Conley-Morse-Forman theory for combinatorial multivector fields on Lefschetz complexes

We introduce combinatorial multivector fields, associate with them multivalued dynamics and study their topological features. Our combinatorial multivector fields generalize combinatorial vector fields of Forman. We define isolated invariant sets, Conley index, attractors, repellers and Morse decompositions. We provide a topological characterization of attractors and repellers and prove Morse inequalities. The generalization aims at algorithmic analysis of dynamical systems through combinatorialization of flows given by differential equations and through sampling dynamics in physical and numerical experiments. We provide a prototype algorithm for such applications

Connection matrices in combinatorial topological dynamics

Connection matrices are one of the central tools in Conley's approach to the study of dynamical systems, as they provide information on the existence of connecting orbits in Morse decompositions. They may be considered a generalisation of the boundary operator in the Morse complex in Morse theory. Their computability has recently been addressed by Harker, Mischaikow, and Spendlove in the context of lattice filtered chain complexes. In the current paper, we extend the recently introduced Conley theory for combinatorial vector and multivector fields on Lefschetz complexes by transferring the concept of connection matrix to this setting. This is accomplished by the notion of connection matrix for arbitrary poset filtered chain complexes, as well as an associated equivalence, which allows for changes in the underlying posets. We show that for the special case of gradient combinatorial vector fields in the sense of Forman \cite{Fo98a}, connection matrices are necessarily unique. Thus, the classical results of Reineck have a natural analogue in the combinatorial setting

Conley-Morse-Forman theory for generalized combinatorial multivector fields on finite topological spaces

We generalize and extend the Conley-Morse-Forman theory for combinatorial multivector fields introduced in Mrozek (Found Comput Math 17(6):1585–1633, 2017). The generalization is threefold. First, we drop the restraining assumption in Mrozek (Found Comput Math 17(6):1585–1633, 2017) that every multivector must have a unique maximal element. Second, we define the dynamical system induced by the multivector field in a less restrictive way. Finally, we also change the setting from Lefschetz complexes to finite topological spaces. Formally, the new setting is more general, because every Lefschetz complex is a finite topological space, but the main reason for switching to finite topologcial spaces is because the latter better explain some peculiarities of combinatorial topological dynamics. We define isolated invariant sets, isolating neighborhoods, Conley index and Morse decompositions. We also establish the additivity property of the Conley index and the Morse inequalities

Morse-Conley-Forman theory for generalized combinatorial multivector fields on finite topological spaces

