Study geometric properties of branched covers of hyperbolic manifolds.
problem Geometric analysis of branched covers of hyperbolic manifolds.
method Analysis of geometric properties of covers of hyperbolic manifolds branched along a totally geodesic submanifold.
result Results on geometric properties of branched covers of hyperbolic manifolds.
Study covers of sphere with homeomorphisms lifting property.
problem Finite abelian covers of sphere with lifting homeomorphisms.
method Completely determined covers with specific lifting property.
result Properties of finite abelian covers with lifting homeomorphisms.
New criterion for branched covers between 2-spheres.
problem Existence of specific types of branched covers between 2-spheres.
method Combining complex analysis and topology techniques.
result Established a complete criterion for the existence of branched covers.
The paper studies branched surfaces and their properties.
problem Global topologies of branched surfaces and their maps.
method Explicit construction of maps and study of topological properties.
result New insights into the geography of branched surfaces and their maps.
The study examines surface branch data on spheres with specific properties and computes the number of realizations.
problem Examining surface branch data on spheres with specific properties and computing the number of realizations.
method Analyzing surface branch data on spheres with three branching points and two partitions of degree d, computing the number of realizations based on arithmetic properties of the entries of the third partition.
result In the only case where n is 0, the entries have a common divisor, supporting conjectures by Edmonds-Kulkarny-Stong and Zieve.
A neural network learns to improve verification of neural networks.
problem Efficient verification of neural networks for safety-critical applications.
method A graph neural network (GNN) learns to imitate strong branching heuristics for effective branching in the Branch and Bound (BaB) formulation.
result Reduces the number of branches and verification time by roughly 50% compared to hand-designed strategies.
New results on hypersurfaces show no branch points, improving smoothness.
problem Analyzing area minimising hypersurfaces mod p without branch points.
method General analysis of immersed stable minimal hypersurfaces with alternating orientation.
result Area minimising hypersurfaces mod p do not admit immersed branch points.
Simplifies surface-knots using chart moves involving black vertices.
problem Simplifying branched covering surface-knots.
method Chart moves involving black vertices to simplify surface-knots.
result Properties of simplified branched covering surface-knots with branch points.
The paper details folding of branched covers of the 3-sphere over knots.
problem Understanding folding of branched covers of the 3-sphere over knots.
method Presenting detailed foldings of dihedral covers of the 3-sphere branched over torus knots.
result Detailed foldings of dihedral covers of the 3-sphere branched over torus knots are presented.
Characterizes sphere covers with lifted homeomorphisms.
problem Characterizing sphere covers with lifted homeomorphisms.
method Characterization through cyclic branched covers and homeomorphisms.
result Every homeomorphism of the sphere lifts to a homeomorphism of the covering surface in these cases.
Simplifies surface-knots by adding 1-handles with chart loops.
problem Complexity of branched covering surface-knots.
method Adding 1-handles with chart loops to simplify charts.
result Properties of simplified charts are investigated.
Study shows indecomposable branched coverings over projective plane for surfaces with low Euler characteristic.
problem Decomposability of branched coverings over the projective plane with low Euler characteristic surfaces.
method Analyzing coverings of degree d odd, over the projective plane, with surfaces M having χ(M)≤0. result Realization of indecomposable branched coverings for given data with even defect greater than d. Improved neural network verification using Lagrangian decomposition and parallel algorithms.
problem Formally proving input-output properties of neural networks efficiently.
method Novel bounding and branching algorithms based on Lagrangian Decomposition and activation-based heuristics.
result Significant reduction in verification times, up to 50x faster on adversarial robustness properties.
The article proves properties of Seifert links and their cyclic branched covers.
problem Properties of Seifert links and their cyclic branched covers.
method Left-orderability, co-oriented taut foliations, and L-space properties. result Proves the ADE link conjecture for Seifert links. The paper explores rational functions with 3 branching points on the Riemann sphere.
problem Existence of rational functions with specific branching points.
method Utilizes complex analysis to establish properties of rational functions.
result Identifies new types of exceptional branching data.
The paper characterizes coverings over the projective plane with minimal defect.
problem Characterizing minimal defect branched coverings over the projective plane.
method Characterization through properties of decomposable and indecomposable coverings.
result Extended family of realizations and generalized results on primitive permutation groups.
