arXiv research

The author connects Poincaré embeddings to Reidemeister traces and diagonal maps.

New tool: relative Hopf invariant for Poincaré surgery.

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.

Hyperbolic embeddings have recently gained attention in machine learning due to their ability to represent hierarchical data more accurately and succinctly than their Euclidean analogues. However, multi-relational knowledge graphs often exhibit multiple simultaneous hierarchies, which current hyperbolic models do not c…

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.

Proposes cone embedding for better graph hierarchical structure representation.

Representation learning has become an invaluable approach for learning from symbolic data such as text and graphs. However, while complex symbolic datasets often exhibit a latent hierarchical structure, state-of-the-art methods typically learn embeddings in Euclidean vector spaces, which do not account for this propert…

Study improves Poincaré-Sobolev inequalities for differential forms.

The study determines lens spaces that can be obtained from surgeries on knots in the Poincaré homology sphere.

New spaces help connect manifold structures on equivariant Poincaré spaces.

New tensors help determine if metrics are related to Poincaré-Einstein ones.

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.

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…

We are concerned with the discovery of hierarchical relationships from large-scale unstructured similarity scores. For this purpose, we study different models of hyperbolic space and find that learning embeddings in the Lorentz model is substantially more efficient than in the Poincaré-ball model. We show that the prop…

A hyperbolic space has been shown to be more capable of modeling complex networks than a Euclidean space. This paper proposes an explicit update rule along geodesics in a hyperbolic space. The convergence of our algorithm is theoretically guaranteed, and the convergence rate is better than the conventional Euclidean gr…

We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincaré profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by Benjamini--Schramm--Timár. In this paper we focus on properties of the Poincaré profiles of groups with …

Study of non-convex potential functions in deep learning with Poincaré inequality.

Matrix Factorization (MF) is a common method for generating recommendations, where the proximity of entities like users or items in the embedded space indicates their similarity to one another. Though almost all applications implicitly use a Euclidean embedding space to represent two entity types, recent work has sugge…

The paper applies Poincaré duality to supergravity, proving its equivalence to other formulations.

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 $\mathcal{E}_n$-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.

The paper proves lower bounds for Gaussian-weighted curvature integrals of self-shrinkers.

In the recent years money laundering schemes have grown in complexity and speed of realization, affecting financial institutions and millions of customers globally. Strengthened privacy policies, along with in-country regulations, make it hard for banks to inner- and cross-share, and report suspicious activities for th…

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…

Study models Indian stock market using hyperbolic geometry for market stability and volatility analysis.

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.

Let M be a simple hyperkahler manifold. Kuga-Satake construction gives an embedding of H^2(M,C) into the second cohomology of a torus, compatible with the Hodge structure. We construct a torus T and an embedding of the graded cohomology space H^*(M,C) \to H^{*+l}(T,C) for some l, which is compatible with the Hodge stru…

The variational auto-encoder (VAE) is a popular method for learning a generative model and embeddings of the data. Many real datasets are hierarchically structured. However, traditional VAEs map data in a Euclidean latent space which cannot efficiently embed tree-like structures. Hyperbolic spaces with negative curvatu…

Improves few-shot learning for hierarchical data using hyperbolic space.

Study vector fields with complex singularities, proving bounds and formulas.

The current article stems from our study on the asymptotic behavior of holomorphic isometric embeddings of the Poincaré disk into bounded symmetric domains. As a first result we prove that any holomorphic curve exiting the boundary of a bounded symmetric domain $Ω$ must necessarily be asymptotically totally geodesic. A…

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 $\Bbb R^3$, 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 $L_p$-Hölder continuous.

Let $X$ be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer $k\geq 2$, we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle $Ω^{\otimes k}_X$, where $Ø$ is the holomorphic cotangent bundle of $X$. Our first main result estimates the corresponding B…

We study the cobordism of manifolds with boundary, and its applications to codimension 2 embeddings $M^m\subset N^{m+2}$, 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.

Study of torus surgeries on knot traces, finding exotic surfaces and traces.

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…

In a flat space, the global topology of comoving space can induce a weak acceleration effect similar to dark energy. Does a similar effect occur in the case of the Poincare dodecahedral space S^3/I^*? Does the effect distinguish the Poincare space from other well-proportioned spaces? The residual acceleration effect in…

Representation learning methods that transform encoded data (e.g., diagnosis and drug codes) into continuous vector spaces (i.e., vector embeddings) are critical for the application of deep learning in healthcare. Initial work in this area explored the use of variants of the word2vec algorithm to learn embeddings for m…

New invariant connects boundary PDEs and conformal geometry.

New Fuchsian groups found with special embedding properties.