arXiv research

Study knot singularities in Bogomolny equation solutions.

Explains how knots relate to 4D shapes.

Notes on Khovanov and knot Floer theories' stable homotopy types.

Lecture notes on Heegaard Floer homology for beginners.

We characterize those closed $2k$-manifolds admitting smooth maps into $(k+1)$-manifolds with only finitely many critical points, for $k\in\{2,4\}$. We compute then the minimal number of critical points of such smooth maps for $k=2$ and, under some fundamental group restrictions, also for $k=4$. The main ingredients ar…

This is a survey on the various notions of Kodaira dimension in low dimensional topology. The focus is on progress after the 2006 survey [78].

Improved likelihood estimation for singular distributions using deep models.

In 1995 the author, Jones, and Segal introduced the notion of "Floer homotopy theory". The proposal was to attach a (stable) homotopy type to the geometric data given in a version of Floer homology. More to the point, the question was asked, "When is the Floer homology isomorphic to the (singular) homology of a natural…

This article sketches various ideas in contact geometry that have become useful in low-dimensional topology. Specifically we (1) outline the proof of Eliashberg and Thurston's results concerning perturbations of foliatoins into contact structures, (2) discuss Eliashberg and Weinstein's symplectic handle attachments, an…

Unified framework for 3D and 4D manifold and knot theory.

In connection with his interest in selfdistributive algebra, Richard Laver established two deep results with potential applications in low-dimensional topology, namely the existence of what is now known as the Laver tables and the well-foundedness of the standard ordering of positive braids. Here we present these resul…

Singular Riemannian Foliations are particular types of foliations on Riemannian manifolds, in which leaves locally stay at a constant distance from each other. Singular Riemannian Foliations in round spheres play a special role, since they provide "infinitesimal information" about general Singular Riemannian Foliations…

Survey on the topology of singular foliations in complex 2-space.

We define an equivalence relation called A-isotopy between finitely determined map-germs, which is a strengthened version of A-equivalence. We consider the number of A-isotopy classes of equidimensional Morin singularities, and some other well-known low-dimensional singularities. We also give an application to stable p…

The primary objects of study in the ``knot theory of complex plane curves'' are C-links: links (or knots) cut out of a 3-sphere in the complex plane by complex plane transverse and totally tangential. Transverse C-links are naturally oriented. There are many natural classes of examples: links of singularities; links at…

Abstracts discuss a common framework for constructing homology theories.

In this paper we discuss algebraic, combinatorial and topological properties of singular virtual braids. On the algebraic side we state the relations between classical and virtual singular objects, in addition we discuss a Birman-like conjecture for the virtual case. On the topological and combinatorial side, we prove …

Study empty polar varieties' impact on singular function-germs.

Paper proves families of singularities can be topologically trivialized.

Survey on categorifying Jones polynomial.

Study on the topology of leaves in singular Riemannian foliations.

We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a di…

We introduce an equivalence relation, called cobordism, for words and study cobordism invariants of words inspired by methods of low-dimensional topology.

This study examines the topology of singularities in optimal semicouplings between unequal spaces.

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

Paper introduces a complete metric topology for low energy spaces.

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.

This research classifies singular foliations and finds a universal deformation.

Paper proves rigidity of de-Sitter tori with conical singularities.

In this short paper, a neural network that is able to form a low dimensional topological hidden representation is explained. The neural network can be trained as an autoencoder, a classifier or mix of both, and produces different low dimensional topological map for each of them. When it is trained as an autoencoder, th…

L. Kauffman conjectured that a particular solution of the Chinese Rings puzzle is the simplest possible. We prove his conjecture by using low-dimensional topology and group theory. We notice also a surprising connection between the Chinese Rings and Habiro moves (related to Vassiliev invariants).

Proves positive mass theorem for AF spin manifolds with conical singularities.

The purpose of this thesis is to study classical combinatorial objects, such as polytopes, polytopal complexes, and subspace arrangements, using tools that have been developed in combinatorial topology, especially those tools developed in connection with (discrete) differential geometry, geometric group theory and low-…

We review localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low-dimensional gauge theories. These are the functional integral counterparts of the Mathai-Quillen formalism, the Duistermaa…

Classical topological concepts are applied to understand high performance computing simulations of molecules writhing in three dimensional space. These simulations produce peta-bytes of floating point data, to describe 3 dimensional changes in molecular structure. A zero-th order analysis is achieved by viewing a compu…

Let X be the moduli space of SL(n,C), SU(n), GL(n,C), or U(n)-valued representations of a rank r free group. We classify the algebraic singular stratification of X. This comes down to showing that the singular locus corresponds exactly to reducible representations if there exist singularities at all. Then by relating a…

Defines Milnor number for foliations and shows its topological invariance.

Detects singularities in complex data to improve machine learning models.

Modeling wormhole creation without singularities in relativity.

We simplify neural networks to 3D to study their topological changes.

New method detects essential tori in mixed singularity links.

Study nondegenerate singularities in mean curvature flow.

Maps with boundary definite fold points restrict manifold structure.

Study on homology groups of cDV singularity links, identifying their topology.

Modeling 3D continua with singular points using Yin sets.

Causal fermion systems and Riemannian fermion systems are proposed as a framework for describing non-smooth geometries. In particular, this framework provides a setting for spinors on singular spaces. The underlying topological structures are introduced and analyzed. The connection to the spin condition in differential…

Khimshiashvili proved a topological degree formula for the Eu-ler characteristic of the Milnor fibres of a real function-germ with an isolated singularity. We give two generalizations of this result for non-isolated singularities. As corollaries we obtain an algebraic formula for the Euler characteristic of the fibres …

The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less then 2π plays an important role in many problems in low dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli of convex hyperbolic and Euclidea…