Related papers: Legendrian persistence modules and dynamics

Based on the contact Hamiltonian Floer theory established by Will J. Merry and the second author that applies to any admissible contact Hamiltonian system $(M, \xi = \ker \alpha, h)$, where $h$ is a contact Hamiltonian function on a…

For any Legendrian knot or link in $\mathbb{R}^3$, we construct an $L_\infty$ algebra that can be viewed as an extension of the Chekanov-Eliashberg differential graded algebra. The $L_\infty$ structure incorporates information from rational…

In this work we introduce contact Hamiltonian mechanics, an extension of symplectic Hamiltonian mechanics, and show that it is a natural candidate for a geometric description of non-dissipative and dissipative systems. For this purpose we…

In this paper, we introduce a geometric description of contact Lagrangian and Hamiltonian systems on Lie algebroids in the framework of contact geometry, using the theory of prolongations. We discuss the relation between Lagrangian and…

We introduce a new notion of persistence modules endowed with operators. It encapsulates the additional structure on Floer-type persistence modules coming from the intersection product with classes in the ambient (quantum) homology, along…

We describe a method, based on "hard" contact topology, of showing the existence of semi-infinite trajectories of contact Hamiltonian flows which start on one Legendrian submanifold and asymptotically converge to another Legendrian…

Motivated by the problem of global stability of thermodynamical equilibria in non-equilibrium thermodynamics formulated in a recent paper [12], we introduce some mechanisms for constructing semi-infinite orbits of contact Hamiltonian…

The theory of persistence modules is an emerging field of algebraic topology which originated in topological data analysis. In these notes we provide a concise introduction into this field and give an account on some of its interactions…

In this paper, we discuss several relations between the existence of invariant volume forms for Hamiltonian systems on Poisson-Lie groups and the unimodularity of the Poisson-Lie structure. In particular, we prove that Hamiltonian vector…

Let $M$ be a smooth closed orientable manifold and $\mathcal{P}(M)$ the space of Poisson structures on $M$. We construct a Poisson bracket on $\mathcal{P}(M)$ depending on a choice of volume form. The Hamiltonian flow of the bracket acts on…

We introduce new invariants associated to collections of compact subsets of a symplectic manifold. They are defined through an elementary-looking variational problem involving Poisson brackets. The proof of the non-triviality of these…

In this work we devise a stochastic version of contact Hamiltonian systems, and show that the phase flows of these systems preserve contact structures. Moreover, we provide a sufficient condition under which these stochastic contact…

We discuss in this paper the canonical structure of classical field theory in finite dimensions within the {\it{pataplectic}} hamiltonian formulation, where we put forward the role of Legendre correspondance. We define the Poisson…

This sketch shows that the usual generating function based capacities have an interpretation in the language of persistent homology as persistences of certain homology classes in the persistence module formed by the corresponding generating…

This article is devoted to a description of the dynamics of the phase flow of monotone contact Hamiltonian systems. Particular attention is paid to locating the maximal attractor (or repeller), which could be seen as the union of compact…

We construct a combinatorial invariant of Legendrian knots in standard contact three-space. This invariant, which encodes rational relative Symplectic Field Theory and extends contact homology, counts holomorphic disks with an arbitrary…

This paper is a survey of persistent homology, primarily as it is used in topological data analysis. It includes the theory of persistence modules, as well as stability theorems for persistence barcodes, generalized persistence,…

Let $L\subset J^1(M)$ be a Legendrian submanifold of the 1-jet space of a Riemannian $n$-manifold $M$. A correspondence is established between rigid flow trees in $M$ determined by $L$ and boundary punctured rigid pseudo-holomorphic disks…

We present a new existence mechanism, based on symplectic topology, for orbits of Hamiltonian flows connecting a pair of disjoint subsets in the phase space. The method involves function theory on symplectic manifolds combined with rigidity…

We investigate combinatorial dynamical systems on simplicial complexes considered as {\em finite topological spaces}. Such systems arise in a natural way from sampling dynamics and may be used to reconstruct some features of the dynamics…