Paper proposes clustering algorithms for data from Union of Polyhedral Cones model.
problem Clustering data from multiple convex polyhedral cones.
method Sparse Subspace Clustering, Least squares approximation, K-nearest neighbor, Spectral Clustering.
result KNN outperforms NCL and LSA in clustering data from UOPC model.
Study shows non-polyhedral structure in moduli spaces for n≥8.
problem Identifying non-polyhedral structure in moduli spaces of pointed stable curves.
method Constructing an extremal non-polyhedral ray via maps on meromorphic strata of differentials.
result Moduli spaces are not Mori Dream Spaces for n≥8.
Conditions for polyhedral Kähler metrics on CP^n with specific singularities.
problem Existence of polyhedral Kähler metrics on complex projective space with specified singularities.
method Parabolic Kobayashi-Hitchin correspondence, linear and quadratic constraints on cone angles.
result Necessary and sufficient conditions for the existence of polyhedral Kähler metrics on CP^n.
The study finds PK cone metrics on complex manifolds near hyperplane arrangements.
problem Finding metrics on complex manifolds near singularities.
method Analyzing flat torsion-free meromorphic connections on \(\mathbb{C}^n\) with simple poles at hyperplanes.
result Metric completion of certain connections yields PK cone metrics on \(\mathbb{C}^n\).
A unique hyperbolic metric is found for each spherical cone-metric on the boundary of a hyperbolizable 3-manifold.
problem Finding a hyperbolic metric for a given spherical cone-metric on the boundary of a hyperbolizable 3-manifold.
method Constructing a strictly polyhedral hyperbolic metric on the 3-manifold such that the given spherical cone-metric is the induced dual metric on the boundary.
result The existence and uniqueness of a strictly polyhedral hyperbolic metric for a given spherical cone-metric on the boundary of a hyperbolizable 3-manifold.
The paper proves regularity for minimal surfaces near polyhedral cones.
problem Understanding the regularity of minimal surfaces near polyhedral cones.
method Adapting Simon's method and establishing C1,α-regularity for minimal varifolds. result Proves C1,α-regularity for minimal varifolds near polyhedral cones. Study convex hyperbolic cone-metrics on 3-manifold boundaries, proving unique bent realizations.
problem Convex hyperbolic cone-metrics on 3-manifold boundaries and their bent realizations.
method Alexandrov-Weyl-type problem, bent metrics, controllably polyhedral, Lipschitz topology.
result Unique bent realizations for convex hyperbolic cone-metrics on 3-manifold boundaries.
We prove that a 3-dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthemore, any hyperbolic metric on the torus with cone singularities of positive curvature can be realized as the induced metric on the boundary of a convex polyhedral cusp. The …
Polyhedral surfaces can be broken down into parallelograms.
problem Decomposing polyhedral surfaces into simpler shapes.
method Analyzing moduli spaces and using geometric properties.
result Polyhedral surfaces with 8 vertices can be decomposed into at most 20 parallelograms.
We investigate the rigidity of hyperbolic cone metrics on 3-manifolds which are isometric gluing of ideal and hyper-ideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyper-ideal hyperbolic polyhedral metrics. It is shown that a hyper-ideal hyperbolic polyhedral metric is determined up to i…
We investigate subgroups of SL (n,Z) which preserve an open nondegenerate convex cone in real n-space and admit in that cone as fundamental domain a polyhedral cone of which some faces are allowed to lie on the boundary. Examples are arithmetic groups acting on selfdual cones, Weyl groups of certain Kac-Moody algebras …
Defines a new version of Turaev-Viro invariants for 3-manifolds with boundaries.
problem Computing the volume of hyperbolic polyhedral 3-manifolds.
method Introduces a relative version of Turaev-Viro invariants for ideally triangulated compact 3-manifolds with boundaries and a coloring on edges.
result Proves the Volume Conjecture for these invariants, suggesting a method to solve the conjecture for hyperbolic 3-manifolds with totally geodesic boundary.
New Calabi-Yau metrics with conical singularities are created near complex lines.
problem Creating Calabi-Yau metrics with conical singularities near complex lines.
method Using branched covering arguments to construct metrics with conical singularities.
result Calabi-Yau metrics with unstable conical singularities are successfully created.
New metrics found with cone singularities along line arrangements.
problem Existence of Calabi-Yau metrics with conical singularities.
method Ricci-flat Kähler metric with cone singularities along weighted line arrangements.
result Existence of a Ricci-flat Kähler metric with cone singularities.
New method for identifying best designs in vector optimization with uncertain feedback.
problem Optimizing vector-valued outcomes with uncertain preferences.
method Stochastic bandit feedback, polyhedral ordering cone, (ε,δ)-PAC Pareto set identification. result Sample complexity characterized and matched by the naïve elimination algorithm.
