arXiv research

We study the topological and differentiable singularities of the configuration space C(Γ) of a mechanical linkage Γin d-dimensional Euclidean space, defining an inductive sufficient condition to determine when a configuration is singular. We show that this condition holds for generic singularities, provide a mechanical…

Solves Plateau-Douglas problem for singular configurations in general metric spaces.

We show that every negative definite configuration of symplectic surfaces in a symplectic 4--manifold has a strongly symplectically convex neighborhood. We use this to show that, if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration, and if the complex si…

Optimizes bounds for multiple T-singularities on surfaces.

Characterizes K-semistability for log Fano cone singularities.

Characterizes closures of test configurations and algebraic singularity types.

Formula proves invariant matches for smooth and orbifold test configurations.

We propose in this paper a constructive procedure that transforms locally, even at singular configurations, the kinematics of a car towing trailers into Kumpera-Ruiz normal form. This construction converts the nonholonomic motion planning problem into an algebraic problem (the resolution of a system of polynomial equat…

New approach approximates c-space geometry of multi-loop linkages.

Study lens spaces' definite fillings, classifying those with specific inequalities.

Let L --> X be a complex line bundle over a compact connected Riemann surface. We consider the abelian vortex equations on L when the metric on the surface has finitely many point degeneracies or conical singularities and the line bundle has parabolic structure. These conditions appear naturally in the study of vortex …

Paper analyzes a three-loop linkage, showing it's overconstrained and shaky.

Suppose that $C=(C_1,..., C_m)$ is a configuration of 2-dimensional symplectic submanifolds in a symplectic 4-manifold $(X,ω)$ with connected, negative definite intersection graph $Γ_C$. We show that by replacing an appropriate neighborhood of $\cup C_i$ with a smoothing $W_S$ of a normal surface singularity $(S, 0)$ w…

We construct simply connected, minimal, symplectic 4-manifolds with exotic smooth structures and each with one Seiberg-Witten basic class up to sign, on the Noether line and between the Noether and half Noether lines by star surgeries introduced by Karakurt and Starkston, and by using complex singularities. We also con…

For a given bundle $ξ\colon E \to M$ over a manifold, configuration-section spaces on $ξ$ parametrise finite subsets $z \subseteq M$ equipped with a section of $ξ$ defined on $M \smallsetminus z$, with prescribed "charge" in a neighbourhood of the points $z$. These spaces may be interpreted physically as spaces of fiel…

New Stein fillings found for non-weighted homogeneous singularities.

Geodesic rays of class C^{1,1} are constructed for any test configuration of a positive line bundle L on X using resolution of singularities. The construction reduces to finding a subsolution of the corresponding Monge-Ampere equation. Geometrically, this is accomplished by the use a positive line bundle on the resolut…

Study meromorphic k-differentials with prescribed singularities on Riemann surfaces.

The configuration space of the mechanism of a planar robot is studied. We consider a robot which has $n$ arms such that each arm is of length 1+1 and has a rotational joint in the middle, and that the endpoint of the $k$-th arm is fixed to $Re^{\frac{2(k-1)π}ni}$. Generically, the configuration space is diffeomorphic t…

Paper rigorously defines Feynman graph integrals on Kähler manifolds.

Non-formal G2 manifold found with holonomy.

Geodesic rays and chordal distances link algebraic and geometric properties of positive metrics.

New framework models high-Hopf-index hopfions using generalized fold maps.

The paper defines biquandles for groups and constructs a homomorphism.

We construct a form of swallowtail singularity in R^3 which uses coordinate transformation on the source and isometry on the target. As an application, we classify configurations of asymptotic curves and characteristic curves near swallowtail.

Study on eigenfunctions on sphere configurations, proving non-existence and construction of critical eigensections.

Study algebraic curves in C^2 using Floer theory.

