The paper solves a problem related to exterior multiplication with singularities.
problem Finding sections of a p-exterior product given sections of a vector bundle.
method Analyzes the sufficiency of a condition involving regular singularities and depth of ideals.
result Shows that the condition is sufficient for existence of a continuous right inverse in smooth, real analytic, and holomorphic categories.
Study pseudo-holomorphic disks in real analytic hypersurfaces using exterior differential systems.
problem Existence of pseudo-holomorphic disks in non-integrable real analytic hypersurfaces.
method Theory of exterior differential systems.
result Non-existence of certain equivalent structures in the non-integrable case.
Let G be a finite group of complex n by n unitary matrices generated by reflections acting on C^n. Let R be the ring of invariant polynomials, and χbe a multiplicative character of G. Let Ω^χbe the R-module of χ-invariant differential forms. We define a multiplication in Ω^χand show that under this multiplication Ω^χha…
A graph multilink is a link with multiplicities in a homology 3-sphere whose exterior is a graph manifold. In this Note, we compute the Novikov homology of graph multilinks. As a corollary, we give a majoration for the number of Novikov modules on a given graph link.
Study exterior maximal surfaces with precise asymptotic behavior.
problem Maximal surfaces in exterior domains with prescribed asymptotic behavior.
method Analyze the maximal surface equation and prove the exterior Dirichlet problem's uniqueness.
result Uniquely solvable exterior Dirichlet problem with given asymptotic behavior.
Introduces exterior differential systems for grad students.
problem No specific problem stated; introduces a topic.
method No specific method mentioned; brief introduction.
result Introduces exterior differential systems.
New knots bound multiple non-isotopic ribbon disks.
problem Finding knots that bound multiple non-isotopic ribbon disks.
method Classification of fibered, homotopy-ribbon disks for generalized square knots.
result Infinitely many knots bound infinitely many pairwise non-isotopic ribbon disks.
Parallelizes DEC on curved meshes using group actions.
problem Efficiently solving DEC operators on curved and 3D meshes.
method Universal block-diagonalization framework for d and ⋆ operators, exploiting group actions. result Block-diagonal structure inherited by operators, enabling parallel solvers.
Characterizes alternating link exteriors using cubed complexes.
problem Understanding the structure of alternating link exteriors.
method Introducing signed BW cubed-complexes and characterizing their homeomorphism to link exteriors.
result Characterization of alternating link exteriors in terms of cubed complexes.
Classifies essential annuli in a genus two handlebody exterior.
problem Classifying essential annuli in a genus two handlebody exterior.
method Building on JSJ-graph classification and essential annuli classification.
result Characterizes the numbers of different types of essential annuli in an infinite family.
The paper classifies symmetries and determines exterior types of knotted handlebodies.
problem Classifying symmetries and determining exterior types of knotted handlebodies.
method Analysis of hyperbolic knots with non-integral toroidal Dehn surgeries.
result Determines symmetries and exterior types of knotted genus two handlebodies.
We give a topological characterisation of alternating knot exteriors based on the presence of special spanning surfaces. This shows that alternating is a topological property of the knot exterior and not just a property of diagrams, answering an old question of Fox. We also give a characterisation of alternating link e…
Discrete exterior calculus shows natural properties of wedge product and averaging.
problem Naturalness of discrete exterior calculus operations.
method Showed naturalness of discrete wedge product and averaging interpretation.
result Discrete wedge product is natural and equals Wilson's cochain product.
The study classifies area-maximizing hypersurfaces with singularities and exterior domains.
problem Classifying area-maximizing hypersurfaces with singularities and exterior domains.
method Complete classification for entire area maximizing hypersurfaces with isolated singularities. Construction of an example. Partial result on asymptotic behavior for exterior domains. Solvability of exterior Dirichlet problems.
result Complete classification and partial results on asymptotic behavior for area maximizing hypersurfaces.
The study finds infinite knot exteriors with meridional surfaces of any genus and boundary components.
problem Existence of meridional essential surfaces in knot exteriors.
method Analyzing knot exteriors and their surfaces.
result Existence of infinitely many knot exteriors with meridional essential surfaces of any genus and boundary components.
