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.
New tool: relative Hopf invariant for Poincaré surgery.
problem Non-simply connected Poincaré surgery.
method Relative Hopf invariant in equivariant setting.
result Established Poincaré embedding results in relative setting.
Poincaré embeddings learn hierarchical symbolic data representations.
problem Learning hierarchical representations for complex symbolic data like text and graphs.
method Embedding into hyperbolic space (Poincaré ball) for efficient Riemannian optimization.
result Poincaré embeddings outperform Euclidean embeddings on data with latent hierarchies.
MuRP embeds multi-relational graphs in hyperbolic space for better hierarchical representation.
problem Current hyperbolic models struggle with multi-relational knowledge graphs that exhibit multiple hierarchies.
method MuRP embeds multi-relational graph data in the Poincaré ball model of hyperbolic space, learning relation-specific parameters for entity embeddings.
result MuRP embeddings outperform Euclidean counterparts and other methods on link prediction tasks, especially at lower dimensions.
We obtain multirelative connectivity statements about spaces of Poincare embeddings, as precursors to analogous statements about spaces of smooth embeddings. The latter are the key to convergence results in the functor calculus approach to spaces of embeddings.
Given Poincare spaces M and X, we study the possibility of compressing embeddings of M x I in X x I down to embeddings of M in X. This results in a new approach to embedding in the metastable range both in the smooth and Poincare duality categories.
This paper investigates the space of codimension zero embeddings of a Poincare duality space in a disk. One of our main results exhibits a tower that interpolates from the space of Poincare immersions to a certain space of "unlinked" Poincare embeddings. The layers of this tower are described in terms of the coefficien…
Study delta invariant of curves on rational surfaces using topological methods.
problem Calculate delta invariant for curves embedded in rational singularities.
method Use topological techniques and Poincaré series.
result Develop formulae for delta invariant in terms of embedded data.
Proposes cone embedding for better graph hierarchical structure representation.
problem Lack of natural and interpretable hierarchical indicators in graph embeddings.
method Metric cone embedding method to capture hierarchical structure.
result Extracts hierarchical structure from other graph embedding outputs.
Paper proposes a new topology for AML analysis using Poincaré embeddings.
problem Complex money laundering schemes and regulatory constraints hinder AML analysis and information sharing.
method Proposes a new topology for AML analysis using Poincaré embeddings.
result Demonstrates improved AML analysis and information sharing through Poincaré embeddings.
The study determines lens spaces that can be obtained from surgeries on knots in the Poincaré homology sphere.
problem Identifying lens spaces that can be obtained from surgeries on knots in the Poincaré homology sphere.
method Developed a lattice embedding obstruction to realize L-space surgeries on knots in the Poincaré homology sphere.
result Identified the only two knots in the Poincaré homology sphere that admit half-integer lens space surgeries.
Study improves Poincaré-Sobolev inequalities for differential forms.
problem Improving Sobolev space embeddings for differential forms.
method Utilizes Lq,p-cohomology and bi-Lipschitz images to estimate embedding norms. result Estimates for embedding norms in Euclidean balls and their images.
Paper proposes a stable update rule in hyperbolic space for better network modeling.
problem Complex network modeling in hyperbolic space.
method Explicit geodesic update rule in hyperbolic space with theoretical convergence guarantees.
result Algorithm convergence rate is better than Euclidean gradient descent and avoids bias.
New spaces help connect manifold structures on equivariant Poincaré spaces.
problem Creating manifold structures on equivariant Poincaré spaces.
method Introducing semifree isovariant G-Poincaré spaces and gap conditions. result Space of isovariant structures on semifree G-Poincaré spaces is highly connected. New tensors help determine if metrics are related to Poincaré-Einstein ones.
problem Determining when metrics on conformally compact manifolds are related to Poincaré-Einstein metrics.
method Developed new tensors and used conformal tractor calculus to analyze metrics.
result The vanishing of these new tensors is a necessary and sufficient condition for a metric to be related to a Poincaré-Einstein metric.
We obtain multirelative connectivity statements about spaces of smooth embeddings, deducing these from analogous results about spaces of Poincare embeddings that were established in our previous paper.
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.
