| Abstract: |
| \begin{abstract}
Topological data analysis (TDA) is usually associated with point clouds and scalar filtrations, but many problems in dynamics naturally come with richer objects: finite trajectory segments, delay-coordinate reconstructions, or vector fields known either analytically or only through sampling. In this talk I will discuss a collection of recent TDA-inspired methods that shift the focus from the topology of static data to the comparison of dynamical systems themselves \cite{Carlsson2009,EdelsbrunnerHarer2010}. On the one hand, I will present tests for topological conjugacy constructed from finite orbit samples and time series, including a formulation based on Takens reconstructions, which makes it possible to assess whether two observed processes encode the same underlying dynamics \cite{Takens1981,DlotkoLipinskiSignerska2024}. On the other hand, I will describe new descriptors for sampled and continuous vector fields---including begin--end point embeddings, densities of directions, Euler characteristic curves, and Euler characteristic profiles---designed to compare dynamics directly at the level of trajectories or vector fields, detect qualitative changes such as bifurcations, and remain computationally feasible in data-driven settings \cite{MarszewskaSignerskaRynkowskaDlotkoInPrep}. Taken together, these methods show that ideas originating in TDA can be turned into practical tools for the quantitative and qualitative analysis of dynamics well beyond the standard point-cloud setting: from finite samples of trajectories to sampled or continuous vector fields, and from discrete observations to analytically defined systems.
\end{abstract}
\begin{thebibliography}{99}
\bibitem{Carlsson2009}
G.~Carlsson,
\newblock Topology and data,
\newblock {\em Bulletin of the American Mathematical Society} 46(2):255--308, 2009.
\bibitem{EdelsbrunnerHarer2010}
H.~Edelsbrunner and J.~Harer,
\newblock {\em Computational Topology: An Introduction},
\newblock American Mathematical Society, 2010.
\bibitem{Takens1981}
F.~Takens,
\newblock Detecting strange attractors in turbulence,
\newblock in {\em Dynamical Systems and Turbulence, Warwick 1980}, Lecture Notes in Mathematics 898, Springer, 1981, pp.~366--381.
\bibitem{DlotkoLipinskiSignerska2024}
P.~D{\l}otko, M.~Lipi{\`n}ski, and J.~Signerska-Rynkowska,
\newblock Testing topological conjugacy of time series,
\newblock {\em SIAM Journal on Applied Dynamical Systems} 23(4):2939--2982, 2024.
\bibitem{MarszewskaSignerskaRynkowskaDlotkoInPrep}
M.~Marszewska, J.~Signerska-Rynkowska, and P.~D{\l}otko,
\newblock Topological Characteristics for the Analysis of Vector Fields: New Stable Characteristics for Discrete and Continuous Dynamical Systems,
\newblock in preparation. |
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