Singular Geometry Research Articles

Singular vectors and geometry at infinity of products of hyperbolic spaces

Singular vectors and geometry at infinity of products of hyperbolic spaces

Nondegenerate locally tame complete intersection varieties and geometry of nonisolated hypersurface singularities

We give a criterion to test geometric properties such as Whitney equisingularity and Thom’s a f a_f condition for new families of (possibly nonisolated) hypersurface singularities that “behave well” with respect to their Newton diagrams. As an important corollary, we obtain that in such families all members have isomorphic Milnor fibrations.

QNMs of branes, BHs and fuzzballs from quantum SW geometries

QNMs govern the linear response to perturbations of BHs, D-branes and fuzzballs and the gravitational wave signals in the ring-down phase of binary mergers. A remarkable connection between QNMs of neutral BHs in 4d and quantum SW geometries describing the dynamics of N=2 SYM theories has been recently put forward. We extend the gauge/gravity dictionary to a large class of gravity backgrounds including charged and rotating BHs of Einstein-Maxwell theory in d=4,5 dimensions, D3-branes, D1D5 ‘circular’ fuzzballs and smooth horizonless geometries; all related to N=2 SYM with a single SU(2) gauge group and fundamental matter. We find that photon-spheres, a common feature of all examples, are associated to degenerations of the classical elliptic SW geometry whereby a cycle pinches to zero size. Quantum effects resolve the singular geometry and lead to a spectrum of quantized energies, labelled by the overtone number n. We compute the spectrum of QNMs using exact WKB quantization, geodetic motion and numerical simulations and show excellent agreement between the three methods. We explicitly illustrate our findings for the case D3-brane QNMs.

A topological introduction to Lipschitz geometry of complex singularities

These are the notes of the three lectures I gave during the IXth Winterbraids School which took place in Reims from 4 to 7th March 2019. The aim of this course was to introduce researchers working in low-dimensional topology to Lipschitz geometry of complex singularities. In these lectures, I focussed on topological points of view on the objects, avoiding as much as possible technical material from algebraic geometry and singularity theory such as resolution of singularities, Nash transform, generic projections of curves and surfaces, etc.

The entanglement entropy of typical pure states and replica wormholes

In a 1+1 dimensional QFT on a circle, we consider the von Neumann entanglement entropy of an interval for typical pure states. As a function of the interval size, we expect a Page curve in the entropy. We employ a specific ensemble average of pure states, and show how to write the ensemble-averaged Rényi entropy as a path integral on a singular replicated geometry. Assuming that the QFT is a conformal field theory with a gravitational dual, we then use the holographic dictionary to obtain the Page curve. For short intervals the thermal saddle is dominant. For large intervals (larger than half of the circle size), the dominant saddle connects the replicas in a non-trivial way using the singular boundary geometry. The result extends the ‘island conjecture’ to a non-evaporating setting.

Curve counting in genus one: Elliptic singularities and relative geometry

We construct and study the reduced, relative, genus one Gromov--Witten theory of very ample pairs. These invariants form the principal component contribution to relative Gromov--Witten theory in genus one and are relative versions of Zinger's reduced Gromov--Witten invariants. We relate the relative and absolute theories by degeneration of the tangency conditions, and the resulting formulas generalise a well-known recursive calculation scheme put forward by Gathmann in genus zero. The geometric input is a desingularisation of the principal component of the moduli space of genus one logarithmic stable maps to a very ample pair, using the geometry of elliptic singularities. Our study passes through general techniques for calculating integrals on logarithmic blowups of moduli spaces of stable maps, which may be of independent interest.

Sum frequency generation from touching wires: a transformation optics approach.

We employ transformation optics to study analytically nonlinear wave mixing from a singular geometry of touching plasmonic wires. We obtain the analytic solution of the near field and complement it with a solution of far-field properties. We find, somewhat surprisingly, that optimal efficiency (in both regimes) is obtained for the degenerate case of second-harmonic generation. We exploit the analytic solution obtained to trace this behavior to the spatial overlap of input fields near the geometric singularity.