Let f: P-->W be an embedding of a compact polyhedron in a closed oriented manifold W, let T be a regular neighborhood of P in W and let C:=closure(W-T) be its complement. Then W is the homotopy push-out of a diagram C<--dT-->P. This homotopy push-out square is an example of what is called a Poincare embedding. We study…
The paper introduces Poincaré profiles for metric spaces and groups, linking them to conformal dimension.
problem Understanding the properties of metric measure spaces and groups with polynomial growth.
method Introducing and analyzing Poincaré profiles for groups and hyperbolic spaces.
result Connection between Poincaré profiles and conformal dimension, leading to non-existence of coarse embeddings.
Efficiently learns high-quality hierarchical embeddings in hyperbolic space.
problem Discovering hierarchical relationships from large-scale similarity scores.
method Used the Lorentz model of hyperbolic geometry to learn embeddings efficiently.
result The proposed approach yields high-quality embeddings that improve over Poincaré embeddings, especially in low dimensions.
Proposes PKG embedding for e-commerce products.
problem Learning product intrinsic relations for e-commerce applications.
method Self-attention-enhanced distributed representation learning model from raw data.
result Compared favorably to baselines in knowledge completion and downstream tasks.
Poincaré VAEs improve hierarchical data representation.
problem Hierarchical data structures are difficult to represent in Euclidean latent spaces.
method Introducing Poincaré ball model of hyperbolic geometry as a latent space for VAEs.
result Better generalization and hierarchical structure recovery in hyperbolic space.
Improved music recommendations using hyperbolic embeddings and Bayesian estimation.
problem Generating personalized music recommendations.
method Hyperbolic Poincaré embeddings for hierarchies, empirical Bayes for link reliability.
result Significant performance improvement in A/B testing.
Study of non-convex potential functions in deep learning with Poincaré inequality.
problem Understanding convergence of stochastic dynamics in non-convex potential landscapes.
method Introduced log-Polyak-Lojasiewicz (log-PL) measures and analyzed their convergence properties.
result Langevin dynamics converges at a rate of O~(1/ε) for sufficiently small ε. The paper applies Poincaré duality to supergravity, proving its equivalence to other formulations.
problem Relating different formulations of supergravity on supermanifolds.
method Proving relative Poincaré duality and using it to connect differential and integral forms.
result Relative Poincaré duality provides a rigorous definition of picture changing operators in supergravity.
We formulate a theory of pointed manifolds, accommodating both embeddings and Pontryagin-Thom collapse maps, so as to present a common generalization of Poincaré duality in topology and Koszul duality in En-algebra.
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…
Paper proves existence of a CMC hypertorus in 4D sphere using numerical methods.
problem Proving the existence of a constant mean curvature (CMC) hypertorus in \(S^4\).
method Employed the round Taylor method with rational arithmetic and the Poincare-Miranda theorem.
result Existence of a constant mean curvature (CMC) hypertorus in \(S^4\).
The paper proves lower bounds for Gaussian-weighted curvature integrals of self-shrinkers.
problem Proving lower bounds for Gaussian-weighted \(L^2\)-curvature integrals of self-shrinkers.
method Combining normal coordinate functions with weighted Poincaré inequalities and first-eigenvalue estimates.
result Explicit lower bounds in terms of entropy for closed self-shrinkers, leading to curvature gaps.
We give the explicit algorithm computing the motivic generalization of the Poincare series of the plane curve singularity introduced by A. Campillo, F. Delgado and S. Gusein-Zade. It is done in terms of the embedded resolution of the curve. The result is a rational function depending of the parameter q, at q=1 it coinc…
The study explores symmetries of graphs in 3-manifolds and their induced homeomorphisms.
problem Understanding when graph automorphisms are induced by homeomorphisms in 3-manifolds.
method Analyzes embeddings of graphs in various 3-manifolds and their symmetries.
result Not all graph automorphisms are induced by homeomorphisms in all 3-manifolds, but many properties hold for homology spheres.
Study models Indian stock market using hyperbolic geometry for market stability and volatility analysis.
problem Identifying market stability and volatility in the Indian stock market.
method Modelled as a heterogeneous scale-free network, embedded in a 2D hyperbolic space, applied coalescent embedding, hyperbolic kmeans, and Bollinger Band analysis.
result Clusters in the embedded network better represent market communities than Euclidean clusters, allowing for early detection of market changes.
It is shown how the coherent states permit to find different geometrical objects as the geodesics, the conjugate locus, the cut locus, the Calabi's diastasis and its domain of definition, the Euler-Poincaré characteristic, the number of Borel-Morse cells, the Kodaira embedding theorem.