We show that the cone associated with a moment map for an action of a torus on a contact compact connected manifold is a convex polyhedral cone and that the moment map has connected fibers provided the dimension of the torus is bigger than 2 and that no orbit is tangent to the contact distribution. This may be consider…
Researchers compute the index of a specific operator on contact manifolds.
problem Computing the index of a twisted Dolbeault operator on toric contact manifolds.
method Using equivariant techniques, they localized the symbol to Reeb orbits and applied polytope decomposition.
result They derived an Atiyah-Bott-Lefschetz type formula for the index.
Anti-de Sitter spacetimes embed cone-metrics as bent Cauchy surfaces.
problem Embedding cone-metrics in anti-de Sitter spacetimes.
method Proving embeddings using Fuchsian representations and GHMC spacetimes.
result Unique embeddings of cone-metrics in GHMC anti-de Sitter spacetimes.
We prove that a 3--dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than 2π is the metric of the Gauss image of som…
The paper extends Gelfand-Kapranov-Zelevinsky construction to hyperbolic Riemann surfaces with punctures.
problem Stratifying the space of weight vectors for hyperbolic Riemann surfaces with punctures.
method Analogous to Gelfand-Kapranov-Zelevinsky construction, associates polyhedral fans to hyperbolic Riemann surfaces with punctures.
result The secondary fan of a hyperbolic Riemann surface with punctures is the normal fan of a convex polyhedron, the secondary polyhedron.
This paper uses combinatorial Ricci flow to tackle Thurston's triangulation conjecture.
problem Thurston's triangulation conjecture for hyperbolic 3-manifolds.
method Combinatorial Ricci flow approach to prove convergence and geometric decompositions.
result Combinatorial Ricci flow converges if and only if the triangulation is geometric.
SOC-ICNN expands neural network representational capacity by using conic optimization.
problem Restrictive representational capacity of ReLU-based ICNNs.
method Proposes SOC-ICNN architecture that uses Second-Order Cone Programming.
result SOC-ICNN strictly expands representational space without increasing complexity.
Method provides bounds for sparse PCA and nuclear norm problems.
problem Semidefinite optimization problems (SDOs).
method Cutting-plane method with focus on initial outer approximation as a second-order cone approximation.
result Method provides bound gaps of 0.5-6.5% for sparse PCA problems with 1000 covariates and solves nuclear norm problems over 500x500 matrices.
Open, connected, saturated sets W without holonomy in codimension one foliations play key roles as fundamental building blocks. Here, for the case of foliated 3-manifolds, we produce a finite system of closed, convex, non-overlapping polyhedral cones in the first cohomology of W with real coefficients such that the iso…
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 $S^1\timesS^2$, S3 or singular S2. The singular set of M(ψ) can have se…
The study constructs periodic surfaces using graph theory and applies cyclically branched coverings to identify their conformal type.
problem Constructing and identifying the conformal type of periodic surfaces with a given geometric structure.
method Graph theory and cyclically branched coverings.
result Explicit cone metrics on compact Riemann surfaces can be realized as the quotient of triply periodic polyhedral surfaces.
This paper studies deformations of hyperbolic surfaces with special structures.
problem Infinitesimal deformations of hyperbolic surfaces with boundary and ideal vertices.
method Description of the admissible cone of deformations in terms of the arc complex.
result Realization of the admissible cone and its faces as arc complexes for specific surface families.
Unique hyperbolic affine spheres associated with convex cones in 3D.
problem Characterizing and classifying self-associated convex cones in 3D.
method Using affine spheres, hyperbolic geometry, and differential equations.
result Complete classification of self-associated convex cones and their associated affine spheres.
David Gabai showed that disk decomposable knot and link complements carry taut foliations of depth one. In an arbitrary sutured 3-manifold M, such foliations F, if they exist at all, are determined up to isotopy by an associated ray [F] issuing from the origin in H^1(M;R) and meeting points of the integer lattice H^1(M…
A method to identify important features without solving the full problem.
problem Identifying important features in high-dimensional data.
method Persistent reduction using extreme ray identification on a polyhedral cone.
result A subset of features can be guaranteed to have zero coefficients in all optimal solutions.
The paper studies flat metrics with cone singularities on surfaces, decomposing moduli spaces into polyhedra.
problem Moduli spaces of flat metrics with cone singularities on surfaces.
method Geometry of Euclidean and Minkowski polyhedra, foliations, hyperbolic and spherical convex polyhedra.
result Moduli spaces of flat metrics with cone singularities have natural decompositions into polyhedra.
