The study examines saddle connections on flat surfaces with poles of higher order, providing bounds and characterizations.
problem Counting saddle connections on flat surfaces with poles of higher order.
method Combinatorial analysis and geometric properties of flat surfaces.
result Lower and upper bounds for the number of saddle connections are provided, and combinatorial characterizations are given for strata with infinite saddle connections.
Pole ladder improves parallel transport in affine spaces, showing exact results in symmetric spaces.
problem Improving numerical stability and accuracy in parallel transport algorithms.
method Developed a third-order parallel transport scheme using pole ladder in affine connection spaces, showing exact results in symmetric spaces.
result Pole ladder is a third-order scheme in general affine connection spaces and is exact in locally symmetric spaces.
We prove the existence of "half-plane differentials" with prescribed local data on any Riemann surface. These are meromorphic quadratic differentials with higher-order poles which have an associated singular flat metric isometric to a collection of euclidean half-planes glued by an interval-exchange map on their bounda…
The study classifies Veech groups of flat surfaces with poles.
problem Classifying Veech groups of flat surfaces with poles.
method Classification through orbit closure and determination of Veech groups.
result Characterization of surfaces with closed orbits and determination of Veech groups.
Extends Teichmüller space parametrization using poles of higher order.
problem Parametrize Teichmüller space of crowned hyperbolic surfaces.
method Use meromorphic quadratic differentials with higher order poles to parametrize.
result Existence of harmonic map from punctured Riemann surface to crowned hyperbolic surface.
We provide a novel proof that the set of directions that admit a saddle connection on a meromorphic quadratic differential with at least one pole of order at least two is closed, which generalizes a result of Bridgeland and Smith, and Gaiotto, Moore, and Neitzke. Secondly, we show that this set has finite Cantor-Bendix…
For a holomorphic family of classical pseudodifferential operators on a closed manifold we give exact formulae for all coefficients in the Laurent expansion of its Kontsevich-Vishik canonical trace. This generalizes a known result identifying the Wodzicki residue with the pole at zero to all higher order terms.
Study moduli space of quadratic differentials with new geometric insights.
problem Understanding the structure of moduli spaces of quadratic differentials.
method Using decorated marked surfaces, Abel-Jacobi map, and 3-Calabi-Yau categories.
result Fundamental group of moduli space equals kernel of Abel-Jacobi map.
We count meromorphic differentials with fixed residues and poles of fixed orders.
problem Counting meromorphic differentials with fixed residues and poles of fixed orders.
method Intersection theory on compactified moduli spaces of differentials.
result Complete solution to the problem with interesting combinatorial properties.
The tautological ring of strata of differentials is characterized based on the presence of poles.
problem Characterizing the tautological ring of strata of differentials.
method Using the κ, ψ, and η classes of moduli spaces of pointed smooth curves.
result The tautological ring is generated by η only if there are no poles of order k, otherwise by the ψ classes corresponding to the poles of order k.
The study connects cubic differentials to convex RP^2-structures and their ends.
problem Understanding the relationship between cubic differentials and convex RP^2-structures.
method Affine sphere construction and analysis of poles of cubic differentials.
result Poles of cubic differentials correspond to ends of convex RP^2-structures.
Following the approach of Carlet et al.(2011)\cite{CDM}, we construct a class of infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy, which are defined on the space of pairs of meromorphic functions with possibly higher-order poles at the origin and at infinity. We also show a connection betw…
New stability conditions identified from quadratic differentials on surfaces.
problem Identifying stability conditions from quadratic differentials.
method Comparison of exchange graphs from tilting hearts and flipping mixed angulations.
result Spaces of stability conditions identified with moduli spaces of quadratic differentials.
Study meromorphic k-differentials with prescribed singularities on Riemann surfaces.
problem Understanding local invariants of meromorphic k-differentials on Riemann surfaces.
method Analyzing orders of zeros and poles, and k-residues at poles.
result For a given pattern of zeros, there exists a primitive holomorphic k-differential with these zeros.
Quadratic differentials induce spiralling foliations on Riemann surfaces.
problem Understanding the structure of foliations induced by quadratic differentials.
method Introduced a space of measured foliations and used harmonic maps to real trees.
result Any measured foliation is realized by a quadratic differential with second order poles at marked points.
This paper shows how to construct Abelian differentials with any prescribed singularities.
problem Constructing Abelian differentials with specific orders and residues.
method Flat representation of Abelian differentials.
result Every pattern of orders and residues can be realized in Abelian differentials, except for two families in genus zero.
