Singular Geometry Research Articles

Local versus global Lipschitz geometry

AbstractIn this article, we prove that for a definable set in an o‐minimal structure with connected link (at 0 or infinity), the inner distance of the link is equivalent to the inner distance of the set restricted to the link. With this result, we obtain several consequences. We present also several relations between the local and the global Lipschitz geometry of singularities. For instance, we prove that two sets in Euclidean spaces, not necessarily definable in an o‐minimal structure, are outer lipeomorphic if and only if their stereographic modifications are outer lipeomorphic if and only if their inversions are outer lipeomorphic.

Geometry of transcendental singularities of complex analytic functions and vector fields

Abstract On Riemann surfaces M M , there exists a canonical correspondence between a possibly multivalued function Ψ X {\Psi }_{X} whose differential is single-valued (i.e. an additively automorphic singular complex analytic function) and a vector field X X . From the point of view of vector fields, the singularities that we consider are zeros, poles, isolated essential singularities, and accumulation points of the above. The theory of singularities of the inverse function Ψ X ‒ 1 {\Psi }_{X}^{‒1} is extended from meromorphic functions to additively automorphic singular complex analytic functions. The main contribution is a complete characterization of when a singularity of Ψ X − 1 {\Psi }_{X}^{-1} is algebraic, is logarithmic, or arises from a zero with non-zero residue of X X . Relationships between analytical properties of Ψ X {\Psi }_{X} , singularities of Ψ X − 1 {\Psi }_{X}^{-1} and singularities of X X are presented. Families and sporadic examples showing the geometrical richness of vector fields on the neighbourhoods of the singularities of Ψ X − 1 {\Psi }_{X}^{-1} are studied. As applications, we have; a description of the maximal univalence regions for complex trajectory solutions of a vector field X X , a geometric characterization of the incomplete real trajectories of a vector field X X , and a description of the singularities of the vector field associated with the Riemann ξ {\rm{\xi }} -function.

The geometry of GTPs and 5d SCFTs

We make progress in understanding the geometry associated to the Generalized Toric Polygons (GTPs) encoding the Physics of 5d Superconformal Field Theories (SCFTs), by exploiting the connection between Hanany-Witten transitions and the mathematical notion of polytope mutations. From this correspondence, it follows that the singular geometry associated to a GTP is identical to that obtained by regarding it as a standard toric diagram, but with some of its resolutions frozen in way that can be determined from the invariance of the so-called period under mutations. We propose the invariance of the period as a new criterion for distinguishing inequivalent brane webs, which allows us to resolve a puzzle posed in the literature. A second mutation invariant is the Hilbert Series of the geometry. We employ this invariant to perform quantitative checks of our ideas by computing the Hilbert Series of the BPS quivers associated to theories related by mutation. Lastly, we discuss the physical interpretation of a mathematical result ensuring the existence of a flat fibration over ℙ1 interpolating between geometries connected by mutation, which we identify with recently introduced deformations of the corresponding BPS quivers.

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.

Joshi–Malafarina–Narayan singularity in weak magnetic field

The importance and significance of magnetic fields in the astrophysical scenario is well known. Many domains of astrophysical black hole physics such as polarized shadow image, high energy emitting processes and jet formation are dependent on the behavior of the magnetic fields in the vicinity of the compact objects. In light of this, we determine the master equation and master differential equation that determine the spatial behavior of the magnetic field inside a matter distribution or vacuum region, of general spherically symmetric metric, which is immersed in a test magnetic field. We also investigate here the case of JMN-1 singularity immersed in a uniform weak magnetic field and determine the behavior of magnetic fields by defining electromagnetic four potential vector. We find that the tangential component of the magnetic field is discontinuous at the matching surface of the JMN-1 singularity with the external Schwarzschild metric, resulting in surface currents. We define the covariant expression of surface current density in this scenario. We also analyze the behavior of center-of-mass energy of two oppositely charged particles in the geometry of the magnetized JMN-1 singularity. We briefly discuss the possible scenarios which would possess a discontinuous magnetic field and implications of the same and future possibilities in the realm of astrophysics are indicated.

Thermodynamic and configurational entropy of quantum Schwarzschild geometries

We study different entropies for coherent states representing the geometry of spherically symmetric compact systems. We show that the thermodynamic entropy reproduces the Bekenstein-Hawking result in the presence of thermal modes at the Hawking temperature if the object is a black hole and saturates the Bekenstein bound for more general compact objects. We also analyse the information entropy of the quantum coherent state without radiation and find further support against the singular Schwarzschild geometry.

