greatly the system’s dynamical property.

Stress-stress correlations reveal force chains in gels.

We investigate the spatial correlations of microscopic stresses in soft particulate gels using 2D and 3D numerical simulations. We use a recently developed theoretical framework predicting the analytical form of stress-stress correlations in amorphous assemblies of athermal grains that acquire rigidity under an external load. These correlations exhibit a pinch-point singularity in Fourier space. This leads to long-range correlations and strong anisotropy in real space, which are at the origin of force-chains in granular solids. Our analysis of the model particulate gels at low particle volume fractions demonstrates that stress-stress correlations in these soft materials have characteristics very similar to those in granular solids and can be used to identify force chains. We show that the stress-stress correlations can distinguish floppy from rigid gel networks and that the intensity patterns reflect changes in shear moduli and network topology, due to the emergence of rigid structures during solidification.

Arbitrarily polarized bound states in the continuum with twisted photonic crystal slabs

Arbitrary polarized vortex beam induced by polarization singularity offers a new platform for both classical optics and quantum entanglement applications. Bound states in the continuum (BICs) have been demonstrated to be associated with topological charge and vortex polarization singularities in momentum space. For conventional symmetric photonic crystal slabs (PhCSs), BIC is enclosed by linearly polarized far fields with winding angle of 2π, which is unfavorable for high-capacity and multi-functionality integration-optics applications. Here, we show that by breaking σz-symmetry of the PhCS, asymmetry in upward and downward directions and arbitrarily polarized BIC can be realized with a bilayer-twisted PhCS. It exhibits elliptical polarization states with constant ellipticity angle at every point in momentum space within the vicinity of BIC. The topological nature of BIC reflects on the orientation angle of polarization state, with a topological charge of 1 for any value of ellipticity angle. Full coverage of Poincaré sphere (i.e., - {{pi over 4}} le chi le {{pi over 4}} and - {{pi over 2}} le psi le {{pi over 2}}) and higher-order Poincaré sphere can be realized by tailoring the twist angles. Our findings may open up new avenues for applications in structured light, quantum optics, and twistronics for photons.

Does the Political Variable Explain the COVID-19 Event in Brazil?

In effectively confronting the Covid-19 pandemic, it is necessary to complement the contributions of Biomedical Sciences to those offered by the Human Sciences. The theoretical foundations derived from the treatment of the interaction of the infectious agent with the host, primates in formatting the guidelines adopted by the World Health Organization, faced strong opposition from government officials in countries such as Brazil, giving rise to critical pronouncements by researchers in the biomedical sciences in space offered by important scientific journals. Despite being fruitful and strategic in nature, we feel encouraged to consider the objective limits of such investiture, a direct reflection of the lack of dialogue with the Human Sciences, in the field of politics and, mainly, of urbanism. Taking as a starting point the unfolding of the phenomenon in Brazil, specifically in the metropolis of Fortaleza-Ceará, we will show how the "denial" posture is inserted in a broader context, encompassing both a representative political framework of a federative government and including postures of insurgency of the Local Governments (of the federative states) in relation to the Central Government as social and economic aspects that characterize a westernized country to establish impactful filtering genre in the implementation of policies of ordering and control of the singular space. In these terms, the drastic expansion of Covid-19 in Brazil denotes a paradoxical picture to represent: a political context of resistance by Local Governments to minimize the controversial actions and attitudes of the Central Government, specifically the creation of the Forum of Governors in the Northeast to guarantee incorporation WHO international guidelines; a trajectory of unequal and perverse modernization that corroborates the indication of a high number of cases and, mainly, deaths in precarious urbanization areas of cities such as Fortaleza, establishing the periphery as the territory of death, registered in areas with a high social vulnerability index of negatively impact the effective implementation of the social isolation policy.

Mapping and Manipulation of Topological Singularities: From Photonic Graphene to T-Graphene

Topological singularities (TSs) in momentum space give rise to intriguing fundamental phenomena as well as unusual material properties, attracting a great deal of interest in the past decade. Recently, we demonstrated universal momentum-to-real-space mapping of TSs and pseudospin angular momentum conversion using photonic honeycomb (graphene-like) and Lieb lattices. Such mapping arises from the nontrivial Berry phase winding and is thus of topological origin. In this paper, after a brief review of previous observations of pseudospin–orbital conversion, we present the first theoretical analysis and experimental demonstration of TS mapping in a new type of Dirac-like structure, namely, the T-graphene lattice. Unlike other lattices, there are two coexisting but distinct TSs located at different high-symmetry points in the first Brillouin zone of T-graphene, which enables controlled topological charge conversion in the same lattice. We show active manipulation of the TS mapping, turning the two TSs into vortices of different helicities, or one into a high-order vortex but the other eliminated and emerged as a quadrupole. Such mapping and manipulation of TSs may find applications in optical communications and quantum information, and may bring insight into the study of other Dirac-like structures with multiple TSs beyond the 2D photonic platform.

