The paper defines invariants for almost graph embeddings and explores their properties.
problem Understanding the properties and limitations of almost graph embeddings in the plane.
method Introducing and analyzing integer invariants (winding number, Wu numbers) for almost embeddings.
result Some values of invariants are realizable for almost embeddings but not for embeddings.
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…
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.
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…
The paper explores invariants of graph drawings in the plane.
problem Understanding the invariants of almost embeddings of graphs in the plane.
method Proves relations between invariants, connects to homology, constructs examples.
result Constructs almost embeddings realizing some values of invariants.
Study the geometry of embedding spaces for almost complex manifolds.
problem Investigate the embeddability of compact almost complex manifolds into foliated algebraic varieties.
method Introduce a more general category of universal embedding spaces and elucidate their geometric structure.
result Characterize the integrability locus of integrable almost complex structures.
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.
Tubular neighborhoods play an important role in differential topology. We have applied these constructions to geometry of almost Hermitian manifolds. At first, we consider deformations of tensor structures on a normal tubular neighborhood of a submanifold in a Riemannian manifold.Further, an almost hyperHermitian struc…
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.
A map f:K→Rd of a simplicial complex is an almost embedding if f(σ)∩f(τ)=∅ whenever σ,τ are disjoint simplices of K. Theorem. Fix integers d,k≥2 such that d=23k+1. (a) Assume that P=NP. Then there exists a finite k-dimensional complex K that does not admit an …
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.
The paper constructs unbounded symplectic embeddings of rational homology balls into surfaces.
problem Bounding symplectic embeddings of rational homology balls into surfaces.
method Using Mori's theory of flips and mutations of polygons.
result Unbounded sequences of symplectically embedded rational homology balls into surfaces.
The paper explores winding numbers of almost embeddings of a 4-vertex graph in the plane.
problem Understanding the winding numbers of almost embeddings of a 4-vertex graph in the plane.
method Constructing examples to show the only relation between the winding numbers of cycles in the graph.
result The sum of winding numbers is odd, and this is the only relation between them.
Identifies metrics on manifolds and their embeddings into spheres, characterizing constant curvature metrics.
problem Characterizing metrics on manifolds and their embeddings into spheres.
method Identifies metrics on manifolds and their embeddings into spheres, characterizes metrics of constant scalar curvature, and uses Yamabe metrics and almost Hermitian structures.
result Characterizes metrics of constant scalar curvature by properties of extrinsic quantities of their embeddings.
Study Markov staircases in symplectic embeddings of rational homology ellipsoids.
problem Symplectic embeddings of rational homology ellipsoids into the complex projective plane.
method Analysis of almost toric fibrations and Hamiltonian isotopies.
result Existence of an infinite staircase for each Markov triple.
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.
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. 3-manifolds can be embedded almost in the spine of a trisected 4-manifold.
problem Embedding 3-manifolds in trisected 4-manifolds.
method Defining and using the spine of a trisected 4-manifold, isotoping 3-manifolds to lie in the spine, and calculating bounds based on graph distances.
result Every 3-manifold can be embedded almost in the spine of a minimal genus trisection of connect sums of S2ildeimesS2. The paper introduces the spirality character of the almost fiber part for a closed essentially immersed subsurface of a closed orientable aspherical 3-manifold, which generalizes an invariant due to Rubinstein and Wang. The subsurface is virtually embedded if and only if the almost fiber part is aspiral, and in this ca…
2D complexes can be almost-embedded in 4D space without self-intersections.
problem Understanding the embedding properties of 2-dimensional complexes in 4-dimensional space.
method Analyzing specific 2-complexes constructed by Freedman-Krushkal-Teichner and showing they can be PL immersed in R4 without self-intersections. result Many 2-complexes can be PL almost-embedded in R4 with singularities only as self-intersections of some 2-cells. Researchers describe a specific type of submanifolds in Euclidean space.
problem Understanding inhomogeneous almost symmetric submanifolds.
method Completely describing submanifolds as unions of parallel symmetric submanifolds.
result Described inhomogeneous properly embedded almost symmetric submanifolds as unions of symmetric submanifolds.
Time-delayed embeddings avoid self-intersections for high enough delay.
problem Analyzing self-intersections in time-delayed embeddings.
method Study of time-delayed coordinate maps for diffeomorphisms on compact manifolds.
result For high enough delay, time-delayed embeddings avoid self-intersections almost everywhere.
We found stronger counterexamples to the topological Tverberg conjecture.
problem Proving the topological Tverberg conjecture for non-prime power values of r and large enough d.
method Using higher-dimensional counterexamples and generalizations of Mabillard-Wagner and Özaydin theorems.
result We produced counterexamples for almost r-embeddings in higher dimensions.
We prove that a compact Riemann surface can be realized as a pseudo-holomorphic curve of (R4,J), for some almost complex structure J if and only if it is an elliptic curve. Furthermore we show that any (almost) complex 2n-torus can be holomorphically embedded in (R4n,J) for a suitable almo…
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.
This paper is the second in a series where we attempt to give a complete description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an emb…
We prove necessary and sufficient conditions for a smooth surface in a 4-manifold X to be pseudoholomorphic with respect to some almost complex structure on X. This provides a systematic approach to the construction of pseudoholomorphic curves that do not minimize the genus in their homology class.
Spheres in curve complexes are almost simply connected.
problem Understanding connectivity of spheres in curve complexes.
method Defining spheres as induced subgraphs and showing almost simple connectivity.
result Spheres in high-complexity surfaces are almost simply connected.