Slice and ribbon disks with identical exteriors found.
problem Identifying slice and ribbon disks with the same exterior.
method Constructed an infinite family of slice disks proving they are also ribbon disks.
result Found an infinite family of slice and ribbon disks with identical exteriors.
Extends exterior diff. sys. to Lie algebroids with examples.
problem Invariant inverse problem of the calculus of variations
method Extends exterior differential systems to Lie algebroids, defines integral manifolds.
result Defines integral manifolds for exterior diff. systems on Lie algebroids.
The study finds infinite meridional essential surfaces in hyperbolic knot exteriors of various genera.
problem Existence of meridional essential surfaces in hyperbolic knot exteriors.
method Essential tangle decompositions and meridional essential embeddings.
result Infinite collection of hyperbolic knots with meridional essential surfaces of arbitrary genus and boundary components.
We extend our earlier work in [TZ1], where an analytic approach to the Guillemin-Sternberg conjecture [GS] was developed, to cases where the Spinc-complex under consideration is allowed to be further twisted by certain natural exterior power bundles. The main result is a weighted quantization formula in the presence…
Algorithm finds knot diagrams from exterior triangulations.
problem Finding knot diagrams from their exteriors.
method First practical algorithm for finding diagrams from triangulations of exteriors.
result First diagrams for 23 link exteriors with over 2,500 crossings.
It is shown that two definitions for an exterior differential in superspace, giving the same exterior calculus, yet lead to different results when applied to the Poisson bracket. A prescription for the transition with the help of these exterior differentials from the given Poisson bracket of definite Grassmann parity t…
Evolutionary forms, as well as exterior forms, are skew-symmetric differential forms. But in contrast to the exterior forms, the basis of evolutionary forms is deforming manifolds (with unclosed metric forms). Such forms possess a peculiarity, namely, the closed inexact exterior forms are obtained from that. The closur…
We extend our earlier work in [TZ1], where an analytic approach to the Guillemin-Sternberg conjecture [GS] was developed, to cases where the Spin^c-complex under consideration is allowed to be further twisted by certain exterior power bundles of the cotangent bundle. The main result is a weighted quantization formula i…
Paper introduces an invariant to distinguish handlebody-knot exteriors.
problem Challenges in distinguishing handlebody-knots with homeomorphic exteriors.
method Defined an invariant (annulus diagram) using characteristic submanifold theory and Koda-Ozawa classification for essential annuli.
result The annulus diagram can differentiate handlebody-knot families.
A new discrete calculus for bundle-valued forms is proposed and validated.
problem Discretization of exterior calculus for bundle-valued forms.
method Discretization of Cartan's exterior calculus for differential forms with values in vector bundles.
result The proposed discrete operator mimics the continuous exterior covariant derivative and ensures numerical convergence.
We describe and construct here pseudo-Hermitian structures θ without torsion (i.e. with transversal symmetry) whose Webster-Ricci curvature tensor is a constant multiple of the exterior differential dθ. We call these structures pseudo-Hermitian Einstein and our result states that they all can be derived locally fro…
Paper shows non-convexity in solutions to Hessian equations.
problem Non-convexity of solutions to k-Hessian equations in exterior domains. method Examples and new proof for quasiconvexity of harmonic functions.
result Solutions to k-Hessian equations are not quasiconvex in exterior domains. Extends Minkowski stability proof to minimal decay assumptions.
problem Global stability of Minkowski spacetime with minimal decay.
method Extends Christodoulou-Klainerman's proof to minimal decay assumptions.
result Exterior stability of Minkowski holds with borderline decay.
A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector spaces of finite and infinite dimension, fractional differential form spaces. The def…
Study of group orderability using tensor and exterior squares.
problem Circular orderability of groups and torsion elements.
method Analysis of tensor and exterior squares of groups.
result Circularly orderable groups have left-orderable tensor and exterior squares under certain conditions.
