Research

Conley-Morse persistence barcodes

Inspired by the previous projects [DLMS2022] revealing strong connections between multivector field theory, Conley index theory, and persistent homology, we develop the Conley-Morse persistence barcode [DLS2025]. It serves as a homological signature for parameterized combinatorial dynamical system, drawing on ideas from topological data analysis. The Conley-Morse persistence barcode captures qualitative changes in Morse sets and their Conley indices. Moreover, it tracks how Conley indices interact when their corresponding isolated invariant sets collide.

Slides from: DyToComp2025

Computation of a connection matrix

The connection matrix is yet another key concept from Conley index theory. It generalizes the construction of the Morse complex and algebraically encodes information about connections between Morse sets in a Morse decomposition.

In [DLMS2024], we introduced an algorithm for the computation of the connection matrix for combinatorial dynamical systems. Later, we simplified the algorithm [DHL2025], making it more closely aligned with the standard persistent homology algorithm.

Morse predecomposition

Morse decomposition can be seen as a compact summary of a dynamical system, which captures the gradient behavior in the system. However the Morse decomposition is useful, it is not well suited for systems exhibiting extensive recurrent sets, which are clumpted into enigmatic black boxes (nodes in the Morse graph). In [LMM2025], we extend the idea and introduce a more general concept called Morse predecomposition, which allows for a further deconstruction of a recurrent invariant set and helps revealing its internal structure. We study the idea of Morse predecomposition both in combinatorial and continuous setting.

A combinatorial model of a Lorenz atractor and its Morse predecomposition.

Reconstructing dynamics from a time series

What can we say about a dynamical system based on just a finite collection of discrete trajectories? In the recent study we develop a score (ConjTest) measuring a topological similarity of time series [DLS2023]. The method is inspired by the notion of topological conjugacy. Thus, the aim is not to simply compare the time series as point clouds, but to find an insight into the processes that generate them.

A presentation on ConjTest: ECMI 2023

Persistence of combinatorial Conley index

What is a counterpart of a small perturbation of a combinatorial dynamical system? Can we study a countinuous-like evolution of such a system?

In this project try to develop a combinatorial version of a continuation of an isolated invariant set in the context of the Multivector Fields Theory (MVF). As the first step in this direction we introduced a simplified definition of continuation and developed an algorithm which allows to track changes of a combinatorial isolated invariant set and its Conley index as the underlying combinatorial dynamical system evolves [DLMS2022].

A presentations on combinatorial continuation from: ComPers 2022, ECM 2024

Multivector field theory (MVF)

Multivector Field Theory (MVF) is a combinatorial counterpart of vector fields generalizing the idea of Forman's combinatorial vector fields. The theory is equipped with a number of fundamental dynamical concepts, among others: combinatorial isolated invariant set, Morse decomposition and the Conley index, making it a solid dynamical theory on its own.

The theory admits natural connections to persistent homology (the main tool of Topological Data Analysis) opening new directions for dynamically inspired data analysis [DJKKLM2019, DLMS2022, DLMS2023].

See an introduction to MVF: LKMW2022 or PhD thesis.

A presentations on MVF from: Oxford seminar 2020, ICIAM 2023.

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