Abstract : Recent years have seen the rapid development of topological data analysis. Via combinatorial modeling of space, TDA enables the study of the geometry of data using persistent homology. In particular, it may be used to analyze the phase space of a sampled dynamical system, thereby providing a static image. To get a dynamic view, a better understanding of classical topological tools in dynamics in the context of data is needed. The aim of this session is to bring together researchers from TDA and dynamical systems to study dynamic aspects of data via topological tools, in particular Morse and Conley theory.
Organizer(s) : Konstantin Mischaikow, Marian Mrozek, Thomas Wanner
00595 (1/3) : 5B @G802 [Chair: Konstantin Mischaikow]
[05133] Analyzing Network Representations of Dynamical Systems Using Persistent Homology
Format : Online Talk on Zoom
Author(s) :
Elizabeth Munch (Michigan State University)
Abstract : Persistent homology, the flagship method of topological data analysis, can be used to provide a quantitative summary of the shape of data. One way to pass data to this method is to start with a finite, discrete metric space (whether or not it arises from a Euclidean embedding) and to study the resulting filtration of the Rips complex. In this talk, we will discuss several available methods for turning a time series into a discrete metric space, including the Takens embedding, and ordinal partition networks. Combined with persistent homology and machine learning methods, we show how this can be used to classify behavior in time series in both synthetic and experimental data.
[01382] Topological Data Analysis of Spatiotemporal Honeybee Aggregation
Format : Online Talk on Zoom
Author(s) :
Elizabeth Bradley (University of Colorado)
Chad Topaz (Williams College)
Golnar Gharooni Fard (University of Colorado)
Varad Deshmukh (University of Colorado)
Orit Peleg (University of Colorado)
Morgan Byers (University of Colorado)
Abstract : We employ topological data analysis to explore honeybee aggregations
in the context of trophallaxis: the exchange of food among nestmates.
Using synthetic and laboratory data, we build topological summaries
called CROCKER plots to capture the shape of the data as a function of
both scale and time. Our results show two distinct regimes
corresponding to successive dynamical regimes: a dispersed phase
before the food is introduced, followed by a food-exchange phase in
which clusters form.
[03128] What does Multivector Fields Theory have to offer?
Format : Talk at Waseda University
Author(s) :
Michał Lipiński (Polish Academy of Sciences)
Abstract : The theory of Multivector Fields (MVF) is a generalization of Forman vector fields. It has been continuously developed since 2017. MVF theory is equipped with a number of fundamental dynamical concepts and matures into a useful combinatorial model for classical vector fields. In the talk I will present the general idea of the MVF theory, its suitability for the analysis in the spirit of topological data analysis, and its usefulness in understanding continuous dynamical systems.
[03410] A Persistence-like Algorithm for Computing Connection Matrices Efficiently
Format : Talk at Waseda University
Author(s) :
Tamal Krishna Dey (Purdue University)
Abstract : Connection matrices are a generalization of Morse boundary operators from the classical Morse theory for gradient vector fields. Developing an efficient computational framework for connection matrices is particularly important in the context of a rapidly growing data science that requires new mathematical tools for discrete data. Toward this goal, the classical theory for connection matrices has been adapted to combinatorial frameworks that facilitate computation. We develop an efficient persistence-like algorithm to compute a connection matrix from a given combinatorial (multi) vector field on a simplicial complex. This algorithm requires a single-pass, improving upon a known algorithm that runs an implicit recursion executing two-passes at each level. Overall, the new algorithm is more simple, direct, and efficient than the state-of-the-art. Because of the algorithm's similarity to the persistence algorithm, one may take advantage of various software optimizations from topological data analysis.
This is a joint work with Michal Lipinski, Marian Mrozek, and Ryan Slechta
[04935] A combinatorial/homological framework for continuous nonlinear dynamics
Format : Talk at Waseda University
Author(s) :
Konstantin Mischaikow (Rutgers University)
Abstract : Computational science and data-driven science suggests the importance of having finite models of dynamics. This raises three questions: 1. How to go from data to appropriate combinatorial models of dynamics. 2. What computations should be performed on these combinatorial models. 3. How to translate the output from the combinatorial models to structures associated with continuous systems. In this talk we will discuss our attempts to provide a coherent approach to addressing these questions.
