Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
problem Embedding maps in higher dimensions without self-intersections.
method Lifting maps to embeddings in product spaces.
result Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
Study of codimension-1 embeddings in 3-manifolds using twist maps and push maps.
problem Embedding 3-manifolds in specific topological spaces.
method Sphere twist maps and push maps to construct codimension-1 spun embeddings.
result Every closed orientable 3-manifold admits a codimension-1 spun embedding in specific spaces.
Article provides conditions for embedding braid groups into mapping class groups.
problem Embedding braid groups into mapping class groups.
method Necessary and sufficient condition for embedding finite index subgroups.
result Conditions for embedding braid groups into mapping class groups.
We introduce (k,l)-regular maps, which generalize two previously studied classes of maps: affinely k-regular maps and totally skew embeddings. We exhibit some explicit examples and obtain bounds on the least dimension of a Euclidean space into which a manifold can be embedded by a (k,l)-regular map. The problem c…
Introduces Hurewicz fibrations for embedding maps of orbifold charts.
problem No specific problem stated; focuses on new concept definition.
method Defines E-fibration embedding and studies its properties.
result Introduces and studies properties of E-fibration embedding.
New subgroups of mapping class groups constructed for infinite-type surfaces.
problem Constructing new subgroups of mapping class groups for infinite-type surfaces.
method Utilization of special homeomorphisms called shift maps and multipush maps.
result Countably (and uncountably in certain cases) many non-conjugate embeddings of subgroups into mapping class groups.
This paper extends boundary embedding results to coarsely convex spaces.
problem Generalizing boundary embedding results to coarsely convex spaces.
method Generalizing Dydak and Virk's work on Gromov hyperbolic spaces to coarsely convex spaces.
result Maps between coarsely convex spaces induce continuous maps between their boundaries.
Study characterizes quasi-isometric embeddings of maps from cusped surfaces into moduli space.
problem Characterizing quasi-isometric embeddings of maps from cusped surfaces into moduli space.
method Investigates shrinking maps from a cusped hyperbolic surface into the moduli space of closed Riemann surfaces, considering quasi-isometric embeddings with respect to Teichmüller distance and intrinsic distance.
result Characterizations of quasi-isometric embeddings are solely determined by the map's monodromy under mild conditions.
Maps on surfaces can be embedded into spheres with minimal dimensions.
problem Embedding periodic maps of surfaces into spheres with the smallest possible dimensions.
method Determining the minimal dimensions m for embeddings of periodic maps of order n on surfaces of genus g into spheres Sm. result For each integer k>1, there exist infinitely many periodic maps such that the smallest possible m is equal to k. Neumann eigenmaps improve landmark-based diffusion map embeddings.
problem Landmark-based diffusion map embeddings can be computationally inefficient and unstable.
method NeuMaps use a renormalized Neumann Laplacian for eigendecomposition, incorporating landmarks as a subgraph.
result NeuMaps offer a computationally efficient and stable embedding method.
Embedding calculus proves convergence for surfaces.
problem Proving convergence of embedding calculus for surfaces.
method Goodwillie-Weiss' embedding calculus for spaces of embeddings into a manifold of dimension at most two.
result Relates Johnson filtration of mapping class group to embedding calculus.
Improves visualization of high-dimensional data by correcting misleading artifacts in neighbor embedding methods.
problem Misleading visual artifacts in t-SNE and UMAP due to lack of data-independent manifold learning interpretations.
method LOO-map framework that extends embedding maps to the entire input space, identifying and correcting map discontinuities.
result Developed point-wise diagnostic scores to detect unreliable embedding points and improve hyperparameter selection.
Maps and embeddings between hyperbolic spaces and their boundaries studied.
problem Understanding relations between maps and embeddings between relatively hyperbolic spaces and their boundaries.
method Establishing correspondences between quasi-isometric embeddings and quasisymmetric embeddings, using polynomial distortion.
result Characterization of hyperbolic relative groups with polynomial distortion embeddings.
The paper explores conditions for lifting maps between graphs to embeddings.
problem Conditions for embedding maps between graphs.
method Combinatorial techniques and satisfiability of 3-CNF formulas.
result Established necessary and sufficient conditions for lifting.
