Study shows no smooth embeddings of rational homology balls into complex projective plane.
problem Embedding rational homology balls into complex projective plane.
method Elementary arguments to prove non-existence of almost complex embeddings.
result No smooth embeddings of rational homology balls into complex projective plane.
CR embeddings in complex spaces for specific Lie groups.
problem Embedding specific Lie groups in complex spaces.
method Using integrable complex structures on subbundles of tangent bundles.
result CR embeddings possible as the edge of wedges in complex domains.
Generalizes embedding complex Grassmannians into quadrics.
problem Holomorphic isometric embeddings of complex Grassmannians into quadrics.
method Generalization of do Carmo-Wallach theory for moduli spaces.
result Moduli spaces of embeddings discussed.
We show that a pseudo-holomorphic embedding of an almost-complex 2n-manifold into almost-complex (2n+2)-Euclidean space exists if and only if there is a CR regular embedding of the 2n-manifold into complex (n+1)-space. We remark that the fundamental group does not place any restriction on the existence of e…
Proves uniqueness of embedding complex manifold into infinite-dimensional space.
problem Balanced embedding of non-compact complex manifold into infinite-dimensional projective space.
method Fine estimates of asymptotics of a balanced embedding.
result Uniqueness of embedding proven.
In this article, we demonstrate methods for the local removal and modification of complex tangents to embeddings of S3 into C3. In particular, given any embedding of S3 and a neighborhood of the complex tangents of the embedding, we show that there exists a (C0-close) totally real embedding which a…
Every 4-manifold can be smoothly embedded in complex projective 3-space.
problem Embedding 4-manifolds in complex projective spaces.
method Analyzing properties of 4-manifolds and complex projective spaces.
result Every closed orientable smooth 4-manifold admits a smooth embedding in $\CP^3$.
Paper addresses hidden faces in configuration space integrals for embeddings.
problem Understanding hidden faces in configuration space integrals for long embeddings.
method Modified configuration space integrals incorporating acyclic bar complex of a dg algebra.
result Cochain map from new graph complex to de Rham complex of embeddings modulo immersions.
Given a simplicial complex K, we consider several notions of geometric complexity of embeddings of K in a Euclidean space Rd: thickness, distortion, and refinement complexity (the minimal number of simplices needed for a PL embedding). We show that any n-complex with N simplices which topologically…
We prove that any convex domain of C^2 carries properly embedded complete complex curves. In particular, we exhibit the first examples of complete bounded embedded complex curves in C^2
We prove that any compact almost complex manifold (M,J) of real dimension 2m admits a pseudo-holomorphic embedding in a Euclidean space of dimension 4m+2, endowed with a suitable non-standard almost complex structure. Moreover, we give a necessary and sufficient condition, expressed in terms of the Segre class…
The paper shows examples of 2-complexes that can't be embedded in R^4, hiding obstructions in higher Milnor invariants.
problem Embedding 2-complexes in R^4 with hidden obstructions.
method Provides examples of 2-complexes and families of PL immersions that hide embedding obstructions.
result Embedding obstructions vanish for the given examples, answering a question in Avramidi-Okun-Schreve's paper.
We show that any compact almost-complex manifold of complex dimension m can be pseudo-holomorphically embedded in R^(6m) equipped with a suitable almost-complex structure.
Random complexes can be embedded linearly if certain conditions on parameters are met.
problem Embedding random simplicial complexes linearly in Euclidean space.
method Established strict inequalities on parameters for linear embedding into R^(2d).
result Necessary and sufficient conditions for linear embedding of random complexes.
Minimal dimensions found for flag manifolds embeddings.
problem Finding the smallest dimensions for flag manifolds embeddings.
method Equivariant embeddings of orthogonal and unitary groups acting on real and complex flag manifolds.
result Minimal dimensions achieved at isospectral models.
We show that, for a closed orientable n-manifold, with n not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex (n-1)-space ensures the existence of a totally real embedding into complex n-space. This implies that a closed orientable (4k+1)-manifold with non-vanishing Kervaire semi-characteri…
Embeds CR manifolds into complex spaces using equivariant actions.
problem Embedding strongly pseudoconvex CR manifolds into complex spaces.
method Equivariant CR maps and quotient maps.
result Universal quotient map property for CR manifolds.