[04988] Computing the Global Dynamics of Parameterized Systems of ODEs
Format : Talk at Waseda University
Author(s) :
Marcio Gameiro (Rutgers University)
Abstract : We present a combinatorial topological method to compute the dynamics of a parameterized family of ODEs. A discretization of the state space of the systems is used to construct a combinatorial representation from which recurrent versus non-recurrent dynamics are extracted. Algebraic topology is then used to validate and characterize the dynamics of the system. We will discuss the combinatorial description and the algebraic topological computations and will present applications to systems of ODEs arising from gene regulatory networks.
[04981] Combinatorics and Topology for Understanding Global Dynamics in Multi-Scale Systems.
Format : Talk at Waseda University
Author(s) :
Ewerton Rocha Vieira (Rutgers University)
Abstract : This talk introduces a new approach to analyzing time-varying systems with multi-scale dynamics, which can be challenging due to poorly measured parameters and numerous variables. Traditionally, these systems are modeled using ordinary differential equations (ODE), but this approach can be difficult to apply directly. The proposed approach is based on combinatorics and algebraic topology, and focuses on describing global dynamics in terms of annotated graphs (Morse graphs) and Conley complexes. The method is based on piecewise linear models and offers a more robust, scalable, and computable description of dynamics than classical ODE analysis, with formal mathematical guarantees that extend to a class of ODE with steep sigmoidal nonlinearities. This approach is particularly useful for modeling complex systems, such as biological systems.
[04843] On the identification of cycling motion using topological tools
Format : Talk at Waseda University
Author(s) :
Ulrich Bauer (Technical University of Munich)
David Hien (Technical University of Munich)
Oliver Junge (Technical University of Munich)
Konstantin Mischaikow (Rutgers University)
Abstract : Nonlinear dynamical systems often exhibit complicated recurrent behaviour. We propose to decompose recurrent sets into elementary oscillations and the connections between them. To this end, we use topological tools that are flexible enough to be computed from data while still providing a comprehensive description of the oscillations. We demonstrate this through several examples. In particular, we identify and analyze 6 oscillations in a 4d hyperchaotic attractor.
[03335] Morse-Smale quadrangulations and persistence of vector fields
Format : Talk at Waseda University
Author(s) :
Claudia Landi (Università di Modena e Reggio Emilia)
Clemens Luc Bannwart (Università di Modena e Reggio Emilia)
Abstract : The goal of this talk is to introduce a persistence barcode for discrete gradient vector fields defined on a combinatorial 2-manifold. The main ingredient is a decomposition of the manifold into quadrangle regions with vertices ordinately given by sinks, saddles, sources, and saddles. We will consider a bottleneck distance and an interleaving distance for such barcodes and we will present results about their stability.
[04607] On the dynamics of the combinatorial model of the real line
Format : Online Talk on Zoom
Author(s) :
Pedro J. Chocano (Rey Juan Carlos University)
Abstract : In this talk, we describe the dynamics that appear when we consider a discrete dynamical system defined on the combinatorial model of the real line. Particularly, we show that there are no periodic points of period greater than or equal to 3, which contrasts with the classical setting (Sharkovski theorem). This fact motivates us to introduce multivalued maps to get richer dynamics than the ones obtained from single valued maps. To conclude we provide some examples.
[04886] Topological Inference of the Conley Index
Format : Online Talk on Zoom
Author(s) :
Vidit Nanda (University of Oxford)
Ka Man Yim (University of Oxford)
Abstract : The Conley index of an isolated invariant set is a fundamental object in the study of dynamical systems. Here we consider smooth functions on closed submanifolds of Euclidean space and describe a framework for inferring the Conley index of any compact, connected isolated critical set of such a function with high confidence from a sufficiently large finite point sample. The main construction of this paper is a specific index pair which is local to the critical set in question. We establish that these index pairs have positive reach and hence admit a sampling theory for robust homology inference. This allows us to estimate the Conley index, and as a direct consequence, we are also able to estimate the Morse index of any critical point of a Morse function using finitely many local evaluations.
[05010] Analysis of solids regarded as compositions of discrete entities
Format : Online Talk on Zoom
Author(s) :
Andrey Jivkov (The University of Manchester)
Abstract : In contrast with the idealisation of materials as continua, their internal structures are regarded here as polyhedral complexes. One approach to formulate conservation laws of scalar (mass, energy, charge) and vector (linear and angular momentum) quantities on such complexes is presented. It uses combi-natorial differential forms, representing physical quantities, and operations with such forms (exterior derivatives, exterior products, and codifferentials), describing processes and conservations. The derived conservation laws are background independent.
