Research
Multivector fields theory
Multivector Fields Theory (MVF) is a discrete counterpart of vector fields. Its idea is based on Forman's combinatorial vector fields. We equipped the theory with fundamental dynamical concepts providing means for exhaustive study of combinatorial dynamics. Among others, we introduced the notion of combinatorial isolated invariant set, Morse decomposition and the Conley index [LKMW2022]. The natural connections of the theory with the main Topological Data Analysis tool, i.e., persistent homology, opens new directions of studying dynamical systems [DJKKLM2019, DLMS2022, DLMS2023]. Moreover, a recent research shows a new type of computer assisted proof for the existence of a periodic orbit with the use of MFV theory (work in review).
Combinatorial continuation theory
In this project we aim to develop the notion of the continuation of an isolated invariant set in the context of the Multivector Fields (MVF) Theory reflecting the continuous theory of Charles Conley. Continuation allows to relate isolated invariant sets as a dynamical system evolves. As the first step in this direction [DLMS2022] we introduced a simplified definition of continuation and developed a tracking algorithm which allows following changes of a combinatorial isolated invariant set as the underlying dynamical system is evolving. The full development of the theory is an on-going work.
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 qualitative 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, but to find an insight into the processes that generates them.