Related papers: Barcodes for Hamiltonian homeomorphisms of surface…

This paper proposes a local search algorithm for a specific combinatorial optimisation problem in graph theory: the Hamiltonian Completion Problem (HCP) on undirected graphs. In this problem, the objective is to add as few edges as possible…

We demonstrate that the proper homotopy equivalence relation for locally finite graphs is Borel complete. Furthermore, among the infinite graphs, there is a comeager equivalence class. As corollaries, we obtain the analogous results for the…

Persistent homology tracks topological features across geometric scales, encoding birth and death of cycles as barcodes. We develop a complementary theory where the filtration parameter is algebraic precision rather than geometric scale.…

We present a braid-theoretic approach to combinatorially computing knot Floer homology. To a knot or link K, which is braided about the standard disk open book decomposition for (S^3,\xi_std), we associate a corresponding multi-pointed nice…

The paper concerns area preserving homeomorphisms of surfaces that are isotopic to the identity. The purpose of the paper is to find a maximal identity isotopy such that we can give a fine descriptions of the dynamics of its transverse…

The periodic Floer homology of a surface symplectomorphism, defined by the first author and M. Thaddeus, is the homology of a chain complex which is generated by certain unions of periodic orbits, and whose differential counts certain…

Persistent homology is a mathematical tool used for studying the shape of data by extracting its topological features. It has gained popularity in network science due to its applicability in various network mining problems, including…

Time dependence is a universal phenomenon in nature, and a variety of mathematical models in terms of dynamical systems have been developed to understand the time-dependent behavior of real-world problems. Originally constructed to analyze…

The purpose of this note is to extend to Brownian loops some homology and holonomy results obtained in the case of discrete loops on a graph

In this paper we present a new approach to computing homology (with field coefficients) and persistent homology. We use concepts from discrete Morse theory, to provide an algorithm which can be expressed solely in terms of simple graph…

Given a compactly supported Hamiltonian diffeomorphism of the plane, one can define a generating function for it. In this paper, we show how generating functions retain information about the braid type of collections of fixed points of…

We find robust obstructions to representing a Hamiltonian diffeomorphism as a full $k$-th power, $k \geq 2,$ and in particular, to including it into a one-parameter subgroup. The robustness is understood in the sense of Hofer's metric. Our…

The purpose of this paper is to study transport equations on the unit tangent bundle of closed oriented Riemannian surfaces and to connect these to the transport twistor space of the surface (a complex surface naturally tailored to the…

We combine standard persistent homology with image persistent homology to define a novel way of characterizing shapes and interactions between them. In particular, we introduce: (1) a mixup barcode, which captures geometric-topological…

We define Floer homology for a time-independent, or autonomous Hamiltonian on a symplectic manifold with contact type boundary, under the assumption that its 1-periodic orbits are transversally nondegenerate. Our construction is based on…

This paper extends the decorated Teichm\"uller theory developed before for punctured surfaces to the setting of ``bordered'' surfaces, i.e., surfaces with boundary, and there is non-trivial new structure discovered. The main new result…

The symplectic Floer homology HF_*(f) of a symplectomorphism f:S->S encodes data about the fixed points of f using counts of holomorphic cylinders in R x M_f, where M_f is the mapping torus of f. We give an algorithm to compute HF_*(f) for…

The computational complexity of the graph isomorphism problem is considered to be a major open problem in theoretical computer science. It is known that testing isomorphism of chordal graphs is polynomial-time equivalent to the general…

While there has been much interest in adapting conventional clustering procedures---and in higher dimensions, persistent homology methods---to directed networks, little is known about the convergence of such methods. In order to even…

We introduce a homotopy theory of digraphs (directed graphs) and prove its basic properties, including the relations to the homology theory of digraphs constructed by the authors in previous papers. In particular, we prove the homotopy…