We study contact manifolds that arise as cyclic branched covers of transverse knots in the standard contact 3-sphere. We discuss properties of these contact manifolds and describe them in terms of open books and contact surgeries. In many cases we show that such branched covers are contactomorphic for smoothly isotopic…
Knots generating infinite subgroup bound rational homology balls.
problem Understanding knots that bound rational homology balls.
method Cyclic branched covers and rational homology balls.
result Infinite two-torsion subgroup in knot concordance group.
New method finds unique branched surfaces in 3-manifolds.
problem Finding unique branched surfaces in 3-manifolds.
method Proving endperiodic maps have periodic splitting sequences.
result Veering branched surfaces are unique up to equivalence.
This is a supplement to [arXiv:1503.03676], where an approach towards a mathematically rigorous definition of the Coulomb branch of a 3-dimensional N=4 SUSY gauge theory was proposed. We ask questions on their expected properties, especially in relation to the corresponding Higgs branch, partly motivated b…
Quantum system linked to knots and branched coverings.
problem Modeling quantum statistical mechanics for knots and branched coverings.
method Construction of a quantum system using knot theory and arithmetic topology.
result The resulting algebra of observables is a noncommutative Bernoulli crossed product.
The paper studies the geometry of Wigner caustics and decomposes curves into parallel arcs.
problem Understanding the geometry and properties of Wigner caustics of curves.
method Decomposing curves into parallel arcs to analyze the Wigner caustic's smooth branches, inflexion points, and singularities.
result New insights into the number of smooth branches, rotation number, inflexion points, and cusp singularities of the Wigner caustic.
Computer proof verifies key differential properties in sphere classifications.
problem Verifying holomorphic properties of meromorphic differentials in sphere classifications.
method Computer-assisted proof using Sage software.
result Holomorphicity of quartic and octic differentials confirmed.
Study finds symplectic fillings' properties for specific contact covers.
problem Determining symplectic fillings' Euler characteristics and signatures.
method Analyzing exact symplectic fillings of contact branched covers.
result Identified links' symplectic fillings' properties.
Analyzes branch points of area-minimizing currents with non-2 planar frequency.
problem Understanding the structure of area-minimizing currents near branch points.
method Intrinsic frequency function and geometric arguments avoiding center manifolds.
result Establishes higher order asymptotics and topological control near branch points.
Study eigenvalues of knot covers to bound knot properties.
problem Bounding knot genera, dealternating number, and alternation number.
method Minimal positive eigenvalues of double branched covers over spanning surfaces.
result Eigenvalue bounds provide estimates for knot properties.
The paper explores representations of specific knot groups and their properties.
problem Investigating representations of branched twist spins with a non-trivial center of order 2.
method Analyzes mSL2(Z3)-representations and dihedral group representations of branched twist spins. result Provides sufficient conditions for the existence of mSL2(Z3)-representations and determines the number of dihedral group representations. The study of universal links in 3-manifolds and their properties.
problem Existence and characterization of universal links in 3-manifolds.
method Analyzing branched coverings and distinguishing between universal and complement universal links.
result Closed spherical 3-manifolds are the only ones admitting universal links.
Study on the geometry of secant caustic of planar curves.
problem Analyzing the secant caustic of planar curves.
method Investigating the geometrical properties of secant caustic, including number of branches, cusps, and inflexion points.
result Detailed investigation of secant caustic properties, especially for rosettes.
Stability of branched immersions with energy constraints.
problem Stability of branched Willmore immersions with bounded energy.
method Refined analysis of fourth-order differential operators with regular singularities.
result Sum of Morse index and nullity is lower semi-continuous.
New model explains low interest rates and large bond market jumps.
problem Explaining recent observations in sovereign bond market.
method Introduces α-CIR model using α-stable Lévy process and branching property. result Unified and parsimonious model for low interest rates and large jumps.
Study determines left-orderable properties of knot covers.
problem Determining left-orderability of knot covers.
method Analyzing 2-bridge knots, including double-twist knots, using cyclic branched covers.
result Identifies specific integers for left-orderability of knot covers.