W niniejszej pracy prezentujemy uogólnioną teorię pól multiwektorowych, która w swej pierwszej postaci została przedstawiona w [21]. Bezpośrednim poprzednikiem teorii pól multiwektorowych jest teoria kombinatorycznych pól wektorowych Robina Formana, która z kolei wywodzi się bezpośrednio z dyskretnej teorii Morse’a. Jednym z celów tej rozprawy jest stworzenie kombinatorycznego odpowiednika pól wektorowych obecnych w teorii ciągłych układów dynamicznych oraz stworzenie odpowiednich narzędzi do ich analizy. U podstaw uogólnienia opisywanej teorii leżą trzy fundamentalne modyfikacje założeń. Po pierwsze, definiujemy pola multiwektorowe dla szerszej rodziny skończonych przestrzeni topologicznych, w przeciwieństwie do [21], gdzie konstrukcja dotyczyła kompleksów Lefschetza. Po drugie, odrzucamy wymaganie istnienia unikalnego elementu maksymalnego w multiwektorze. W rezultacie jedynym elementem definicji multiwektora jest założenie o jego lokalnej domkniętości. Uzyskujemy w ten sposób znaczną elastyczność w konstruowaniu pola multiwektorowego. Po trzecie, została uproszczona definicja odwzorowania wielowartościowego indukowanego przez pole multiwektorowe i reprezentującego kombinatoryczną dynamikę. Przekłada się to na uproszczenie dowodów i algorytmicznego aspektu wyznaczania rozkładów Morse’a oraz prowadzi do nowej interpretacji multiwektora jako dynamicznej "czarnej skrzynki". Przy nowych założeniach teorii definiujemy kombinatoryczne odpowiedniki obiektów znanych z teorii ciągłych układów dynamicznych oraz badamy ich własności. Wśród nich mamy: zbiór izolowany niezmienniczy, parę indeksową, indeks Conley’a, zbiory graniczne, atraktor, czy rozkład Morse’a. Pokazujemy również pożądane własności jakich oczekiwalibyśmy od wymienionych wyżej obiektów, m.in. addytywność indeksu Conley’a oraz nierówności Morse’a. Nowe założenia pociągają za sobą konieczność przeprowadzenia nowych dowodów wszystkich własności. W dalszej części pracy korzystamy z podstawowego narzędzia topologicznej analizy danych, tj. homologii persystetnych, do analizy strukturalnej trwałości zbiorów Morse’a. W tym celu konstruujemy moduł persystentny zygzak dla słabych rozkładów Morse’a oraz rozkładów Morse’a. Następnie prezentujemy eksperymenty numeryczne bazujące na omawianej w tej rozprawie teorii. Przedstawiamy algorytm konstrukcji pola multiwektorowego z chmury wektorów. W szczególności uzyskujemy je poprzez próbkowanie wybranych ciągłych układów dynamicznych. W jednym z eksperymentów odtwarzamy graf Conley’a-Morse. Natomiast w kolejnych przykładach korzystamy z homologii persystentnych w celu zbadania ewolucji struktury zbiorów Morse’a względem wybranego parametru modyfikującego dynamikę. Eksperymenty prezentują potencjał dalszego wykorzystania wypracowanych narzędzi do analizy danych o dynamicznej naturze.In this work, we present a generalization of the theory of multivector fields first introduced in [21]. The direct predecessor of the multivector fields theory is the theory of combinatorial vector fields by Robin Forman. His work, in turn, is a natural consequence of a discrete Morse theory. One of the main goals of this thesis is to construct a combinatorial counterpart of vector fields induced by continuous dynamical systems and to create tools for its analysis. The generalization involves three fundamental changes in the setting of the theory. First, we define multivector fields for a broader family of finite topological spaces, in comparison to [21] where Lefschetz complexes are used. Secondly, we lift the assumption that a multivector must have a unique maximal element. Thus, a multivector simply becomes a locally closed subset of space. This results in a greater flexibility in constructing multivector fields. Finally, we define less restrictively the multivalued map induced by a multivector field that defines a combinatorial dynamical system. Consequently, we can simplify the computational aspects of the theory, and we can introduce a new interpretation of a multivector as a dynamical "black box." With a new setting of the multivector fields theory, we define combinatorial counterparts of multiple objects from the classical theory of dynamical systems; among others: isolated invariant set, index pair, Conley index, limit set, attractor, or Morse decomposition. We also show that the desirable properties as additivity of a Conley index and Morse inequalities hold. Even though the theory’s general structure is preserved, new proves and ideas are required by the new setup. In the further part, we use persistent homology – the topological data analysis main tool, to study the robustness of the structure of Morse sets. In particular, we construct a zigzag persistence module for weak Morse decomposition and Morse decomposition for multivector fields. Finally, we show some numerical experiments based on the presented theory. We discuss the algorithm for constructing the multivector field from a vector cloud. As a proof of concept, we study vector clouds obtained by sampling chosen continuous vector fields. In the first experiment, we algorithmically reconstruct the Conley-Morse graph. In the further experiments, we use the persistence homology to study Morse sets’ evolution with respect to a parameter modifying a dynamic. These experiments show the potential of the multivector fields theory as a new analysis tool for data with a dynamical nature

Unsupervised Features Learning for Sampled Vector Fields

In this paper we introduce a new approach to computing hidden features of sampled vector fields. The basic idea is to convert the vector field data to a graph structure and use tools designed for automatic, unsupervised analysis of graphs. Using a few data sets we show that the collected features of the vector fields are correlated with the dynamics known for analytic models which generates the data. In particular the method may be useful in analysis of data sets where the analytic model is poorly understood or not known

The Depth Poset of a Filtered Lefschetz Complex

Taking a discrete approach to functions and dynamical systems, this paper integrates the combinatorial gradients in Forman's discrete Morse theory with persistent homology to forge a unified approach to function simplification. The two crucial ingredients in this effort are the Lefschetz complex, which focuses on the homology at the expense of the geometry of the cells, and the shallow pairs, which are birth-death pairs that can double as vectors in discrete Morse theory. The main new concept is the depth poset on the birth-death pairs, which captures all simplifications achieved through canceling shallow pairs. One of its linear extensions is the ordering by persistence

The isomorphism problem for directed acyclic graphs: an application to multivector fields

This thesis is based on a project developed by a group of researchers at the Faculty of Mathematics and Computer Science at the Jagiellonian University of Krakow. They study sampled dynamics using combinatorial multivector fields. Applying a decomposition into strongly connected components, it is possible to create a directed acyclic graph, called Morse graph, which is a description of the multivector field's global dynamics. Therefore the purpose of this thesis is to compare directed acyclic graphs. In the first chapter we describe the creation process of a Morse graph and an algorithm to study the graph isomorphism problem. The second chapter is dedicated to our personal work, so we describe four Python tests we developed to establish whether two directed acyclic graphs are definitely not isomorphic. In the third chapter we sum up many examples. The last chapter aims to present a possible way for the future work, that is to treat a combinatorial multivector filed as a finite topological space