Study describes how to realize periods of meromorphic differentials with specific properties.
problem Realizing meromorphic differentials with given zeros, poles, and topological constraints.
method Complete description of period representations for specified conditions on Riemann surfaces.
result A comprehensive method for realizing meromorphic differentials with prescribed characteristics.
The paper examines how polarized curves behave near singular points.
problem Analyzing the behavior of polarized curves near singular points.
method Investigates the limiting behavior of Darboux and Calapso transforms of polarized curves in the conformal n-dimensional sphere.
result For a pole of first order, all transforms converge to the original curve. For a pole of second order, a generic Darboux transform converges, but a Calapso transform has a limit point or circle.
The paper defines and computes volumes of meromorphic differentials with simple poles.
problem Defining and computing volumes of strata of meromorphic differentials with simple poles.
method Definition of volume as an integral of a tautological class, computation by induction, and solution of an integrable system.
result Algebraic constants of volumes can be computed and shown to be solutions of integrable systems.
Study meromorphic k-differentials on Riemann surfaces, focusing on their singularities.
problem Characterize the local invariants of meromorphic k-differentials on Riemann surfaces.
method Analyzing orders of zeroes and poles, and k-residues at poles, for different genera.
result For genus g ≥ 2, every expected tuple of k-residues appears as actual residues.
The isoresidual fibration maps Riemann sphere strata to resonance arrangements.
problem Mapping Riemann sphere strata to resonance arrangements.
method Defining isoresidual fibration and studying its properties using tree structures.
result The isoresidual fibration is an unramified cover of degree a!/(a+2-p)! above the complement of a hyperplane arrangement.
Proves theorem about meromorphic projective structures with complex poles.
problem Proving a theorem about meromorphic projective structures with complex poles.
method Using coordinates on the moduli space of framed representations from Fock and Goncharov.
result Proves the analogue of a theorem of Gallo-Kapovich-Marden.
We present an explicit formula relating volumes of strata of meromorphicquadratic differentials with at most simple poles on Riemann surfacesand counting functions of the number of flat cylinders filled by closedgeodesics in associated flat metric with singularities. This generalizes the resultof Athreya, Eskin and Zor…
We study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion a∼∑j=0∞am−j,am−j(x,ξ)=∑l=0kam−j,l(x,ξ)logl∣ξ∣, where am−j,l is homogeneous in ξ of degree m−j. We will explain why this algebra of pseudo…
Non-asphericity of strata of genus-one differentials
problem Strata of genus-one differentials
method Non-asphericity
result Infinitely many counterexamples to conjectures
Study projective structures with poles, mapping them to local systems.
problem Understanding projective structures with poles on surfaces.
method Define a monodromy map from parameter space to framed local systems, proving containment in cluster chart domains.
result The image of the monodromy map is contained in the union of cluster chart domains.
System identifies power grid location from media recordings.
problem Identifying the origin of power distribution grid from media recordings.
method Cascaded SVM and pole-matching classifiers for grid identification.
result Cascaded system improves accuracy by 15.57%.
Proposes an algorithm for infinite-dimensional sparse learning in system identification.
problem System identification without known model structures.
method Atomic norm regularization and greedy algorithm for solving an infinite-dimensional group lasso problem.
result The proposed algorithm outperforms benchmark methods in impulse response fitting and pole location estimation.
Study on monodromy of cyclic opers on Riemann sphere with single pole.
problem Analyzing the monodromy of meromorphic cyclic opers on the Riemann sphere.
method Developed a method based on isomonodromic deformations and properties of structure constants for sl(n,C).
result Monodromy map is an immersion when the order of the pole is a multiple of n.
The Ward equation, also called the modified 2+1 chiral model, is obtained by a dimension reduction and a gauge fixing from the self-dual Yang-Mills field equation on R2,2. It has a Lax pair and is an integrable system. Ward constructed solitons whose extended solutions have distinct simple poles. He also used a li…
The paper proves that certain geodesics pass through timelike poles on specific Lorentzian 2-tori.
problem Existence of closed timelike geodesics through timelike poles on class A Lorentzian 2-tori.
method Study of isometries on globally hyperbolic planes to prove the existence of geodesics.
result Existence of closed timelike geodesics through timelike poles in the interior of the stable time cone.
The paper examines affine properties of differential strata on curves.
problem Characterizing affine structures in strata of differentials.
method Analyzing affine varieties and Teichmüller dynamics.
result Affine strata do not contain complete curves with certain pole orders.