Study on embeddings of surfaces in 4-manifolds and their mapping classes.
problem Characterizing and understanding embeddings of surfaces in 4-manifolds and their mapping classes.
method Analyzing smooth proper embeddings and mapping classes induced by diffeomorphisms of 4-manifolds.
result Most surfaces do not admit flexible embeddings in 4-manifolds with specific homology types.
The author connects Poincaré embeddings to Reidemeister traces and diagonal maps.
problem Existence of Poincaré embeddings for specific spaces.
method Relates total obstruction to Reidemeister trace and uses Poincaré duality.
result Diagonal maps admit Poincaré embeddings under certain conditions.
We construct several families of embeddings of braid groups into mapping class groups of orientable and non-orientable surfaces and prove that they induce the trivial map in stable homology in the orientable case, but not so in the non-orientable case. We show that these embeddings are non-geometric in the sense that t…
Defines coupled embeddability for maps on products of spaces, generating examples and nonexamples.
problem Understanding when maps on products of spaces can be embedded.
method Uses known results for nonsingular biskew and bilinear maps, studies genericity properties, extends Whitney embedding theorems, and relates to Z/2-coindex of embedding spaces. result Generates strong obstructions to coupled embeddability in terms of combinatorics of triangulations.
Study on embeddings and their topological properties in R^d.
problem Topology of embeddings and their bounds in R^d.
method Combinatorial formula for upper bounds of embeddings into R^d.
result Simple combinatorial formula for upper bounds of embeddings into R^d.
We obtain a criterion for approximability by embeddings of piecewise linear maps of a circle to the plane, analogous to the one proved by Minc for maps of a segment to the plane. Theorem. Let S be a triangulation of a circle with s vertices. Let f be a simplicial map of the graph S to the plane. The map f is approximab…
The abstract shows how embeddings inscribe trapezoids or map three points to a line, proving nonexistence of certain maps.
problem Proving the nonexistence of affinely 3-regular maps in infinitely many dimensions.
method Elementary proof using embeddings and nonsingular bilinear maps.
result Recovery of nonexistence results for affinely 3-regular maps without complex algebraic techniques.
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…
Embeddings of mapping tori for end-periodic graph maps are proven.
problem Embedding mapping tori of end-periodic graph maps into finite complexes.
method Flowline-preserving homotopy equivalence and π1-injective map. result Every mapping class of Γ arising from an end-periodic homotopy equivalence contains a representative whose mapping torus realizes such an embedding.
Nonorientable surface mapping class group embeds quasi-isometrically in its orientable cover.
problem Embedding nonorientable surface mapping class group in orientable surface mapping class group.
method Utilized semihyperbolicity of orientable surface mapping class group and orientation double covering properties.
result Injective homomorphism is a quasi-isometric embedding.
Segre embedding was introduced by C. Segre (1863--1924) in his famous 1891 article \cite{segre}. The Segre embedding plays an important roles in algebraic geometry as well as in differential geometry, mathematical physics, and coding theory. In this article, we survey main results on Segre embedding in differential geo…
Earth observation embeddings can convert discrete biome maps into continuous representations that better capture ecological variation.
problem Biome maps impose categorical boundaries that compress continuous variation in biotic communities.
method Fit a linear classifier on Earth observation embeddings to predict biome labels.
result Continuous biome representation outperforms discrete biome labels for predicting species occurrence.
We consider finite-sheeted, regular, possibly branched covering spaces of compact surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides with the entire mapping class group of the surface. As a consequence, we construct …
Study embeddings of manifolds via acyclic maps and surgery.
problem Relationship between embeddings of manifolds in spheres.
method Solve Poincaré duality variant, apply surgery machine, focus on homology spheres.
result Deduce results about homotopy type of embedding spaces.