Structural Foundation and Geometry of the Material Singularity (and Its Quantum Entanglement)

Structural Foundation and Geometry of the Material Singularity (and Its Quantum Entanglement)

Modelling of an In-Line Bladder-Type Hydraulic Suppressor for Pressure Ripple Reduction in Positive Displacement Pumps

Positive displacement pumps are widely employed for their characteristics, but the pulsating flow they produce is a major and well-known drawback. To reduce the flow ripple produced by the pump, which in turn generates a pressure ripple, many methods have been investigated, from optimising the pump geometrical features to the introduction of active and passive systems to the delivery side. A passive system that has demonstrated to be particularly effective is the in-line bladder-type hydraulic pulsation suppressor. This device, consisting of a bladder gas-charged accumulator with a singular geometry, has been the subject of several studies. This paper describes a model based on the lumped parameter method for simulating and predicting the reduction effect on the pressure ripple achieved by the hydraulic suppressor. To validate the model, an experimental study was conducted, which confirmed the good potential of the model proposed thanks to the good agreement between the modelling results and empirical data.

On Definitions of Finsler Spaces and Axiomatics of Singular Finsler Geometry

A review of various approaches to the concept of a Finsler space based on different definitions is given. In particular, the axiomatics of a singular Finsler space is given.

Transforming singular black holes into regular black holes sourced by nonlinear electrodynamics

We present a way to generate regular black hole spacetimes that are a solution of General Relativity with nonlinear electrodynamics (NLED) as a source, out of a particular metric that inherits the “mass function” of a singular black hole geometry. The singular metric may not be a vacuum, or electro-vacuum, solution of General Relativity.

Singular matrices and geometry at infinity of products of symmetric spaces

Let $\mathbf{k}$ be a number field and $\mathbf{k}_M$ the Minkowski space associated to $\mathbf{k}$. Dirichlet's theorem in Diophantine approximation is generalized to the case of systems of linear forms with coefficients in $\mathbf{k}_M$. We study the set of singular systems in this setting. We generalize the transference principle, Dani's correspondence and give an estimate of the Hausdorff dimension of the set of singular systems from below.

Geometric Hydrodynamics in Open Problems

Geometric Hydrodynamics has flourished ever since the celebrated 1966 paper of V. Arnold. In this paper we present a collection of open problems along with several new constructions in fluid dynamics and a concise survey of recent developments and achievements in this area. The topics discussed include variational settings for different types of fluids, models for invariant metrics, the Cauchy and boundary value problems, partial analyticity of solutions to the Euler equations, their steady and singular vorticity solutions, differential and Hamiltonian geometry of diffeomorphism groups, long-time behaviour of fluids, as well as mechanical models of direct and inverse cascades.

Singular cotangent models in fluids with dissipation

In this article we analyze several mathematical models with singularities where the classical cotangent model is replaced by a b-cotangent model. We provide physical interpretations of the singular symplectic geometry underlying in b-cotangent bundles featuring two models: the canonical (or non-twisted) model and the twisted one. The canonical one models systems on manifolds with boundary and the twisted one represents Hamiltonian systems with a singularity on the fiber. The twisted cotangent model includes (for linear potentials) the case of fluids with dissipation. We prove (non)-existence of cotangent lift dynamics and show the existence of an infinite number of escape orbits in this model. We also discuss more general physical interpretations of the twisted and non-twisted b-symplectic models. Twisted b-symplectic models yield in a natural way escape orbits that go to the critical set. Under compactness assumptions those escape orbits are continued as singular periodic orbits in the sense of Miranda and Oms (2021) and Miranda (2020). These models offer a Hamiltonian formulation for systems which are dissipative, extending the horizons of Hamiltonian dynamics and opening a new approach to study non-conservative systems.

A singular Riemannian geometry approach to deep neural networks II. Reconstruction of 1-D equivalence classes

We proposed in a previous work a geometric framework to study a deep neural network, seen as sequence of maps between manifolds, employing singular Riemannian geometry. In this paper, we present an application of this framework, proposing a way to build the class of equivalence of an input point: such class is defined as the set of the points on the input manifold mapped to the same output by the neural network. In other words, we build the preimage of a point in the output manifold in the input space. In particular. We focus for simplicity on the case of neural networks maps from n–dimensional real spaces to (n−1)–dimensional real spaces, we propose an algorithm allowing to build the set of points lying on the same class of equivalence. This approach leads to two main applications: the generation of new synthetic data and it may provides some insights on how a classifier can be confused by small perturbation on the input data (e.g. a penguin image classified as an image containing a chihuahua). In addition, for neural networks from 2D to 1D real spaces, we also discuss how to find the preimages of closed intervals of the real line. We also present some numerical experiments with several neural networks trained to perform non-linear regression tasks, including the case of a binary classifier.