Riemann moduli spaces are quantum ergodic

In this note we show that the Riemann moduli spaces $M_{\gamma,n}$ equipped with the Weil–Petersson metric are quantum ergodic for $3 \gamma + n \geq 4$. We also provide other examples of singular spaces with ergodic geodesic flow for which quantum ergodicity holds.

스피노자-들뢰즈적 배움과 성장 서사

I examine how Gilles Deleuze’s discussion of “learning” is predicated on Spinoza’s “common notion,” and how Jane Austen’s Pride and Prejudice, a representative classical Bildungsroman in Britain, also works as a narrative of “Spinozist-Deleuzian learning.” Deviating from Lukacs’s definition of Bildungsroman as the maturation of “a problematic individual” and his/her “reconciliation” with the society, narratives of Spinozist-Deleuzian learning remind us of the importance of learning (apprenticeship) as an indispensable part of our lives even in the 21st century. Spinoza’s Ethics presents learning or apprenticeship as a crucial facet of his ethical project of active liberty where common notion as “a strange harmony of reason and imagination” plays a decisive role. Deleuze’s account of “learning” reformulates the differences between Spinozist common notion and Cartesian concept of truth (i.e. correspondence of an object with the mind’s representation of it) into the distinction of “learning” and “knowledge.” Simply put, while learning is a problem or a problematic field, knowledge is a solution; they are as distant as possible in nature. Pride and Prejudice exemplifies a process of learning in Spinozist-Deleuzian sense where Elizabeth and Mr. Darcy encounter to realize passive affects like pride as the core part of their selves and proceed to joy and affirmation through the formation of common notions. Furthermore, Austen’s novel evidences that the novel reader’s forrmation of common notions with characters and also with the narrator through sympathy and distancing is the mechanism of novel reading more pivotal than identification or sympathy. The novel is a singular space where imagination as the necessary condition of human knowledge is unfolded as well as what Spinoza calls the virtue (eminence) of the mind, i.e. the mind’s meta-power of being aware that it imagines as it imagines.

On the global dynamics of path-following control of automated passenger vehicles

The nonlinear dynamics of the path-following control of passenger cars is analyzed in this paper. The effect of specific modeling aspects, such as tire deformation, steering dynamics, feedback delay and controller saturation, is considered. Possible equilibrium points and singularities in the state space are uncovered and analyzed for different vehicle model and controller designs. The equilibrium of stable path following is then analyzed in greater detail: The domains of stabilizing control gains are presented in stability charts and the basin of attraction of the equilibrium along the stable domain is approximated with the help of numerical continuation. Unsafe zones of control gains are highlighted, where the stable equilibrium is surrounded by low-amplitude unstable limit cycles. Finally, it is shown how specific modifications of the control law can remove unwanted equilibrium points and increase the basin of attraction of stable path following, resulting in safer and more reliable control of the vehicle.

Nonabelian Hodge theory for stacks and a stacky P=W conjecture

We introduce a version of the P=W conjecture relating the Borel–Moore homology of the stack of representations of the fundamental group of a genus g Riemann surface with the Borel–Moore homology of the stack of degree zero semistable Higgs bundles on a smooth projective complex curve of genus g. In order to state the conjecture we propose a construction of a canonical isomorphism between these Borel–Moore homology groups. We relate the stacky P=W conjecture with the original P=W conjecture concerning the cohomology of smooth moduli spaces, and the PI=WI conjecture concerning the intersection cohomology groups of singular moduli spaces. In genus zero and one, we prove the conjectures that we introduce in this paper.

Algebraic Symplectic Reduction and Quantization of Singular Spaces

The algebraic method of singular reduction is applied for non regular group action on manifolds which provides singular symplectic spaces. The problem of deformation quantization of the singular surfaces is the focus. For some examples of singular Poisson spaces the deformation quantization is explicitly constructed. In is shown that for the flat phase space with the classical moment map and the orthogonal group action the deformation quantization converges for the entire arguments of exponential type.

Construction of New Infinite-Series Exact Solitary Wave Solutions and Its Application to the Korteweg–De Vries Equation

The Korteweg–de Vries (KDV) equation is one of the most well-known models in nonlinear physics, such as fluid physics, plasma, and ocean engineering. It is very important to obtain the exact solutions of this model in the process of studying these topics. In the present paper, using distinct function iteration relations in two ways, namely, squaring infinitely and extracting the square root infinitely, which have not been reported in other documents, we construct abundant types of new infinite-series exact solitary wave solutions using the auxiliary equation method. Most of these solutions have not been reported in previous papers. The numerical analysis of some solutions shows complex solitary wave phenomena. Some solutions can have stable solitary wave structures, while others may have singularities in certain space–time positions. The results show that the analysis model we use is very simple and effective for the construction of new infinite-series solutions and new solitary wave structures of nonlinear models.