Affine connections on singular warped products

In this paper, we introduce semi-symmetric metric Koszul forms and semi-symmetric non-metric Koszul forms on singular semi-Riemannian manifolds. Semi-symmetric metric Koszul forms and semi-symmetric non-metric Koszul forms and their curvature of semi-regular warped products are expressed in terms of those of the factor manifolds. We also introduce Koszul forms associated with the almost product structure on singular almost product semi-Riemannian manifolds. Koszul forms associated with the almost product structure and their curvature of semi-regular almost product warped products are expressed in terms of those of the factor manifolds. Furthermore, we generalize the results in [O. Stoica, The geometry of warped product singularities, Int. J. Geom. Methods Mod. Phys. 14(2) (2017) 1750024, arXiv:1105.3404.] to singular multiply warped products.

Easy and efficient preconditioning of the isogeometric mass matrix

This paper deals with the fast solution of linear systems associated with the mass matrix, in the context of isogeometric analysis. We propose a preconditioner that is both efficient and easy to implement, based on a diagonal-scaled Kronecker product of univariate parametric mass matrices. Its application is faster than a matrix–vector product involving the mass matrix itself. We prove that the condition number of the preconditioned matrix converges to 1 as the mesh size is reduced, that is, the preconditioner is asymptotically equivalent to the exact inverse. Moreover, we give numerical evidence of its good behaviour with respect to the spline degree and the (possibly singular) geometry parametrization. We also extend the preconditioner to the multipatch case through an Additive Schwarz method.

Resonant cavity modes in nickel oxide microcrystals

Fabry-Perot resonant modes in the visible range have been detected and analysed for NiO microcrystals. The singular geometry and morphology of the well-faceted NiO microstructures favour the light confinement and the optical resonances. Quality and finesse factors around 207 and 1.8, respectively, as well as a refractive index n = 2.3 ± 0.2 have been estimated for the probed microstructures. The achievement of optical modulations in NiO microstructures can lead to improved functionality and applicability of this oxide in optical devices, a field in which its use has not been exploited so far.

Projectively nonsingular horizons in Kerr-NUT–de Sitter spacetimes

It was recently discovered that Killing horizons in the generic Kerr-NUT-(anti) de Sitter spacetimes are projectively singular, i.e. their spaces of the null generators have singular geometry. Only if the cosmological constant takes the special value determined by the Kerr and NUT parameters, and the radius of the horizon, then the corresponding horizon does not suffer that problem. In the current paper, the projectively non-singular horizons are investigated. They are found to be cosmological and non-extremal. Every projectively non-singular horizon can be used to define a global completion of the Kerr-NUT-de Sitter spacetime it is contained in. The resulting spacetime extends from $\mathcal{I}^-$ to $\mathcal{I}^+$, has the topology of $\mathbb{R}\times S_3$ and is smooth except for a possible Kerr-like singularity.

Quadratic differentials, measured foliations, and metric graphs on punctured surfaces

A meromorphic quadratic differential on a punctured Riemann surface induces horizontal and vertical measured foliations with pole singularities. In a neighborhood of a pole, such a foliation comprises foliated strips and half-planes, and its leaf space determines a metric graph. We introduce the notion of an asymptotic direction at each pole and show that for a punctured surface equipped with a choice of such asymptotic data, any compatible pair of measured foliations uniquely determines a complex structure and a meromorphic quadratic differential realizing that pair. This proves the analogue of a theorem of Gardiner–Masur for meromorphic quadratic differentials. We also prove an analogue of the Hubbard–Masur theorem; namely, for a fixed punctured Riemann surface there exists a meromorphic quadratic differential with any prescribed horizontal foliation, and such a differential is unique provided we prescribe the singular flat geometry at the poles.

Ehrhart polynomials of polytopes and spectrum at infinity of Laurent polynomials

Gathering different results from singularity theory, geometry and combinatorics, we show that the spectrum at infinity of a tame Laurent polynomial counts (weighted) lattice points in polytopes. We deduce an effective algorithm in order to compute the Ehrhart polynomial of a simplex containing the origin as an interior point.