The paper proves properties of branched covers of specific knots and tori.
problem Investigating the smoothness and diffeomorphism of specific 4-manifolds.
method Analyzing double branched covers of twist-roll spun knots and turned twisted tori, applying techniques to show diffeomorphism.
result Proves that certain 4-manifolds are diffeomorphic to standard manifolds.
The paper introduces a new method to infer phylogenetic trees without bifurcations.
problem Inferring phylogenetic trees with zero-length branches and polytomies.
method Adaptive LASSO-type regularization estimators for phylogenetics.
result Regularization is a practical approach for phylogenetics, revealing zero-length branches.
Paper explores transport maps and measure rigidity in metric spaces.
problem Existence and uniqueness of transport maps in non-branching metric measure spaces.
method Investigates the relationship between transport maps and essentially non-branching measures.
result Essentially non-branching metric measure spaces have unique transport maps under certain conditions.
Study of knotoids using double branched covers and geometric tools.
problem Characterizing and distinguishing knotoids and knots.
method Using double branched covers and geometric invariants.
result Established a 1-1 correspondence between knotoids and knots with strong inversion.
Study normal curves in sub-Finsler Lie groups with specific norms, focusing on branching and face stability.
problem Analyzing normal curves in sub-Finsler Lie groups with different norms.
method Using tools from convex analysis, the Pontryagin Maximum Principle is revisited to express the normal equation as a differential inclusion involving the subdifferential of the dual norm.
result Normal curves in polyhedral norms have controls that locally take values in a single face of a sphere with respect to the norm.
New knots bound rational homology balls, using Alexander polynomials.
problem Finding sliceness obstructions for knots.
method Computing twisted Alexander polynomials and simplifying their calculation.
result New non-slice knots with rational homology ball bounds.
Uniform bounds on ends for non-branching CD spaces with nonnegative curvature outside a compact set.
problem Bounding the number of ends of non-branching CD spaces with nonnegative curvature outside a compact set.
method Adapting Z.-D. Liu's work to prove a ball covering property.
result Uniform bounds on the number of ends of such spaces.
For distinct points p and q in a two-dimensional Riemannian manifold, one defines their mediatrix Lpq as the set of equidistant points to p and q. It is known that mediatrices have a cell decomposition consisting of a finite number of branch points connected by Lipschitz curves. This paper establishes addi…
Improved particle pricing methods for path-dependent options.
problem Efficient simulation of spot price and volatility for path-dependent options.
method Sequential Monte Carlo with branching and resampling.
result Branching algorithms improve pricing performance for path-dependent options.
New findings on convexity of special Lagrangian geodesics.
problem Convexity of special Lagrangian geodesics in space-time.
method Space-time coordinate transformation preserving Lagrangian angle, leading to C2 estimate. result Subsolutions in all branches of the degenerate special Lagrangian equation are bi-convex.
The Birman-Hilden theory is extended to infinite type surfaces and branched covers.
problem Extending Birman-Hilden theory to surfaces of infinite type and branched covers of infinite degree.
method Proving the Birman-Hilden property for fully ramified branched covering maps.
result The mapping class group of a non-orientable surface of infinite type can be realized as a subgroup of the mapping class group of its orientable double cover.
The study proves a conjecture about arborescent links with many twigs.
problem Proving the meridional rank conjecture for arborescent links.
method Using an upper bound on the bridge number in terms of the maximal number of link components of the underlying tree.
result Proves the meridional rank conjecture for arborescent links with specific properties.
The Blum medial axis rigidity is studied in terms of cross ratios and differential geometry.
problem Examining rigidity properties of the Blum medial axis under diffeomorphisms.
method Using cross ratios from projective geometry and differential geometry.
result Cross ratios and differential geometry uniquely determine the angles between smooth sheets.
We give examples of harmonic maps between negatively curved manifolds with special properties. These negatively curved manifolds do not have the homotopy type of a locally symmetric space.
New class of links with specific homology properties and instanton computations.
problem Understanding homology properties of links and their instanton invariants.
method Introduced a new class of links, computed instanton homology, and discussed spectral sequences.
result Computed framed instanton homology for double branched covers of new links.
Proofs for Gauss map properties on surfaces with finite geometric type.
problem Properties of Gauss maps on surfaces with finite geometric type.
method Topological proof and generalization of the little Picard theorem.
result Gauss map can not omit three or more points for minimal and no flat surfaces.