Smooth but not symplectic embeddings of rational balls in complex projective plane found.
problem Finding smooth embeddings of rational balls in complex projective plane that are not symplectic.
method Infinite family of rational homology balls, lattice embedding obstruction from Donaldson's diagonalisation theorem.
result No two examples may be embedded disjointly.
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 $\CP^3$.
Smooth lens spaces embed in complex projective planes but not in finite copies.
problem Embedding lens spaces in complex projective planes.
method Analyzes smooth and locally flat embeddings of lens spaces in complex projective planes.
result No finite number of copies of complex projective plane can embed every lens space smoothly.
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.
Paper tackles multi-label zero-shot learning, improving label embedding projection for unseen classes.
problem Challenges in transferring knowledge from seen to unseen classes in multi-label zero-shot learning.
method Proposes a transfer-aware embedding projection approach to project label embeddings into a low-dimensional space for better inter-label relationships and explicit information transfer.
result Demonstrates the efficacy of the proposed approach through experiments on zero-shot multi-label image classification.
Paper proves Chow stability implies balanced embedding.
problem Chow stability and balanced embeddings in projective varieties.
method Continuity method conditional on technical hypothesis.
result Chow stability implies balanced embedding.
Estimates spectral projections restricted to uniformly embedded submanifolds.
problem Estimating spectral projections on submanifolds of manifolds with nonpositive curvature.
method Estimates the L2(M)oLq(Σ) norm of spectral projection operators. result Sharp spectral projection estimates for small spectral windows.
The paper examines how well node similarities are preserved by random projections in graph embeddings.
problem The preservation of node similarities under random projections in graph embeddings.
method Investigation of dot product and cosine similarity preservation by random projections over graph matrix rows.
result Random projections produce unreliable embeddings for dot product, especially for high-degree nodes.
In this paper, we study asymptotic behavior of projective embeddings of Kummer varieties given by theta functions, and their amoebas. We prove that a Lagrangian fibration of the Kummer variety can be approximated by moment maps of the projective spaces.
Characterizes minor-minimal separating projective planar graphs and their generalizations.
problem Understanding projective planar graphs and their properties.
method Analyzing minors, embeddings, and specific link types.
result Partial characterization of minor-minimal separating projective planar graphs and their generalizations.
New constraints on embedded spheres and projective planes in 4-manifolds from Seiberg-Witten theory.
problem Constraints on configurations of embedded spheres and real projective planes in 4-manifolds.
method Equivariant Seiberg-Witten invariants and gluing formula for relative Seiberg-Witten invariants.
result Existence of certain configurations of surfaces leads to 4-manifolds of non-simple type.
By the quantization condition compact quantizable Kaehler manifolds can be embedded into projective space. In this way they become projective varieties. The quantum Hilbert space of the Berezin-Toeplitz quantization (and of the geometric quantization) is the projective coordinate ring of the embedded manifold. This all…
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.
Let n be aninteger>4. There is a smoothly knotted n-dimensional sphere in (n+2)-space such that the singular point set of its projection in (n+1)-space consists of double points and that the components of the singular point set are two. (The sphere is knotted in the sense that it does not bound any embedded (n+1)-ball …
We discuss holomorphic isometric embeddings of the projective line into quadrics using a generalisation of the theorem of do Carmo--Wallach to provide a description of their moduli spaces up to image and gauge--equivalence. Moreover, we show rigidity of the real standard map from the projective line into quadrics.
The Whitney embedding theorem gives an upper bound on the smallest embedding dimension of a manifold. If a data set lies on a manifold, a random projection into this reduced dimension will retain the manifold structure. Here we present an algorithm to find a projection that distorts the data as little as possible.
Optimizes t-SNE for high-dimensional data with random projections.
problem High computational cost of t-SNE for high-dimensional data.
method Use random projections to reduce high-dimensional data to a few dimensions, then apply t-SNE.
result Random projections preserve clustering while significantly reducing t-SNE runtime.
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.
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.
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.
New IPL graphs identified and conditions for their projective embeddings established.
problem Characterizing and identifying intrinsically projectively linked graphs.
method Applying Δ-Y exchanges and analyzing projective planar graphs.
result No minor-minimal IPL graphs on 16 edges exist, and new ones are identified.
We embed projective spaces into spheres using telescopic constructions.
problem Embedding projective spaces into spheres isometrically.
method Inductive telescopic construction extending previous embeddings.
result Isometric embeddings defined for projective real and complex spaces.
Minimal hypersurfaces in real projective spaces have at least n+2 Morse index.
problem Understanding the Morse index of minimal hypersurfaces in real projective spaces.
method Analyzing unstable one-sided and two-sided minimal hypersurfaces in real projective spaces.
result The Morse index of minimal hypersurfaces is at least n+2, with specific examples provided.
Introduces Kähler duality between domains in complex space.
problem None explicitly stated; focuses on concept introduction.
method None explicitly stated; focuses on concept introduction.
result Introduces Kähler duality between domains in complex space.
Paper proposes a new method for sentence embeddings using weighted word vectors.
problem Improving sentence embeddings for natural language processing tasks.
method A simple sentence embedding method using weighted average of word vectors followed by soft projection.
result Demonstrates effectiveness on clinical semantic textual similarity task.
