#### Hugo Chu (Imperial College London)

Rigorous calculation of Lyapunov exponents of stochastic flows

## Abstract

I will present a very simple method to rigorously enclose ergodic averages of stochastic flows under mild assumptions. This method is applied to prove the positivity of Lyapunov exponents (and hence chaos) for some systems that were previously out of reach. This is joint work with Maxime Breden, Jeroen Lamb and Martin Rasmussen.

#### Brittany Gelb (Rutgers University)

Rigorous Machine Learning of Homological Dynamics

## Abstract

We present a machine-learning framework for rigorously characterizing the nonlinear dynamics generated by a continuous map. The key elements are polyhedral cell complexes and piecewise linear approximations, both determined by neural networks. We describe the gradient-like dynamics by a Morse graph and the recurrent dynamics by Conley indices (homological invariants). We show theoretically that such computations can be carried out to recover dynamics at any level of resolution within this framework, motivating the problem of an efficient implementation.

#### Joan Gimeno (Universitat de Barcelona)

Computation of normal forms in discrete systems

## Abstract

I will describe a semi-analytical method for computing normal forms in discrete dynamical systems, such as Poincaré maps. This approach involves calculating high-order derivatives, applying coordinate transformations to simplify the local dynamics, and retaining resonance terms essential to the system’s behavior.

The method is general, requiring only regularity assumptions, and is robust under parameter variations. As an application, I will demonstrate how normal form computations can be used to construct high-dimensional twist maps and introduce a frequency recovery technique for visualizing high-dimensional tori within these maps.

This has been a joint work with À. Jorba, M. Jorba-Cuscó, and M. Zou.

#### Jun Okamoto (Kyoto University)

Spatiotemporal reconstruction of gene expression on the human axioloid using optimal transport theory

## Abstract

In this study, we aim to identify genes for which spatiotemporal structure in expression patterns is essential and to elucidate their functions, using scRNA-seq data and spatial data from cell tissues. Here, scRNA-seq is a technology capable of extracting expression information of all genes within a single cell. However, scRNA-seq requires the dissociation of multicellular tissues into single cells and the subsequent measurement after cell death, resulting in the loss of spatial and temporal information. To address this issue, we developed a method for spatiotemporal reconstruction of gene expression based on optimal transport theory. Furthermore, we created a clustering method for spatiotemporal patterns to identify key genes from the reconstructed expression patterns. In this presentation, we report the results of applying this method to the time-series data of a 3D cell culture model that recapitulates human somitogenesis using iPS cells, generated by the Alev group at Kyoto University’s ASHBi. We will discuss the spatiotemporal reconstruction of expression patterns for key genes and the exploration of similar genes.

#### Jose Perea (Northeastern University)

Topological detection and parametrization of toroidal dynamics

## Abstract

The use of time-delay embeddings alongside persistent (co)homology, has proven to be highly effective in quantifying the topology of attractors given observed time series data. In this talk, I will describe some of the challenges, recent theoretical results and applications of delay embeddings and persistence, when trying to detect and parametrize toroidal attractors.

#### Justyna Signerska-Rynkowska (Gdansk University of Technology & Dioscuri Centre in TDA)

Testing topological conjugacy of time-series

## Abstract

We consider a problem of testing topological conjugacy of two trajectories coming from dynamical systems (X,f) and (Y,g) and deliver a number of tests to check if the corresponding trajectories of f and g are topologically conjugate. The values of the tests are close to zero for systems conjugate and large for systems that are not. For our main developed method, ConjTest, the convergence of the test values, in case when sample size goes to infinity, is established. Various numerical examples indicate scalabilit