A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations

Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. The strategies employed to investigate their theoretical properties mainly rely on Euclidean geometry, but in the last years new approaches based on Riemannian geometry have been developed. Motivated by some open problems, we study a particular sequence of maps between manifolds, with the last manifold of the sequence equipped with a Riemannian metric. We investigate the structures induced through pullbacks on the other manifolds of the sequence and on some related quotients. In particular, we show that the pullbacks of the final Riemannian metric to any manifolds of the sequence is a degenerate Riemannian metric inducing a structure of pseudometric space. We prove that the Kolmogorov quotient of this pseudometric space yields a smooth manifold, which is the base space of a particular vertical bundle. We investigate the theoretical properties of the maps of such sequence, eventually we focus on the case of maps between manifolds implementing neural networks of practical interest and we present some applications of the geometric framework we introduced in the first part of the paper.

Modular compactifications of ℳ2,n with Gorenstein curves

We study the geometry of Gorenstein curve singularities of genus two, and of their stable limits. These singularities come in two families, corresponding to either Weierstrass or conjugate points on a semistable tail. For every $1\leq m <n$, a stability condition - using one of the markings as a reference point, and therefore not $\mathfrak S_n$-symmetric - defines proper Deligne-Mumford stacks $\overline{\mathcal M}_{2,n}^{(m)}$ containing the locus of smooth curves as a dense open substack.

Direct geometric probe of singularities in band structure.

A quantum system's energy landscape may have points where multiple energy surfaces are degenerate and that exhibit singular geometry of the wave function manifold, with major consequences for the system's properties. Ultracold atoms in optical lattices have been used to indirectly characterize such points in the band structure. We measured the non-Abelian transformation produced by transport directly through the singularities. We accelerated atoms along a quasi-momentum trajectory that enters, turns, and then exits the singularities at linear and quadratic band-touching points of a honeycomb lattice. Measurements after transport identified the topological winding numbers of these singularities to be 1 and 2, respectively. Our work introduces a distinct method for probing singularities that enables the study of non-Dirac singularities in ultracold-atom quantum simulators.

On the M2–Brane Index on Noncommutative Crepant Resolutions

On certain M-theory backgrounds which are a circle fibration over a smooth Calabi–Yau the quantum theory of M2 branes can be studied in terms of the K-theoretic Donaldson–Thomas theory on the threefold. We extend this relation to noncommutative crepant resolutions. In this case the threefold develops a singularity and classical smooth geometry is replaced by the algebra of paths of a certain quiver. K-theoretic quantities on the quiver representation moduli space can be computed via toric localization and result in certain rational functions of the toric parameters. We discuss in particular the case of the conifold and certain orbifold singularities.

Singular geometry perturbation based method for shape-topology optimization in unsteady Stokes flow

This paper concerns the topological derivatives and its applications for solving shape and topology optimization problems in fluid mechanics. The fluid flow is governed by the unsteady incompressible Stokes equations in the two dimensional case. We derive a topological sensitivity analysis for this parabolic-type operator. The proposed approach is based on a preliminary estimate describing the variation of the velocity field caused by the presence of a small obstacle inside the fluid flow. We obtain a topological asymptotic expansion for the unsteady Stokes operator valid for a large class of shape functions and an arbitrarily shaped geometric perturbation. Then, the topological gradient is exploited for building an efficient and accurate topology optimization algorithm. Finally, we present some numerical investigations showing the efficiency of the proposed approach.

Weak-strong Uniqueness for the Navier–Stokes Equation for Two Fluids with Ninety Degree Contact Angle and Same Viscosities

We consider the flow of two viscous and incompressible fluids within a bounded domain modeled by means of a two-phase Navier–Stokes system. The two fluids are assumed to be immiscible, meaning that they are separated by an interface. With respect to the motion of the interface, we consider pure transport by the fluid flow. Along the boundary of the domain, a complete slip boundary condition for the fluid velocities and a constant ninety degree contact angle condition for the interface are assumed. In the present work, we devise for the resulting evolution problem a suitable weak solution concept based on the framework of varifolds and establish as the main result a weak-strong uniqueness principle in 2D. The proof is based on a relative entropy argument and requires a non-trivial further development of ideas from the recent work of Fischer and the first author (Arch. Ration. Mech. Anal. 236, 2020) to incorporate the contact angle condition. To focus on the effects of the necessarily singular geometry of the evolving fluid domains, we work for simplicity in the regime of same viscosities for the two fluids.