We give an elegant formulation of the structure equations (of Cartan) and the Bianchi identities in terms of exterior calculus without reference to a particular basis and without the exterior covariant derivative. This approach allows both structure equations and the Bianchi identities to be expressed in terms of forms…
New disks found with similar outer shapes.
problem Finding similar outer shapes for Lagrangian disks.
method Modified construction of Abe and Tange.
result Arbitrarily large families of disks found.
Paper solves overdetermined k-Hessian equation in exterior domains.
problem Overdetermined problem for k-Hessian equation in exterior domains. method Combining integral identities and geometric inequalities, derived general monotone formulas.
result Established general monotone formulas for k-admissible solutions. Survey revisits vector calculus results using exterior derivative and provides a new formulation of Stokes' theorem.
problem Classical results in vector calculus and analysis.
method Generalised perspective on the exterior derivative and a higher-dimensional Mean Value Theorem.
result Provides a natural formulation of Stokes' theorem and a practical algorithm for exterior differentiation.
Maxwell meets Korn: A New Coercive Inequality for Tensor Fields with Square-Integrable Exterior Derivative
Geometric DEC solves Poisson on general triangulations.
problem Solving Poisson equation on arbitrary triangulations.
method Revisited Discrete Exterior Calculus (DEC) for general triangulations using Vector Calculus and Matrix Algebra.
result DEC solutions match FEML for Poisson equation.
Extends Young integral to Hölder differential forms in arbitrary dimensions.
problem Extending the Young integral to Hölder differential forms in arbitrary dimensions.
method Introducing a complex of cochains, α-fractional charges, and defining the exterior product between them.
result The exterior product between α-fractional and β-fractional charges is defined when α + β > 1.
Exterior differential systems are given, and their Cartan characters calculated, for Maxwell and SU(2)-Yang-Mills equations in dimensions from three to six.
The Whitehead link exterior lacks most Euler class taut foliations.
problem Existence of co-orientable taut foliations with specific Euler classes.
method Combining foliation theory techniques and topological obstructions.
result Most lattice points in the dual Thurston ball cannot be Euler classes of co-orientable taut foliations.
Study of Lee form exterior derivative in almost Hermitian manifolds.
problem Understanding the exterior derivative of Lee form in almost Hermitian manifolds.
method Analyzes the Lee form θ of almost Hermitian manifolds and its exterior derivative dθ in terms of intrinsic torsion and Riemannian curvature tensor components. result Proves that the Rω-component of dθ is always zero and provides expressions for other components. We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, i…
We construct a two-parameter covariant differential calculus on the quantum h-exterior plane. We also give a deformation of the two-dimensional fermionic phase space.
Brunnian theta curves in 3D spheres have hyperbolic exteriors.
problem Characterizing Brunnian theta curves in 3D spheres.
method Sutured manifold theory, classification of annuli, and Thurston's hyperbolic geometry.
result Brunnian theta curves of low bridge number have hyperbolic exteriors with totally geodesic boundary.
Researchers prove constant mean curvature graphs in hyperbolic 3-space for specific domains.
problem Existence of hyperbolic Killing graphs with constant mean curvature in exterior domains.
method Existence proof using CMC graphs and Killing vector fields.
result Existence of hyperbolic Killing graphs of constant mean curvature H in exterior domains.
The paper extends circle pattern theory to include obtuse angles.
problem Existence and rigidity of circle patterns with non-obtuse exterior intersection angles.
method Topological degree theory, variational principle, Teichmüller theory, Sard's Theorem.
result The Circle Pattern Theorem is generalized to include obtuse angles.
We provide a classification of the essential surfaces of non-negative Euler characteristic in the exteriors of genus two handlebodies embedded in the 3-sphere.
I describe work in progress with Baryshnikov and Zharnitsky on periodic billiard orbits that leads one to an exterior differential system (EDS). I then give a brief introduction to EDS illustrated by several examples.
Characterizes 3-punctured spheres in non-hyperbolic link exteriors.
problem Characterizing 3-punctured spheres in non-hyperbolic link exteriors.
method Analyzes non-hyperbolic 3-component links and multibranched surfaces.
result Existence of embeddings of multibranched surfaces in the 3-sphere.