Study convex hulls of orbits for compact groups, defining new invariants related to polynomial degrees.
problem Understanding properties of convex hulls of coadjoint orbits of compact groups.
method Introduce partial convex hulls and use them to define numerical invariants.
result Orbits with new invariants form rational convex polyhedral cones related to Littlewood-Richardson cones.
Study positive scalar curvature on manifolds with skeleton singularities.
problem Positive scalar curvature on manifolds with skeleton singularities.
method Polyhedral comparison theory and edge metrics.
result Edge singularities do not affect the Yamabe type in all dimensions.
Study on likelihood functions, associative equations, and Frobenius manifolds.
problem Maximum likelihood estimation and associativity equations in statistical models.
method Analyzes the cone of concentration matrices, log-likelihood function, and Frobenius manifolds.
result Maximum likelihood degree is indexed by components of Frobenius residuals.
Non-rigidity degree of a lattice L, nrdL, is dimension of the L-type domain to which L belongs. We complete here the table of nrd's of all root lattices and their duals; namely, the hardest remaining case of Dn∗, and the case of E7∗ are decided. We describe explicitly the L-type domain D(Dn∗)…
Let M be a compact hyperkahler manifold with maximal holonomy (IHS). The group H2(M,R) is equipped with a quadratic form of signature (3,b2−3), called Bogomolov-Beauville-Fujiki (BBF) form. This form restricted to the rational Hodge lattice H1,1(M,Q), has signature (1,k). This gives a hyperbolic Rieman…
Let a torus T act effectively on a compact connected cooriented contact manifold, and let Psi be the natural momentum map on the symplectization. We prove that, if dim T > 2, the union of the origin with the image of Psi is a convex polyhedral cone, the non-zero level sets of Psi are connected (while the zero level set…
New theory assesses smoothness of polyhedral surfaces.
problem Evaluate smoothness of polyhedral surfaces.
method Incorporates geometry of polyhedral surfaces, proposes new notions of smoothness.
result Seemingly mild conditions significantly limit polyhedral surface shapes.
Study of hyperbolic polyhedral surfaces with regular faces.
problem Understanding the properties of hyperbolic polyhedral surfaces with regular faces.
method Combinatorial and geometric analysis of hyperbolic polyhedral surfaces with regular faces.
result There is a gap between the areas of non-smooth hyperbolic polyhedral surfaces and smooth hyperbolic surfaces.
Study on discrete Gaussian curvature for polyhedral surfaces.
problem Discretization of Gaussian curvature for polyhedral surfaces.
method Generalization of discrete conformal equivalence to define discrete Gaussian curvature and classify polyhedral surfaces.
result Existence of polyhedral surfaces with constant discrete Gaussian curvature in every discrete conformal class.
A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying …
Polyhedral semantics for intermediate logics; Nerve Criterion ensures completeness.
problem Characterize polyhedrally-complete intermediate logics.
method Developed Nerve Criterion to characterize polyhedrally-complete logics combinatorially.
result Nerve Criterion provides a necessary and sufficient condition for polyhedrally-completeness.
Locally finite complexes with polyhedral CAT(0) metrics are arborescent.
problem Characterizing locally finite complexes with CAT(0) metrics. method Proving arborescence for complexes with polyhedral CAT(0) metrics. result Locally finite complexes with polyhedral CAT(0) metrics are arborescent. Computational knot theory and 3-manifold topology have seen significant breakthroughs in recent years, despite the fact that many key algorithms have complexity bounds that are exponential or greater. In this setting, experimentation is essential for understanding the limits of practicality, as well as for gauging the …
The paper proves rigidity of bordered polyhedral surfaces using variational principles.
problem Determining the rigidity of bordered polyhedral surfaces.
method Using the variational principle, the paper shows that bordered polyhedral surfaces are determined by boundary values and discrete curvatures on interior edges.
result The paper re-proves the classical result that two Euclidean or hyperbolic cyclic polygons are congruent if their side lengths are equal.
Characterizes pseudo-Anosov mapping classes on general marked surfaces.
problem Stability of mapping classes on marked surfaces.
method Cluster algebraic description and reduction procedure of mapping classes.
result Characterizes pseudo-Anosov mapping classes in terms of uniform sign stability.
We develop a method to find a set of diminimal polyhedral maps on the torus from which all other polyhedral maps on the torus may be generated by face splitting and vertex splitting. We employ this method, though not to its completion, to find 53 diminimal polyhedral maps on the Torus.
Characterizes metrics on triangulated surfaces using glued Euclidean triangles.
problem Describing metrics on triangulated surfaces constructed from glued Euclidean triangles.
method Carefully constructing polyhedral metrics and proving their uniqueness.
result Polyhedral metrics are the only intrinsic metrics preserving Euclidean triangle lengths.