Birational geometry of symplectic quotient singularities

For a finite subgroup Gamma subset mathrm {SL}(2,mathbb {C}) and for nge 1, we use variation of GIT quotient for Nakajima quiver varieties to study the birational geometry of the Hilbert scheme of n points on the minimal resolution S of the Kleinian singularity mathbb {C}^2/Gamma . It is well known that X:={{,mathrm{{mathrm {Hilb}}},}}^{[n]}(S) is a projective, crepant resolution of the symplectic singularity mathbb {C}^{2n}/Gamma _n, where Gamma _n=Gamma wr mathfrak {S}_n is the wreath product. We prove that every projective, crepant resolution of mathbb {C}^{2n}/Gamma _n can be realised as the fine moduli space of theta -stable Pi -modules for a fixed dimension vector, where Pi is the framed preprojective algebra of Gamma and theta is a choice of generic stability condition. Our approach uses the linearisation map from GIT to relate wall crossing in the space of theta -stability conditions to birational transformations of X over mathbb {C}^{2n}/Gamma _n. As a corollary, we describe completely the ample and movable cones of X over mathbb {C}^{2n}/Gamma _n, and show that the Mori chamber decomposition of the movable cone is determined by an extended Catalan hyperplane arrangement of the ADE root system associated to Gamma by the McKay correspondence. In the appendix, we show that morphisms of quiver varieties induced by variation of GIT quotient are semismall, generalising a result of Nakajima in the case where the quiver variety is smooth.

On certain type of singular varieties with smooth subvarieties

Let k an algebraically closed field, $$char\ k=0.$$ Let C be an irreducible nonsingular curve such that $$kC=S\cap X$$, $$k\in {\mathbb {N}},$$ where S and X are two surfaces and all the singularities of X are of the form $$z^{p}=x^{s}-y^{s},$$$$p, \, s$$ primes, $$s=pt+m,$$$$t\in {\mathbb {N}}$$, $$m=1$$ or 2. We also study the cases, for $$p, \, s$$ primes, $$p=4r+1,$$$$r \in {\mathbb {N}},$$$$s=pt+2r+1,$$$$t \in {\mathbb {N}},$$ and $$p=4r+3,$$$$r \in {\mathbb {N}},$$$$s=pt+2r+2,$$$$t \in {\mathbb {N}}.$$ We prove that C can never pass through such kind of singularities of a surface, unless $$k=pa,$$$$a\in {\mathbb {N}}$$. We study multiplicity-k structures on varieties, $$k\in {\mathbb {N}}.$$ Let Z be a reduced irreducible nonsingular $$(n-1)$$-dimensional variety such that $$kZ=X\cap S$$, where S is a $$(N-1)$$-fold in $${\mathbb {P}}^{N},$$X is a normal n-fold with certain type of singularities, like linear compound $$V_{ps}$$ singularity or (d,l) complete intersection compound $$V_{ps}$$ singularity, We study when $$Z\cap \text {Sing} (X)\ne \emptyset .$$ These results generalize some results in Gonzalez-Dorrego (On singular varieties with smooth subvarieties, singularities in geometry, topology, foliations and dynamics. Birkhauser, Basel, pp 125–134, 2017). Seifert invariants of $$z^{p}=x^{s}-y^{s},$$$$p, \, s$$ primes, $$s=pt+m,$$$$t\in {\mathbb {N}}$$, $$m=1,$$ 2 and for $$p=4r+1,$$$$r \in {\mathbb {N}},$$$$s=pt+2r+1,$$$$t \in {\mathbb {N}},$$ and$$p=4r+3,$$$$r \in {\mathbb {N}},$$$$s=pt+2r+2,$$$$t \in {\mathbb {N}}.$$ are studied (Prop. 34).