Study holomorphic isometric embeddings of a Grassmannian into quadrics.
problem Holomorphic isometric embeddings of complex Grassmannian into quadrics.
method Generalization of do Carmo--Wallach theory to study moduli space.
result Moduli space of embeddings up to equivalence discussed.
We construct examples of complex algebraic surfaces not admitting normal embeddings (in the sense of semialgebraic or subanalytic sets) with image a complex algebraic surface.
Minimal simplicial complexes in high dimensions always contain complex links.
problem Existence of complex links in high-dimensional embeddings.
method Demonstrated through minimal simplicial complexes in R2n. result Minimal simplicial n-complexes inevitably contain a nonsplittable two-component link. Study of pseudoconvex 3-manifolds in complex surfaces.
problem Understanding pseudoconvex 3-manifolds in complex surfaces.
method Develop tools for constructing topologically pseudoconvex embeddings and classify almost-complex structures.
result Every closed, oriented 3-manifold can be embedded in a compact complex surface realizing any homotopy class of almost-complex structures.
In statistical relational learning, the link prediction problem is key to automatically understand the structure of large knowledge bases. As in previous studies, we propose to solve this problem through latent factorization. However, here we make use of complex valued embeddings. The composition of complex embeddings …
Hardness proven for embedding simplicial complexes in R^d, especially for k-dimensional ones.
problem Recognizing almost embeddability of k-dimensional complexes in R^d.
method NP-hardness proof using configuration spaces and preimage cycle properties.
result Embedding obstruction is incomplete for k-dimensional complexes in R^d.
New embeddings capture local structure in complex networks.
problem Embeddings cannot capture local structure in complex networks.
method Logistic Principal Component Analysis (LPCA) algorithm for exact low-rank representations.
result Exact low-rank representations of real-world networks are possible.
Study local and global aspects of complex plane curve embeddings.
problem Local and global problems of complex plane curve embeddings.
method Braid monodromy, local and global analysis.
result Historical progress in understanding complex plane curve embeddings.
3D Schoenflies theorem for simply-connected 2-complexes.
problem Embedding simply-connected 2-complexes in 3-space uniquely.
method Proving a 3-dimensional Schoenflies theorem for 3-connected link graphs.
result Essentially unique locally flat embedding into 3-sphere.
Every lens space has a locally flat embedding in a connected sum of 8 copies of the complex projective plane and a smooth embedding in n copies of the complex projective plane for some positive integer n. We show that there is no n such that every lens space smoothly embeds in n copies of the complex projective plane.
Unsupervised domain adaptation aims to generalize the hypothesis trained in a source domain to an unlabeled target domain. One popular approach to this problem is to learn domain-invariant embeddings for both domains. In this work, we study, theoretically and empirically, the effect of the embedding complexity on gener…
Embeds complex into higher-dimensional pseudomanifold.
problem Embedding complex structures into higher-dimensional spaces.
method Deformation retraction and embedding into pseudomanifolds.
result Finite d-dimensional simplicial complex can be embedded as a retract in a closed (2d−1)-dimensional pseudomanifold. In statistical relational learning, knowledge graph completion deals with automatically understanding the structure of large knowledge graphs---labeled directed graphs---and predicting missing relationships---labeled edges. State-of-the-art embedding models propose different trade-offs between modeling expressiveness, …
New research shows graph embeddings fail to capture key network properties.
problem Graph embeddings fail to capture salient properties of complex networks.
method Mathematical proof and empirical study of various embedding techniques.
result Any successful graph embedding must have a rank nearly linear in the number of vertices.
We prove that 2-dimensional simplicial complexes whose first homology group is trivial have topological embeddings in 3-space if and only if there are embeddings of their link graphs in the plane that are compatible at the edges and they are simply connected.
