arXiv research

Study on stable Hamiltonian topology finds non-density of certain structures.

New stable minimal hypersurfaces found in 4-manifolds, proving topology results.

We investigate how Viro's integral calculus applies for the study of the topology of stable maps. We also discuss several applications to Morin maps and complex maps.

New stable homotopy refinement of quantum annular Khovanov homology.

Study the topology of stable vector fields and Lyapunov functions on R^n.

This paper characterizes stable polynomial mappings in a specific set.

A faster, more stable method for optimizing topological functions.

We investigate link homology theories for stable equivalence classes of link diagrams on orientable surfaces. We apply (1+1)-dimensional unoriented topological quantum field theories to Bar-Natan's geometric formalism to define new theories for stable equivalence classes.

The study classifies semiaffine stable planes into affine, projective, or punctured projective planes.

The paper explores density of stable mappings and their properties.

Defines super stable maps and proves quotient superorbifolds for genus zero.

Study finds topological restrictions for stable free boundary CMC surfaces in negatively curved settings.

Paper proves convex domains have one maximum for semi-stable solutions.

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

We show that any complete $f$-stable translating soliton $M$ admits no codimension one cycle which does not disconnect $M$. As a corollary, it follows that any two dimensional complete $f$-stable translating soliton has genus zero.

Topology of non-orientable spaces without boundary is studied.

Under a simple assumption on Seifert surfaces, we characterise knots whose stable topological 4-genus coincides with the genus.

Paper introduces stable vectorization for multiparameter PH using signed barcodes.

We show that the difference between the genus and the stable topological 4-genus of alternating knots is either zero or at least 1/3.

Given an injective amalgam at the level of fundamental groups and a specific 3-manifold, is there a corresponding geometric-topological decomposition of a given 4-manifold, in a stable sense? We find an algebraic-topological splitting criterion in terms of the orientation classes and universal covers. Also, we equivari…

We provide a new topological obstruction for complete stable minimal hypersurfaces in R^n. For $n\geq 4$, we prove that any complete orientable stable hypersurfaces in R^n has only one end. This follows from a more general analytic theorem we prove in the paper.

Non-compact manifolds prevent $C^0$-stable mappings from being dense.

New bounds set for stable 2-systole in specific geometric spaces.

Proves conjecture on graph configuration spaces' complexity.

The gradient flow of the Yang-Mills action acts pointwise on closed loops of gauge fields. We construct a topologically nontrivial loop of SU(2) gauge fields on S4 that is locally stable under the flow. The stable loop is written explicitly as a path between two gauge fields equivalent under a topologically nontrivial …

In this paper, we study stability and instability problem for type-II partitioning problem. First, we make a complete classification of stable type-II stationary hypersurfaces in a ball in a space form as totally geodesic $n$-balls. Second, for general ambient spaces and convex domains, we give some topological restric…

This paper studies the canonical Chow quotient of a smooth projective variety by a reductive algebraic group. The main purpose is to give some topological interpretations and characterization of Chow quotient which have the advantage to be more intuitive and geometric. This is to be done over the field of complex numbe…

Stable knots and links can exist in electromagnetic fields.

Stable actions of hyperbolic groups on their boundaries.

Generic smooth plane-to-plane map germs are topologically equivalent to cones of mappings of the circle. We carry out a complete topological classification of smooth stable mappings of the circle and show how this classification leads, via the result mentioned above, to a topological classification of finitely determin…

Paper proves no specific CMC hypersurfaces in hyperbolic space.

New invariant for classifying 4-manifolds up to cobordism.

Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can stabilize invariants characterizing these objects. We outline how so called contour fun…

Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persist…

Visual construction of maps linking to two-bridge links.

Computes mapping class groups of 4-manifolds with boundary.

We study here some aspects of the topology of the space of smooth, stable, genus 0 curves in a Riemannian manifold $X$, i.e. the Kontsevich stable curves, which are not necessarily holomorphic. We use the Hofer-Wysocki-Zehnder polyfold structure on this space and some natural characteristic classes, to show that for $X…

In this paper, we study stable weighted minimal hypersurfaces in manifolds with nonnegative Bakry-Emery Ricci curvature. We will give some geometric and topological applications. In particular, we give some partial classification of complete 3-manifolds with nonnegative Bakry-Emery Ricci curvature assuming that $f$ is …

Smoothly isotopic surfaces in a 4-manifold are stable isotopic if trivial in the 2-homology.

We show that, up to topological conjugation, the equivalence class of a Morse-Smale diffeomorphism without heteroclinic curves on 3-manifold is completely defined by an em- bedding of two-dimensional stable and unstable heteroclinic laminations to a characteristic space.

Stable topological summary captures evolving dependency structure in dynamic Bayesian networks.

Research describes all possible gradient vector fields on a sphere with up to ten singular points.

Paper proves no stable Yang-Mills fields on spheres.

Hyperbolic groups act on spheres, and nearby actions are semi-conjugate.

We show that stable commutator length is rational on free products of free Abelian groups amalgamated over $\mathbb{Z}^k$, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups and parameterize all surfaces with specified boundary mapping to th…

Many attempts have been made in recent decades to integrate machine learning (ML) and topological data analysis. A prominent problem in applying persistent homology to ML tasks is finding a vector representation of a persistence diagram (PD), which is a summary diagram for representing topological features. From the pe…

New invariants derived from random matrices for words in free groups.

We study the topology of the tropical moduli space parametrizing stable tropical curves of genus g with n marked points in which the bounded edges have total length 1, and prove that it is highly connected. Using the identification of this space with the dual complex of the boundary in the moduli space of stable algebr…