arXiv research

The article consists of a survey on analytic and topological torsion. Analytic torsion is defined in terms of the spectrum of the analytic Laplace operator on a Riemannian manifold, whereas topological torsion is defined in terms of a triangulation. The celebrated theorem of Cheeger and Müller identifies these two noti…

The main purpose of this paper is to give a topological and symplectic classification of completely integrable Hamiltonian systems in terms of characteristic classes and other local and global invariants.

Study enhances neural network predictions for wave height using topological features.

Paper tackles dynamic behavior of variable topology mechanisms, presenting new transition conditions.

This paper proposes a new method for learning covers of geometric datasets to improve topological inference and visualization.

Topology-GS improves 3D GS for better structural and feature integrity.

In this paper we study a notion of topological complexity for the motion planning problem. The topological complexity is a number which measures discontinuity of the process of motion planning in the configuration space X. More precisely, it is the minimal number k such that there are k different motion planning rules,…

We provide an upper bound on the topological complexity of twisted products. We use it to give an estimate $TC(X)\le TC(π_1(X))+\dim X$ of the topological complexity of a space in terms of its dimension and the complexity of its fundamental group.

We consider several diffeomorphism invariant field theories of 2- and 3-forms in six dimensions. They all share the same kinetic term $BdC$, but differ in the potential term that is added. The theory $BdC$ with no potential term is topological - it describes no propagating degrees of freedom. We show that the theory co…

Topological proof of Weil-Petersson symplectic form using Fenchel-Nielsen coordinates.

Topology-enhanced loss improves 3D object reconstruction from 2D images.

A topological groupoid G is K-pointed, if it is equipped with a homomorphism from a topological group K to G. We describe the homotopy groups of such K-pointed topological groupoids and relate these groups to the ordinary homotopy groups in terms of a long exact sequence. As an application, we give an obstruction to pr…

Characterizes continuity of monotone functionals in mixed topology.

New method detects uncertainty in neural networks for out-of-distribution detection.

Extends topological results for nonpositive curvature spaces.

Study delta invariant of curves on rational surfaces using topological methods.

String topology coproduct and Turaev cobracket computed for surfaces.

Paper introduces a new topological loss for better convergence.

We propose a novel approach for preserving topological structures of the input space in latent representations of autoencoders. Using persistent homology, a technique from topological data analysis, we calculate topological signatures of both the input and latent space to derive a topological loss term. Under weak theo…

This work deals with the topological classification of germs of singular foliations on $(\mathbb C^{2},0)$. Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and the projective holonomy representations and we compute the moduli space of topo…

We construct an infinite family of smoothly slice knots that we prove are topologically doubly slice. Using the correction terms coming from Heegaard Floer homology, we show that none of these knots is smoothly doubly slice. We use these knots to show that the subgroup of the double concordance group consisting of smoo…

We present two paradigms relating algebraic, topological and quantum computational statistics for the topological model for quantum computation. In particular we suggest correspondences between the computational power of topological quantum computers, computational complexity of link invariants and images of braid grou…

We use an estimate on the Thurston--Bennequin invariant of a Legendrian link in terms of its Kauffman-polynomial to show that links of topological unknots, e.g. the Borromean rings or the Whithead link, may not be represented by Legendrian links of Legendrian unknots.

Study monopole h-invariants from a topological viewpoint.

We extend the topological field theory (``itsy bitsy topological field theory"') of our previous work from mod-2 to twisted coefficients. This topological field theory is derived from sutured Floer homology but described purely in terms of surfaces with signed points on their boundary (occupied surfaces) and curves on …

A new method tracks index using topological data analysis for sparse portfolios.

New topology shows Morse boundaries are topologically invariant.

Graph braid groups' complexity stabilizes for most graphs.

The boundary of hyperbolic groups is locally simply connected.

We prove integral curvature bounds in terms of the Betti numbers for compact submanifolds of the Euclidean space with low codimension. As an application, we obtain topological obstructions for $δ$-pinched immersions. Furthermore, we obtain intrinsic obstructions for minimal submanifolds in spheres with pinched second f…

Proposes a method to infer complex network topologies from multiple graphs.

Determining when two knots are equivalent (more precisely isotopic) is a fundamental problem in topology. Here we formulate this problem in terms of Predicate Calculus, using the formulation of knots in terms of braids and some basic topological results. Concretely, Knot theory is formulated in terms of a language with…

The paper explores the topology of polygonal meshes and their properties.

We develop a tighter implementation of basic PL topology, which keeps track of some combinatorial structure beyond PL homeomorphism type. With this technique we clarify some aspects of PL transversality and give combinatorial proofs of a number of known results. New results include a combinatorial characterization of c…

3D topological order linked to Seifert manifolds and gauge groups.

We consider intersecting hypersurfaces in curved spacetime with gravity governed by a class of actions which are topological invariants in lower dimensionality. Along with the Chern-Simons boundary terms there is a sequence of intersection terms that should be added in the action functional for a well defined variation…

Study bounds topological entropy of toroidal attractors.

The bipolar filtration of Cochran, Harvey and Horn presents a framework of the study of deeper structures in the smooth concordance group of topologically slice knots. We show that the graded quotient of the bipolar filtration of topologically slice knots has infinite rank at each stage greater than one. To detect nont…

Canonical quantization of abelian BF-type topological field theory coupled to extended sources on generic d-dimensional manifolds and with curved line bundles is studied. Sheaf cohomology is used to construct the appropriate topological extension of the action and the topological flux quantization conditions, in terms …

Study index theory on Lie group homogeneous spaces using topological and analytic methods.

We use a knot invariant, namely the Tristram--Levine signature to study deformations of singular points of plane curves. We find a bound on the sum of M numbers over all singularities of a generic fiber in terms of the M number of the singularity at the central fiber and some topological data.

We present a new, more elementary proof of the Freedman-Teichner result that the geometric classification techniques (surgery, s-cobordism, and pseudoisotopy) hold for topological 4-manifolds with groups of subexponential growth. In an appendix Freedman and Teichner give a correction to their original proof, and reform…

Improved modeling of persistence diagrams for data analysis.

It is known that, for Dirac operators on Riemann surfaces twisted by line bundles with Hermitian-Einstein connections, it is possible to obtain estimates for the first eigenvalue in terms of the topology of the twisting bundle \cite{JL2}. Attempts to generalize topological estimates for higher rank bundles or higher di…

The correlation functions of supersymmetric gauge theories on a four-manifold X can sometimes be expressed in terms of topological invariants of X. We show how the existence of superconformal fixed points in the gauge theory can provide nontrivial information about four-manifold topology. In particular, in the example …

This article introduces descriptive cellular homology on cell complexes, which is an extension of J.H.C. Whitehead's CW topology. A main result is that a descriptive cellular complex is a topology on fibres in a fibre bundle. An application of two forms of cellular homology is given in terms of the persistence of shape…

New framework improves worst-case generalization bounds for stochastic optimization.

Study characterizes compact Einstein-type manifolds with boundary.