Unified representation of homogeneous and quasi-homogenous systems in geometrothermodynamics

We analyze homogeneous and quasi-homogeneous thermodynamic systems within the formalism of geometrothermodynamics (GTD). A generalized Euler identity is used to obtain the explicit form of the three Legendre invariant metrics that are known in GTD for the equilibrium space. In so doing, we fix all the arbitrary parameters that enter the GTD metrics in terms of the quasi-homogeneous coefficients. We obtain quite general results that relate the curvature singularities of the equilibrium space with the thermodynamic stability conditions and the phase transition structure of the system. This result allows us to avoid the appearance of non-physical singularities at the level of the equilibrium space.

Dynamics of diverse polarization singularities in momentum space with far-field interference

Polarization singularities in momentum space, including bound states in the continuum (BICs) and circularly polarized states (CPSs), render great potential in singular and chiral optics. As far-field topological signatures of Bloch modes in photonic crystal slab, the dynamic evolution of the polarization singularities is observed by tuning the geometric parameters of the slab itself. Here, we combine the photonic crystal slab with a reflected multilayer substrate in which interference between the reflected optical field and the upward radiation of the Bloch mode brings us an extra degree of freedom to manipulate the polarization singularities without breaking the structural symmetry. As the polarization singularities evolve with the spacing between the photonic crystal slab and the substrate, the CPSs and extrinsic BICs are both generated from charge reversal of the at-\ensuremath{\Gamma} intrinsic BIC and then collide with the off-\ensuremath{\Gamma} intrinsic BICs. Importantly, all BICs are topologically protected from splitting throughout the whole dynamic process owing to the preservation of structural symmetry. Our findings suggest an alternative route to pursue ultrahigh-$Q$ CPS resonances and merge diverse BICs, which is potentially applicable in chiral optics, topological photonics, and beam modulation.

Control problem for quadratic parabolic differential equations with sparse sensor sets of finite volume or anisotropically decaying density

We prove observability and null-controllability for quadratic parabolic differential equations. The sensor set is allowed to be sparse and have finite volume if the generator has trivial singular space S. In the case of generators with singular space S ≠ {0} the sensor set is permitted to decay in directions determined by S. The proof is based on dissipation estimates for the quadratic differential operator with respect to spectral projections of partial harmonic oscillators and corresponding uncertainty relations.

Intersection theory and Chern classes in Bott–Chern cohomology

In this article, we investigate an axiomatic approach introduced by Grivaux for the study of rational Bott-Chern cohomology, and use it in that context to define Chern classes of coherent sheaves. This method also allows us to derive a Riemann-Roch-Grothendieck formula for a projective morphism between smooth complex compact manifolds. In the general case of complex spaces, the Poincar\'e and Dolbeault-Grothendieck lemmas are not always valid. For this reason, and to simplify the exposition, we only consider non singular complex spaces. The appendix presents a calculation of integral Bott-Chern cohomology in top degree for a connected compact manifold.

LCK SPACES. A SHORT SURVEY

In this short survey, we present the results obtained so far in the geometry of lcK singular spaces.

Numerical characterization of complex torus quotients

This article gives a characterization of quotients of complex tori by finite groups acting freely in codimension two in terms of a numerical vanishing condition on the first and second Chern class. This generalizes results previously obtained by Greb–Kebekus–Peternell in the projective setting, and by Kirschner and the second author in dimension three. As a key ingredient to the proof, we obtain a version of the Bogomolov–Gieseker inequality for stable sheaves on singular spaces, including a discussion of the case of equality.

Perfect absorption and phase singularities induced by surface lattice resonances for plasmonic nanoparticle array on a metallic film.

The formation of pairs of perfect absorption associated with phase singularities in the parameter space using the hybridized structure constructed with a metallic nanoparticle array and a metallic film is promising to enhance light-mater interactions. However, the localized plasmon resonances of the array possess strong radiative losses, which is an obstacle to improve the performances for many applications. On the contrary with the subwavelength array hybridized structure, this study shows that by enlarging the lattice spacing, the oscillator strength of the nanoparticles can be enhanced with the formation of surface lattice resonance, thereby leading to similar but much narrower pairs of perfect absorption due to the interactions with the Fabry-Pérot cavity modes. Furthermore, when the surface plasmon polariton mode shift to the same spectral range associated with the enlarged lattice spacing, the coupling and mode hybridization with the surface lattice resonance result in an anticrossing in the spectra. Although the resonance coupling does not enter the strong coupling regime, the quality factors (∼ 134) and near-field enhancements (∼ 44) are strongly enhanced for the hybridized resonance modes due to the effectively suppressed radiative losses compared with that of the localized plasmon resonances, which make the hybridized structure useful for the design of functional nanophotonic device such as biosensing, multi-model nanolasing, and high-quality imaging.