The transition amplitudes between coherent states on a coherent state manifold are expressed in terms of the embedding of the coherent state manifold into a projective Hilbert space. Consequences for the dimension of projective Hilbert space and a simple geometric interpretation of Calabi's diastasis follows.
The first examples of complete projective connections are uncovered: normal projective connections on surfaces whose geodesics are all closed and embedded are complete, as are normal projective connections induced from complete affine connections with slowly decaying positive Ricci curvature.
Two strategies for embedding new data points from proximity data are explored.
problem Embedding new data points using proximity data.
method Two competing strategies: projection and restricted reconstruction.
result Projection and restricted reconstruction can be derived from kernel methods.
Veronese minimizes normal curvatures to sphere.
problem Bounding normal curvatures of submanifolds.
method Veronese embeddings of projective planes.
result Optimal bound on normal curvatures guarantees sphere.
By computing certain cohomology of Vect(M) of smooth vector fields we prove that on 1-dimensional manifolds M there is no quantization map intertwining the action of non-projective embeddings of the Lie algebra sl(2) into the Lie algebra Vect(M). Contrariwise, for projective embeddings sl(2)-equivariant quantization ex…
Symplectic embedding extended to stratified spaces.
problem Symplectic embedding theorem for stratified spaces.
method Defined symplectic structure on stratified spaces, demonstrated embedding in complex projective space.
result Symplectic embedding theorem extended to stratified spaces.
New biharmonic submanifolds found in complex projective spaces.
problem Identifying biharmonic submanifolds in complex projective spaces.
method Analyzing Segre embeddings and studying biharmonicity of submanifolds.
result Two new large classes of biharmonic submanifolds discovered.
Develops a new framework for temporal anchoring in deep embedding spaces.
problem Temporal anchoring in deep embedding spaces, especially drift and convergence issues.
method Operator-theoretic framework with drift maps and event-indexed blocks, proving convergence theorems and equivalence theorems.
result Proves convergence theorems and equivalence theorems for the proposed framework.
Survey of Laplacian-based methods for data dimensionality reduction and embedding.
problem Efficiently reducing high-dimensional data to lower dimensions while preserving important features and structures.
method Laplacian-based methods including spectral clustering, Laplacian eigenmap, locality preserving projection, graph embedding, and diffusion map.
result Comprehensive overview of various optimization variants and applications of Laplacian-based techniques.
The paper describes hyperkähler geometry of cotangent bundles using rank-1 projections.
problem Understanding hyperkähler geometry of cotangent bundles via algebraic methods.
method Algebraic description via the scheme of rank-1 projections, isometric embeddings, and generalizations.
result Explicit isometric embeddings and generalizations of hyperkähler geometry.
The paper studies stability of CR structures on compact manifolds.
problem Stability of CR structures on compact manifolds.
method Smooth deformations and projective embeddings.
result Nearby structures still admit projective CR embeddings.
Kodaira embedding theorem provides an effective characterization of projectivity of a Kähler manifold in terms the second cohomology. Recently X. Yang [21] proved that any compact Kähler manifold with positive holomorphic sectional curvature must be projective. This gives a metric criterion of the projectivity in terms…
The Farey tree helps embed rational balls and lens spaces into complex projective space.
problem Embedding rational homology balls and lens spaces into complex projective space.
method Recursive Kirby calculus argument using the Farey tree.
result Explicit constructions of embeddings of triples of rational homology balls into homotopy CP2. Network embedding, which learns low-dimensional vector representation for nodes in the network, has attracted considerable research attention recently. However, the existing methods are incapable of handling billion-scale networks, because they are computationally expensive and, at the same time, difficult to be accele…
New knots found that can only fit in non-reduced projections.
problem Finding knots that can only fit in non-reduced projections.
method Infinite family of knots, systematic flype finding tool.
result Knots with hexagonal mosaic number realized only in non-reduced projections.
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.
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.
Let L^ be the projective completion of an ample line bundle L over D, a smooth projective manifold. Hwang-Singer \cite{HwangS} have constructed complete CSCK metric on L^\D. When the corresponding \kahler form is in the cohomology class of a rational divisor A and when L has negative CSC…
The fields of compressed sensing (CS) and matrix completion have shown that high-dimensional signals with sparse or low-rank structure can be effectively projected into a low-dimensional space (for efficient acquisition or processing) when the projection operator achieves a stable embedding of the data by satisfying th…
Extends Tian theorem to Vaisman manifolds for approximations.
problem Approximating Vaisman metrics by immersions/embeddings.
method Study Vaisman metrics on compact manifolds.
result Extend Tian's theorem to Vaisman manifolds.
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.
Study foliations in PSL(4,R)-Teichmüller theory, proving two invariant foliations.
problem Structure of domains of discontinuity in PSL(4,R)-Hitchin representations.
method Detailed study and proof of foliations by projective line segments and convex domains.
result Foliated component has exactly two group-invariant foliations.
Recent advances suggest that encoding images through Symmetric Positive Definite (SPD) matrices and then interpreting such matrices as points on Riemannian manifolds can lead to increased classification performance. Taking into account manifold geometry is typically done via (1) embedding the manifolds in tangent space…