Abstracts (August 8)

Topological Data Analysis (TDA) generally refers to utilizing topological features from data. A central topic in TDA is persistent homology, which observes data at various resolutions and summarizes topological features that persistently appear. TDA has been proven valuable in enhancing machine learning applications.  This work explores how TDA can enhance machine learning workflows, focusing on two areas: feature extraction and model evaluation. Persistent homology, while rich in structural information, is often challenging to integrate directly into statistical and machine learning frameworks. To address this, various featurization techniques map persistence-based information into Euclidean or functional spaces, enabling its incorporation into neural networks and other learning algorithms. I will examine different approaches that efficiently transform topological summaries into differentiable layers and leverage geometric representations for visualization and dimensionality reduction. In addition to feature extraction, TDA has recently been applied to evaluate data quality and model performance. By quantifying the topological structure of generated or transformed data, TDA-based methods provide robust evaluation metrics that improve the reliability of model assessment. I will present how this is done in particular in generative modeling scenarios.

In this talk, we introduce a new method to estimate Stiefel–Whitney classes—topological invariants that detect features like orientability and embedding obstructions—directly from point cloud data. We first extend classical vector bundle theory to persistent vector bundles in the setting of topological data analysis. By applying cohomology operations to persistent cohomology, we compute these classes in a persistent setting. We demonstrate the method with applications in image analysis, molecular conformation, and high-dimensional data.

In this talk, we analyze ensemble forecasting trajectories from a geometric point of view. We focus on the oriented turning angles to cluster and distinguish different weather scenarios. Ensemble forecasting is a method used in weather prediction, which consists of running multiple forecast simulations, each with slightly varied initial conditions or model parameters, to capture the inherent uncertainty in weather forecasting. Collectively, these outputs map the range and likelihood of future states, and it is crucial to identify and label them to take the proper preventive actions in each situation. We quantify the shape of the trajectory with the Frenet frame, which is a coordinate system attached to a moving point along a curve. In two dimensions, the curvature at each point of the curve defines the frame; in three dimensions, the torsion is additionally included; and analogous quantities extend to higher dimensions. As a first step in this research, we emphasize the oriented turning angle (the cumulative signed change of direction) as a feature for grouping ensemble forecasting data. We apply this to heavy‑rain datasets and show that the turning angle helps distinguish different meteorological scenarios. We think that studying these techniques further will improve interpretation of ensemble information and uncertainty assessment.