Testing the reliability of bonded joints in material assemblies is one of the major subjects in the aeronautical industry. One of the methods for characterizing adhesion is based on the study of the roughness of the fracture surfaces of assemblies. In this study, the interest is focused on the quantification of the adhesion of the bonded structure by the corner cleavage test, allowing the study of crack propagation within bonded assemblies. Optical profilometry measurements, obtained by scanning the surface of the fractured surfaces of monoadhesives: ABT M52, copolymer-doped and non-nanostructured AHT are used. The aim of the profilometric study is to quantify surface roughness. To go further than traditional methods of characterizing roughness, we analyze the profiles using mathematical methods to confirm the experimental studies and extract more geometric information. We apply various techniques of topological data analysis (TDA) to extract the topological features of the profiles. These techniques recover experimental elements in a quantitative manner. Persistence diagrams give us a multi-scale characterization of the rupture facies, attesting the voids, micro-cracks at nano-level, quantifying the maximum amplitude of the peaks at micro level. We extract features vectors using persistent images for each profile and we look for similarities in persistence diagrams using Bottleneck and Wasserstein distances in perspective of a machine learning application in a future study. Then the profiles were studied by Takens embedding which will produce a point cloud analyzed by homological persistence in order to produce indications of periodicity or quasi-periodicity. This computational topological approach makes it possible to extract the essential characteristics of the surface and roughness of the profile of the adhesives in order to conclude on their toughness and fracture resistance.

We propose a geometry topological framework to analyze storm dynamics by coupling persistent homology with Anti-de Sitter (AdS)-inspired metrics. On radar images of a bow echo event, we compare Euclidean distance with three compressive AdS metrics ($\alpha$ = 0.01, 0.1, 0.3) via time-resolved $H_1$ persistence diagrams for the arc and its internal cells. The moderate curvature setting ($\alpha =0.1$) offers the best trade-off: it suppresses spurious cycles, preserves salient features, and stabilizes lifetime distributions. Consistently, the arc exhibits longer, more dispersed cycles (large-scale organizer), while cells show shorter, localized patterns (confined convection). Cross-correlations of $H_1$ lifetimes reveal a temporal asymmetry: arc activation precedes cell activation. A differential indicator $\Delta \left(t\right)$ based on Wasserstein distances quantifies this divergence and aligns with the visual onset in radar, suggesting early warning potential. Results are demonstrated on a rapid Corsica bow echo; broader validation remains future work.

This study examines the extremely severe storm that hit Corsica in the early hours of August 18, 2022. A "derecho" is a powerful storm system known for producing long-lasting and widespread destructive winds. In this case, the storm was driven by a large cluster of thunderstorms that organized into a fast-moving structure, leading to intense wind gusts across the region. The associated radar signatures generally exhibit linear characteristics, often displaying "bow echo" shapes. In this work, we study the formation and evolution of the bow echo that struck the Corsican coast, causing significant damage and multiple fatalities. Bow echoes are notoriously difficult to predict, as they can form quickly and remain quite localized. We propose using Topological Data Analysis (TDA, specifically, persistent homology, to characterize the shape evolution of a bow echo over time. From weather radar images of reflectivity, we generate point clouds corresponding to the most intense precipitation cells and compute their persistence diagrams via Vietoris–Rips filtrations. We compare persistence diagrams using the Bottleneck distance, the Wasserstein distance, and the kernel approach: the Persistence Weighted Gaussian Kernel (PWGK), the Sliced Wasserstein Kernel (SWK) and the Pertinence Fisher Kernel (PFK). We then apply hierarchical clustering, principal component analysis (PCA), and change-point detection to these distances in order to highlight major structural transitions and detect the moment when the bow echo forms. Additionally, we incorporate a curvature-based metric by measuring the ratio between the arc length and the straight length of the arc shape. We show that combining TDA-based distances with curvature measurements provides valuable insight into the storm's evolution. Finally, we introduce a discussion on triggering an alert based on the topological analysis of the bow echo. Our results suggest that TDA-based methods can help identify distinct structural changes in complex meteorological phenomena.

Vertical wind shear plays a crucial role in the organization and persistence of mesoscale convective systems, yet its geometrical and topological effects remain challenging to quantify. In this study, we introduce a shear-induced anisotropic metric, denoted $d_S$, which embeds the direction and magnitude of environmental wind shear directly into the framework of persistent homology. The metric deforms the ambient geometry by weighting distances differently along and across the shear direction, enabling topological descriptors to respond dynamically to the flow environment. We establish the analytical properties of $d_S$, and demonstrate its compatibility with Vietoris–Rips filtrations. The method is applied to the Corsican bow–echo event of 18 August 2022, where shear vectors are derived from ERA5 reanalysis data. Two complementary topological analyses are performed: a transport analysis on $H_0$ using Wasserstein distances, and a structural analysis on $H_1$ persistent generators under parallel and perpendicular shear metrics. The results reveal distinct topological evolutions associated with different shear orientations, highlighting the sensitivity of persistent homology to shear-induced deformation. Overall, the framework provides a mathematically consistent bridge between dynamical meteorology and topological data analysis, extending persistent homology to anisotropic metric spaces.

We propose an exploratory framework based on Topological Digital Twins (TDTs) for the monitoring and short-term forecasting of spatial dynamical systems. The approach represents the system through a reduced state built from topological descriptors obtained via persistent homology. These descriptors capture features such as connected components, cycles, and large-scale structure. The framework combines three components: an observation operator mapping spatial fields to a low-dimensional state, a reduced dynamical model evolving this state in time, and a data assimilation step aimed at improving robustness. This construction maps persistence diagrams to a finite-dimensional Euclidean space. This makes the model tractable but does not preserve the full algebraic structure of the original topological objects. We provide theoretical results supporting the stability of the representation under perturbations of the input field. The method is illustrated on a bow-echo convective system observed over Corsica on 18 August 2022, where the reduced state captures the main structural organization of the system over time. A comparison with standard nowcasting methods shows complementary behavior: pixel-based approaches provide better local accuracy, while the TDT framework better preserves the global spatial structure, as reflected by Wasserstein distances and persistence-based comparisons. Additional tests also indicate that the topological observables remain stable under small perturbations of the input field. The present study is based on a single case and should be understood as a proof of concept, rather than as a definitive validation. Future work will focus on validation on larger datasets and on the use of more advanced dynamical models.