In this note we use the Calabi ansatz, in the context of metrics with conical singularities along a divisor, to produce regular Calabi-Yau cones and Kähler-Einstein metrics of negative Ricci with a cuspidal point. As an application, we describe singularities and cuspidal ends of the completions of the complex hyperboli…

Discovering the underlying low dimensional structure of high dimensional data has attracted a significant amount of researches recently and has shown to have a wide range of applications. As an effective dimension reduction tool, singular value decomposition is often used to analyze high dimensional matrices, which are…

Study of ants' movement rules on a 6D space, revealing distribution structures and singular trajectories.

Defines renormalised energies for singular harmonic maps into compact manifolds.

In this paper, we study the structure of the singular set for a $C^{1}$ smooth surface in the $3$-dimensional Heisenberg group $\boldsymbol{H}_{1}$. We discover a Codazzi-like equation for the $p$-area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation …

In earlier work, we provided a general description of the forces of attraction and repulsion, encountered by two parallel vertical plates of infinite extent and of possibly differing materials, when partially immersed in an infinite liquid bath and subject to surface tension forces. In the present study, we examine som…

The purpose of the present paper is to set up a formalism inspired from non-Archimedean geometry to study K-stability. We first provide a detailed analysis of Duistermaat-Heckman measures in the context of test configurations, characterizing in particular the trivial case. For any normal polarized variety (or, more gen…

In continuing the study of harmonic mapping from 2-dimensional Riemannian simplicial complexes in order to construct minimal surfaces with singularity, we obtain an a-priori regularity result concerning the real analyticity of the free boundary curve. The free boundary is the singular set along which three disk-type mi…

New BDEs reveal singular surfaces from line congruences.

This paper extends previous work on genus two fibrations by studying and resolving singular fibers.

We associate to each stable Higgs pair $(A_0,Φ_0)$ on a compact Riemann surface $X$ a singular limiting configuration $(A_\infty,Φ_\infty)$, assuming that $\det Φ$ has only simple zeroes. We then prove a desingularization theorem by constructing a family of solutions $(A_t,tΦ_t)$ to Hitchin's equations which converge t…

We are interested in comparing properties of symplectic mapping class groups of symplectic manifolds of dimension four or higher with properties of classical mapping class groups of surfaces. For $n \geq 2$, consider a configuration of Lagrangian $S^n$s in a Weinstein domain $M^{2n}$. If it is analogous, in some sense …

Mathematical formulas for elliptic curve integrals solve anomaly equations.

The paper studies parallel surfaces of cuspidal cross caps and their degeneracy.

We use invariants of Hendricks and Manolescu coming from involutive Heegaard Floer theory to find constraints on possible configurations of singular points of a rational cuspidal curve of odd degree in the projective plane. We show that the results do not carry over to rational cuspidal curves of even degree.

Let $\M$ be a classical Riemannian globally symmetric space of rank one and non-compact type. We prove the existence and uniqueness of solutions to the Dirichlet problem for harmonic maps into $\M$ with prescribed singularities along a closed submanifold of the domain. This generalizes our previous work where such maps…

We propose a new method of computing cohomology groups of spaces of knots in $\R^n$, $n \ge 3$, based on the topology of configuration spaces and two-connected graphs, and calculate all such classes of order $\le 3.$ As a byproduct we define the higher indices, which invariants of knots in $\R^3$ define at arbitrary si…

Study singular fibers in genus 2 algebraic fibrations and their monodromy factorizations.

Two single parameter families of polyhedra $P(ψ)$ are constructed in three dimensional spaces of constant curvature $C(ψ)$. Identification of the faces of the polyhedra via isometries results in cone manifolds $M(ψ)$ which are topologically $S^1\timesS^2$, $S^3$ or singular $S^2$. The singular set of $M(ψ)$ can have se…

Study of limiting configurations for SU(1,2) Hitchin equation solutions.

The Einstein/Maxwell equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities phi: R^3Σ-> H^2_C, where Sigma is a subset of the axis of symmetry, and H^2_C is the complex hyperbolic plane. Motivated by this problem, we prove the existence and uniqueness of harmonic m…