Improves few-shot learning for hierarchical data using hyperbolic space.
problem Few-shot class-incremental learning for hierarchical data.
method Contrastive learning in hyperbolic space, Poincaré ball model, hyperbolic contrastive loss, maximum entropy distribution.
result Effective improvement of coarse and fine class accuracies in few-shot conditions.
Study Bergman kernel and Kähler metrics on Riemann surfaces and symmetric products.
problem Estimating metrics on Riemann surfaces and symmetric products.
method Investigates Bergman kernels and Kähler metrics on Riemann surfaces and symmetric products.
result Estimates the Bergman metric and Kähler metric on symmetric products in terms of the Bergman kernel and Poincaré metric.
Study vector fields with complex singularities, proving bounds and formulas.
problem Understanding the Milnor number of vector fields with specific singularities.
method Global and local formulas expressing Milnor/Poincare-Hopf contributions, sharp lower bounds under perturbations.
result Sharp lower bounds for Milnor number contributions under holomorphic perturbations.
We prove that a metric space does not coarsely embed into a Hilbert space if and only if it satisfies a sequence of Poincaré inequalities, which can be formulated in terms of (generalized) expanders. We also give quantitative statements, relative to the compression. In the equivariant context, our result says that a gr…
For a compact differentiable surface with boundary embedded in R3, we give simple proofs of the Gauss-Bonnet theorem, Poincaré-Hopf theorem, and several other integral formulas. We complete all of the proofs without using fundamental or differential forms.
To an inclusion topological groups H->G, we associate a naive G-spectrum. The special case when H=G gives the dualizing spectrum D_G introduced by the author in the first paper of this series. The main application will be to give a purely homotopy theoretic construction of Poincare embeddings in stable codimension.
Derivatives of sub-Riemannian geodesics are always Lp-Hölder continuous.
problem Smoothness of sub-Riemannian geodesics
method Proving Lp-Hölder continuity of derivatives result Derivatives of sub-Riemannian geodesics are Lp-Hölder continuous The paper proves inequalities for varifolds on Riemannian manifolds.
problem Proving inequalities for functions on varifolds in Riemannian manifolds.
method Developed techniques to handle functions with compact support on k-rectifiable varifolds in Riemannian manifolds with positive injectivity radius and sectional curvature bounded above. result Proved Poincaré and Sobolev type inequalities for varifolds.
Embeds cohomology of hyperkahler manifolds into torus cohomology.
problem Embedding cohomology of hyperkahler manifolds into torus cohomology.
method Kuga-Satake construction and embedding of graded cohomology spaces.
result Compatibility of embeddings with Hodge structures and Lie algebra actions.
We study the cobordism of manifolds with boundary, and its applications to codimension 2 embeddings Mm⊂Nm+2, using the method of the algebraic theory of surgery. The first main result is a splitting theorem for cobordisms of algebraic Poincaré pairs, which is then applied to describe the behaviour on the c…
Enhances graph modeling with hyperbolic geometry and variational inference.
problem Challenges in modeling relational data with complex dependencies.
method Semi-implicit hierarchical variational Bayes with Poincaré embedding and mutual information regularization.
result Improves graph representation quality and flexibility in edge prediction and node classification.
Holomorphic curves exiting bounded symmetric domains are asymptotically totally geodesic.
problem Understanding the asymptotic behavior of holomorphic curves in bounded symmetric domains.
method Proof by contradiction and rescaling, using the Poincaré-Lelong equation.
result Holomorphic curves exiting a bounded symmetric domain are asymptotically totally geodesic.
Study of torus surgeries on knot traces, finding exotic surfaces and traces.
problem Understanding exotic surfaces and traces through torus surgeries.
method Realizing annulus twisting as torus surgery, using key technical insight.
result Exotic elliptic surfaces and traces discovered, improving known geography.
We compute the Heegaard-Floer link homology of algebraic links in terms of the multivariate Hilbert function of the corresponding plane curve singularities. The main result of the paper identifies four homologies: (a) the Heegaard-Floer link homology of the local embedded link of the germ, (b) the lattice homology asso…
The paper constructs thin Loewner carpets and their embeddings in S2.
problem Understanding the properties of Loewner carpets and their embeddings.
method Admissible quotiented inverse system construction for Loewner carpets and explicit embeddings.
result Explicit construction of infinitely many pairwise quasi-symmetrically distinct Q-Loewner carpets that admit quasisymmetric embeddings into S2.