Obtaining continuous representations of structural data such as directed acyclic graphs (DAGs) has gained attention in machine learning and artificial intelligence. However, embedding complex DAGs in which both ancestors and descendants of nodes are exponentially increasing is difficult. Tackling in this problem, we de…
We consider an embedding of a 2-dimensional CW complex into the 3-sphere, and construct it's dual graph. Then we obtain a homogeneous system of linear equations from the 2-dimensional CW complex in the first homology group of the complement of the dual graph. By checking that the homogeneous system of linear equa…
Engel structures on bundles over 3-manifolds in complex 3-space.
problem Embedding bundles over 3-manifolds into complex 3-space with Engel structures.
method Sufficient condition for S1-bundles to admit immersions/embeddings with complex tangencies defining Engel structures. result Every oriented S1-bundle over a closed, oriented 3-manifold admits an immersion with complex tangencies defining Engel structures. New Fuchsian groups found with special embedding properties.
problem Finding new Fuchsian groups with specific embedding properties.
method Using period domains and properties of complex hyperbolic surfaces.
result First cocompact nonarithmetic Fuchsian groups with modular embedding not commensurable with triangle groups.
Unified framework for complex-valued eigenfunctions on Riemannian symmetric spaces.
problem Finding a unified scheme for complex-valued eigenfunctions on Riemannian symmetric spaces.
method Employing the Cartan embedding for classical compact Riemannian symmetric spaces and quaternionic Grassmannians.
result Construction of new eigenfunctions on quaternionic Grassmannians.
Existence of balanced embedding proved for complex manifold into infinite-dimensional space.
problem Balanced embedding of non-compact complex manifolds into infinite-dimensional projective space.
method Gradient flow in a Hilbert space, long-time existence established by perturbation, convergence depends on a priori bounds.
result Existence of balanced embedding proved in a model case.
Symplectic embeddings of balls into specific manifolds are studied, with restrictions and obstructions identified.
problem Understanding symplectic embeddings of balls into complex projective spaces, tori, and K3 surfaces.
method Analyzing embeddings with respect to complex structures compatible with the symplectic form and identifying obstructions.
result Symplectic volume is the primary obstruction for the existence of embeddings of balls into certain manifolds.
The paper uses Seshadri constants to construct symplectic ellipsoid embeddings.
problem Constructing symplectic embeddings of ellipsoids.
method Exploring weighted blow-ups and Seshadri constants to demonstrate symplectic embeddings.
result Illustrates constructions of ellipsoid fillings and embeddings.
We exhibit an infinite family of rational homology balls which embed smoothly but not symplectically in the complex projective plane. We also obtain a new lattice embedding obstruction from Donaldson's diagonalisation theorem, and use this to show that no two of our examples may be embedded disjointly.
Classifies 4-manifolds with genus one horizontal handlebody decomposition.
problem Classifying 4-manifolds with specific properties.
method Classification using Euler characteristic and handlebody decomposition.
result Found a large family of rational homology balls that embed into CP2. Explains Hodge theory and Kodaira embedding theorem for complex manifolds.
problem Understanding complex manifold properties and their geometric implications.
method Expository review of harmonic forms, Hodge theory, and Kodaira embedding theorem.
result Establishes connections between de Rham cohomology, Dolbeault cohomology, and projective varieties.
Study smooth embeddings of line configurations in complex projective plane.
problem Realizing line configurations as smooth 2-spheres.
method Lattice-theoretic arguments based on Donaldson's diagonalization theorem.
result Established a stronger obstruction in the smooth category.
In the mid-1980's, M. Gromov used his machinery of the h-principle to prove that there exists totally real embeddings of S3 into C3. Subsequently, Patrick Ahern and Walter Rudin explicitly demonstrated such a totally real embedding. In this paper, we consider the generic situation for such embeddings, …
For a given embedded Lagrangian in the complement of a complex hypersurface we show existence of a holomorphic disc in the complement having boundary on that Lagrangian.
New symplectic caps and embeddings found in complex projective plane.
problem Embeddings of homology balls in complex projective plane.
method Handlebody construction of symplectic caps and embeddings.
result First examples of symplectic handlebody decompositions of a closed symplectic 4-manifold.
Study of metrics on spheres and their complex structure properties.
problem Identifying metrics on spheres and their complex structure properties.
method Identify metrics via Nash isometric embeddings, use isotopic extension theorem, and analyze extrinsic quantities.
result No sphere of dimensions 6 or higher can be diffeomorphic to a complex manifold.