The notion of ideal embeddings was introduced in [B.-Y. Chen, {Strings of Riemannian invariants, inequalities, ideal immersions and their applications.} The Third Pacific Rim Geometry Conference (Seoul, 1996), 7-60, Int. Press, Cambridge, MA, 1998]. Roughly speaking, an ideal embedding (or a best of living) is an isome…
Electronic health record (EHR) systems are used extensively throughout the healthcare domain. However, data interchangeability between EHR systems is limited due to the use of different coding standards across systems. Existing methods of mapping coding standards based on manual human experts mapping, dictionary mappin…
The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a …
We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmuller space is a quasi-isometric embedding for both of the standard metrics. As a …
Mapping and translating professional but arcane clinical jargons to consumer language is essential to improve the patient-clinician communication. Researchers have used the existing biomedical ontologies and consumer health vocabulary dictionary to translate between the languages. However, such approaches are limited b…
Let X be a CR manifold with transversal, proper CR G-action. We show that X/G is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold factorises uniquely over a holomorphic map on X/G. We then use this result and complex …
Given a closed oriented 3-manifold M, we establish an isomorphism between the Heegaard Floer homology group HF^+(-M) and the embedded contact homology group ECH(M). Starting from an open book decomposition (S,h) of M, we construct a chain map Φ^+ from a Heegaard Floer chain complex associated to (S,h) to an embedded co…
Homomorphisms between pure mapping class groups are classified for certain genus surfaces.
problem Classifying homomorphisms between pure mapping class groups for specific genus surfaces.
method Proving every multitwist-preserving map is induced by a multi-embedding, then applying to classify homomorphisms.
result All homomorphisms between pure mapping class groups for g≥4 and g′≤6⋅2g−4 are classified. Study proves rigidity of harmonic maps from 2-torus to complex projective space.
problem Rigidity of isotropic harmonic maps from a 2-torus to a complex projective space.
method Proves rigidity through holomorphic embeddings and complete linear systems.
result Ensures rigidity of harmonic bands in condensed matter physics.
Szűcs introduced cobordism of singular maps to compute groups of immersions and embeddings.
problem Computing cobordism groups of immersions and embeddings in dimensions where classical theory fails.
method Investigation of classifying spaces constructed by Szűcs and Rimányi.
result Collection and organization of results towards computation of cobordism groups of singular maps.
Unified view of SOMs and SNE from a common framework.
problem Comparing and understanding SOMs and SNE.
method Unified mathematical framework, quantitative comparison on datasets.
result SOMs and SNE can be derived from a common framework.
WEGL embeds graphs in a vector space for faster machine learning.
problem Efficiently embedding graphs for machine learning tasks.
method Wasserstein distance for node embedding similarity, Monge maps for graph representation.
result State-of-the-art classification performance with superior computational efficiency.
We give a necessary and sufficient condition for the mapping class group of the pair of the 3-sphere and a graph embedded in it to be isomorphic to the topological symmetry group of the embedded graph.
Maps embed manifolds using heat kernels of connection Laplacian.
problem Embedding manifolds in Euclidean space.
method Using heat kernels of the connection Laplacian and truncated heat kernels.
result Maps can be made arbitrarily close to isometries.
Study on embedding tree products into groups, distinguishing them.
problem Quasi-isometric embedding of tree products into various groups.
method Using coarse embeddings of products of bushy trees into hierarchically hyperbolic spaces.
result Quasi-isometrically distinguish and rule out embeddings between groups.
In this article, we reformulate the cobordism map of embedded contact homology, which is induced by exact symplectic cobordism and defined as direct limit of homomorphisms called filtered ECH cobordism map. The filtered ECH cobordism map is defined by counting embedded holomorphic curves with zero ECH index and we prov…
The study connects specific circle embeddings to 4-manifold diffeomorphisms.
problem Understanding diffeomorphisms of 4-manifolds from specific circle embeddings.
method Using a parameterised surgery map to relate framed embeddings of S^1 to mapping class groups.
result Established connections between grasper families and known diffeomorphisms.
Embedding right-angled Artin groups in mapping class groups of nonorientable surfaces.
problem Embedding right-angled Artin groups in mapping class groups for nonorientable surfaces.
method Proving embedding for finite full subgraphs and some non-full subgraphs of curve graphs.
result Existence of non-full subgraphs that can be embedded in mapping class groups.
This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorit…
This paper reviews MDS, Sammon mapping, and Isomap, explaining their theory and applications.
problem Exploring multidimensional data structures and mappings.
method Explains classical MDS, metric MDS, kernel classical MDS, Sammon mapping, Isomap, and their applications.
result Detailed understanding of MDS, Sammon mapping, and Isomap methods.