FREDE efficiently embeds graphs using linear space and guarantees quality.
problem Efficiently embedding graphs with quality guarantees and linear space complexity.
method FREDE combines matrix sketching with a nonlinear transform of PageRank similarities to achieve linear space and quality guarantees.
result FREDE provides column-covariance approximation guarantees that are nearly as good as SVD, even with limited node similarities.
We develop a new approach to the classical problem on isotopy classification of embeddings of manifolds into Euclidean spaces. This approach involves studying of a new embedding invariant, of almost-embeddings and of smoothing, as well as explicit constructions of embeddings. Using this approach we obtain complete conc…
Proves min-max theory for constant geodesic curvature curves on closed surfaces.
problem Prescribing mean curvature on surfaces with constant geodesic curvature.
method Min-max theory applied to classify blowups and ensure almost embedded solutions.
result Produces a solution with constant geodesic curvature c on closed surfaces. Given an integral symplectic manifold, we construct a family of "coherent state" maps into complex projective space. The maps are built from sections of the tensor powers of a hermitian line bundle whose curvature is a multiple of the symplectic form. We show that this family is an almost-complex version of the Kodiara…
The paper defines quasi-isometry for almost contact metric manifolds and explores its properties.
problem Understanding quasi-isometry in almost contact metric manifolds.
method Definition and study of quasi-isometry for almost contact metric manifolds.
result Established a relation between scalar curvature and quasi-isometric constants.
This work introduces a method for almost equivariance in neural networks using Lie algebra convolutions.
problem Real-world data often does not conform to strict group equivariances, leading to underperformance in models.
method Definition and practical implementation of almost equivariance through Lie algebra convolutions.
result Demonstrated the validity of the approach through benchmarking against fully equivariant settings.
This research shows how to learn shared representations from unpaired data.
problem Learning shared representations from unpaired data.
method Spectral embeddings of random walk matrices from each unimodal representation.
result Shared representations can be learned almost exclusively from unpaired data.
SepNE separates network embedding for efficiency and scalability.
problem Inseparable network embedding methods waste resources on irrelevant nodes.
method Formalized separated matrix factorization, SepNE learns node embeddings independently.
result SepNE outperforms state-of-the-art methods on large networks.
Given a constant k>1 and a real valued function K on the hyperbolic plane H2, we study the problem of finding, for any ε≈0, a closed and embedded curve uε in H2 having geodesic curvature k+εK(uε) at each point.
Threshold found for embedding 2D complexes into random 2-complexes.
problem Embedding 2D simplicial complexes into random 2-complexes.
method Multi-parameter model with independent simplex probabilities; geometric subdivisions and inequalities.
result Threshold $p_0 p_1^3 p_2^2 = rac{1}{n}$ for embedding 2D complexes into random 2-complexes.
Motivated by Tverberg-type problems in topological combinatorics and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R^d without higher-multiplicity intersections. We focus on conditions for the existence of almost r-embe…
The abstract discusses embedding theorems for pseudo-Kähler manifolds.
problem Embedding theorems for pseudo-Kähler manifolds.
method Using quantizable pseudo-Kähler manifolds and Hermitian line bundles, the asymptotic expansion of Bergman kernels is analyzed.
result The asymptotic expansion of Bergman kernels implies analogues of Kodaira embedding theorem and Tian's almost-isometry theorem.
Study on submanifolds of Euclidean space, classifying their symmetry types.
problem Classifying symmetry types of submanifolds in Euclidean space.
method Analyzing properties of full irreducible almost symmetric submanifolds and their cohomogeneity.
result Classification of almost symmetric submanifolds into specific types.
The paper generalizes free boundary min-max theory to equivariant settings.
problem Existence of minimal hypersurfaces with free boundary in manifolds with boundary.
method Free boundary min-max theory generalized to equivariant settings.
result Existence of nontrivial smooth almost properly embedded G-invariant minimal hypersurfaces with free boundary. Study of minimal surfaces in 4D with specific ends.
problem Characterize minimal surfaces in R4 with specific ends. method Modification of Costa and Hoffman-Meeks method, generalized Weierstrass representation.
result Minimal surfaces with specific ends are J-holomorphic under certain conditions. In this note, we investigate the relation between double points and complex points of immersed surfaces in almost-complex 4-manifolds and show how estimates for the minimal genus of embedded surfaces lead to inequalities between the number of double points and the number of complex points of an immersion. We also provi…
Study Kodaira dimensions on almost complex manifolds, proving integrability and structural descriptions.
problem Understanding Kodaira dimensions on almost complex manifolds.
method Using pseudoholomorphic pluricanonical maps, defining new dimensions, and applying probabilistic combinatorics.
result Almost complex structures with top Kodaira dimension are integrable, and for compact 4-manifolds, they have elliptic fibration structures.
We develop a geometric and explicit construction principle that generates classes of Poincare-Einstein manifolds, and more generally almost Einstein manifolds. Almost Einstein manifolds satisfy a generalisation of the Einstein condition; they are Einstein on an open dense subspace and, in general, have a conformal scal…
New criterion for almost-complex 4-manifolds using polyhedral decompositions.
problem Bounding the genus of surfaces in almost-complex 4-manifolds.
method Polyhedral decompositions and adjunction criterion.
result Established adjunction inequality for almost-complex 4-manifolds.
It is proved that the moduli space of all connected compact orientable embedded minimal affine Lagrangian submanifolds of a complex equiaffine space constitutes an infinite dimensional Frechet manifold (if it is not the empty set). The moduli space of all connected compact orientable metric Lagrangian embedded surfaces…