Back(reaction) to the future in the Unruh-de Sitter state

Motivated by black hole physics, we define the Unruh state for a scalar field in de Sitter space. Like the Bunch-Davies state, the Unruh-de Sitter state appears thermal to a static observer. However, it breaks some of the symmetries of de Sitter space. We calculate the expectation value of the energy-momentum tensor in the Unruh-de Sitter state in two dimensions and find a non-vanishing flux of outgoing negative energy. Extrapolating the result to four dimensions, we argue that this backreacts on the initial de Sitter geometry semi-classically. Notably, we estimate that de Sitter space is destabilized on a timescale set by the gravitational entropy; analogous to black hole evaporation, the endpoint of this instability is a singular geometry outside the regime of effective field theory. Finally, we suggest that the Unruh-de Sitter state may be a natural initial state for patches of de Sitter space, and discuss the implications for slow-roll and eternal inflation, and for de Sitter thermodynamics.

Geometrothermodynamics as a singular conformal thermodynamic geometry

In this letter, we first redefine our formalism of the thermodynamic geometry introduced in [1,2] by changing coordinates of the thermodynamic space by means of Jacobian matrices. We then show that the geometrothermodynamics (GTD) is conformally related to this new formalism of the thermodynamic geometry. This conformal transformation is singular at unphysical points were generated in GTD metric. Therefore, working with our metric neatly excludes all unphysical points without imposing any constraints.

The two-to-infinity norm and singular subspace geometry with applications to high-dimensional statistics

The singular value matrix decomposition plays a ubiquitous role throughout statistics and related fields. Myriad applications including clustering, classification, and dimensionality reduction involve studying and exploiting the geometric structure of singular values and singular vectors. This paper provides a novel collection of technical and theoretical tools for studying the geometry of singular subspaces using the two-to-infinity norm. Motivated by preliminary deterministic Procrustes analysis, we consider a general matrix perturbation setting in which we derive a new Procrustean matrix decomposition. Together with flexible machinery developed for the two-to-infinity norm, this allows us to conduct a refined analysis of the induced perturbation geometry with respect to the underlying singular vectors even in the presence of singular value multiplicity. Our analysis yields singular vector entrywise perturbation bounds for a range of popular matrix noise models, each of which has a meaningful associated statistical inference task. In addition, we demonstrate how the two-to-infinity norm is the preferred norm in certain statistical settings. Specific applications discussed in this paper include covariance estimation, singular subspace recovery, and multiple graph inference. Both our Procrustean matrix decomposition and the technical machinery developed for the two-to-infinity norm may be of independent interest.

Architectural singularities of parallel mechanisms with prismatic joints due to special designs of platform shapes

Abstract. Singularity is an inherent property of robotic manipulators. A manipulator becomes singular when it gains or losses degrees of freedom at a particular configuration. In this work, a type of singularities caused by special shapes of platforms, either the mobile or the base platform, is addressed. This type of singularities pertains to the architecture singularity, but associated only with special shape designs of base and mobile platforms and spans in the whole workspace, which is referred as shape singularity. The paper provides formulations of shape singularity. The geometry and algebra properties of shape singularity are analyzed. Three examples of shape singularity identification for parallel mechanisms with prismatic joints are included, one for 3-DOF planar mechanisms, the others for 3-DOF and 6-DOF spatial mechanisms. The application of shape singularity in adjustable compliance mechanism design is illustrated.

CT lesion recognition algorithm based on improved particle reseeding method

In order to improve the performance of CT image's lesion recognition algorithm and improve the diagnosis accuracy of doctors, a CT lesion recognition algorithm based on improved particle reseeding method is proposed. First of all, aiming at the non- uniformity of topography, the Lagrangian labeled particles are calculated before the level set formula is calculated to reconstruct the embedded interface, thus improving the quality conservation characteristics of the level set algorithm. Secondly, in view of the uncertainty of the traditional particle method in dealing with interface singularity and complex geometry- related problems, the convergence of velocity fields at singular points and topological change points is promoted by adding velocity vectors and unit normal vectors. Finally, the effectiveness of the proposed algorithm is verified by simulation experiments.