This paper proves a bound on the energy of Nahm pole solutions on S3imesR+.
problem Bounding the energy of Nahm pole solutions on a specific manifold.
method Proving an energy bound using the Kapustin-Witten equation with Nahm pole boundary conditions.
result There exists a constant C>0 such that ∥FA∥L2≤C for any Nahm pole solution (A,φ). This paper shows how to create quadratic differentials with any given singularities.
problem Creating quadratic differentials with prescribed singularities.
method Using the flat metric induced by the differentials, the authors classify and construct quadratic differentials with specific singularities.
result Every pattern of local invariants can be obtained by a quadratic differential on some Riemann surface, with exceptions in genera zero and one.
Improved estimates for p-Green functions near poles in Euclidean and Riemannian settings.
problem Asymptotic behavior of p-Green functions near poles.
method Asymptotic expansion and integrability properties for derivatives.
result Improved estimates and asymptotic expansions for p-Green functions.
Under several geometric conditions imposed below, the existence of the discrete spectrum below the essential spectrum is shown for the Dirichlet Laplacian on the quantum layer built over a spherically symmetric hypersurface with a pole embedded in the Euclidean space R4. At the end of this paper, we also show the advan…
Study of Nahm pole solutions over 3-manifolds, proving smoothness at boundary.
problem Analyzing Nahm pole solutions over 3-manifolds.
method Examining polyhomogeneous solutions and their expansions.
result Smoothness of sub-leading terms at boundary for Einstein 3-manifolds.
Paper proves non-existence of certain convex functions on a Riemannian manifold with a pole.
problem Proving non-existence of specific convex functions on a Riemannian manifold with a pole.
method Developed notions of odd and even functions on a Riemannian manifold with a pole, proved non-existence of non-trivial and non-negative convex functions.
result Deduced non-existence of non-trivial and non-negative differentiable odd convex functions whose gradient is complete.
Classifies components of strata of k-differentials on Riemann surfaces.
problem Classifying connected components of strata of k-differentials.
method Developed new techniques to study connected components of strata of k-differentials for general k.
result Complete classification of connected components of the strata of quadratic differentials with arbitrary poles.
The paper constructs solutions to Kapustin-Witten equations with Nahm poles over manifolds with boundaries.
problem Constructing solutions to Kapustin-Witten equations with Nahm poles over manifolds with boundaries.
method Establishing a gluing construction for Nahm pole solutions over manifolds with boundaries and cylindrical ends.
result There exists an obstruction class for gluing Nahm pole solutions along cylindrical ends.
Finite groups can be automorphism groups of translation surfaces with poles.
problem Existence of finite automorphism groups on translation surfaces with poles.
method Analyzing translation surfaces with poles and extending results to branched projective structures.
result Finite groups can be automorphism groups of translation surfaces with poles.
Extends knotoid theory to include multiple poles and intervals.
problem No new problem introduced.
method Definition of generalized knotoids and graphs, exploration of invariants.
result Theory subsumes various topological objects and introduces new cases.
We consider compact minimal surfaces f:M→S3 of genus 2 which are homotopic to an embedding. We assume that the associated holomorphic bundle is stable. We prove that these surfaces can be constructed from a globally defined family of meromorphic connections by the DPW method. The poles of the meromorphic co…
The paper classifies solutions to Kapustin-Witten equations with specific singularities.
problem Classifying solutions to Kapustin-Witten equations with Nahm pole singularities.
method Analytical classification of solutions with detailed singularity analysis.
result Classification of solutions with Nahm pole and generalized Nahm pole singularities.
Continuous functions on Riemannian manifolds with poles have fixed points.
problem Extending continuous functions on Riemannian manifolds with poles.
method Simple geometrical technique to generalize Brouwer fixed point theorem.
result Any continuous function on the boundary of a convex domain of a 2D Riemannian manifold with a pole has a fixed point that can be extended to the domain.
Study of 0-instantons on hyperbolic manifolds, proving invariants and energy formulas.
problem Understanding 0-instantons on hyperbolic manifolds and their properties.
method Analyzing asymptotic expansions and using Fefferman-Graham expansion for Poincaré-Einstein metrics.
result The 0-instanton obstruction tensor is a conformal invariant related to Weyl curvature, vanishing for smooth 0-instantons.
After recalling the Dirichlet problem at infinity on a Cartan-Hadamard manifold, we discuss what is known and the difference between the two-dimensional and higher-dimensional cases. Turning our attention to the two-dimensional case, we prove that the Dirichlet problem at infinity on a two-dimensional Cartan-Hadamard m…
Normal forms and moduli stacks for flat connections on complex manifolds.
problem Understanding singular flat connections on complex manifolds.
method Introducing homogeneous Lie groupoids and studying their representation theory to prove normal form theorems and moduli space structures.
result Moduli spaces of singular flat connections admit the structure of algebraic quotient stacks.