arXiv research

Study shows no smooth embeddings of rational homology balls into complex projective plane.

Paper proves Chow stability implies balanced embedding.

Estimates spectral projections restricted to uniformly embedded submanifolds.

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.

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.

The paper examines how well node similarities are preserved by random projections in graph embeddings.

Characterizes minor-minimal separating projective planar graphs and their generalizations.

New constraints on embedded spheres and projective planes in 4-manifolds from Seiberg-Witten theory.

Zero-shot learning transfers knowledge from seen classes to novel unseen classes to reduce human labor of labelling data for building new classifiers. Much effort on zero-shot learning however has focused on the standard multi-class setting, the more challenging multi-label zero-shot problem has received limited attent…

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.

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.

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.

Study proves rigidity of harmonic maps from 2-torus to complex projective space.

New symplectic caps and embeddings found in complex projective plane.

The paper uses Seshadri constants to construct symplectic ellipsoid embeddings.

New IPL graphs identified and conditions for their projective embeddings established.

Minimal hypersurfaces in real projective spaces have at least n+2 Morse index.

Introduces Kähler duality between domains in complex space.

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.

Veronese minimizes normal curvatures to 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.

New biharmonic submanifolds found in complex projective spaces.

Develops a new framework for temporal anchoring in deep embedding spaces.

Survey of Laplacian-based methods for data dimensionality reduction and embedding.

The paper describes hyperkähler geometry of cotangent bundles using rank-1 projections.

The paper studies stability of CR structures on compact manifolds.

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.

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…

Study Markov staircases in symplectic embeddings of rational homology ellipsoids.

In this article we show that every closed orientable smooth $4$--manifold admits a smooth embedding in the complex projective $3$--space.

Let $\hat{L}$ 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 $\hat{L}\backslash D$. 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.

Proves uniqueness of embedding complex manifold into infinite-dimensional space.

Study foliations in PSL(4,R)-Teichmüller theory, proving two 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…

New criteria for Cantor set tameness and wildness via projections.

Let $X$ be a union of a sequence of symplectic manifolds of increasing dimension and let $M$ be a manifold with a closed $2$-form $ω$. We use Tischler's elementary method for constructing symplectic embeddings in complex projective space to show that the map from the space of embeddings of $M$ in $X$ to the cohomology …

Paper presents a novel hyperbolic neural network for efficient data representation.

The study connects projective codes to the distribution of zeros of odd maps.

We extend Thurston's metric to projective filling currents, embedding Teichmüller space into the larger space.