Topic: Topological Data Analysis for Time-Varying Manifolds: Stability, Computation, and Applications to Real-World Signals

Project Abstract

Topological Data Analysis (TDA) has emerged as a powerful framework for extracting robust geometric and topological features from complex data. This abstract presents a comprehensive study on the stability, computation, and application of TDA to time-varying manifolds within real-world signal contexts. We develop a theoretical foundation for persistent homology on evolving shapes, drawing connections between changes in topology and dynamical behavior of signals. Our work introduces a novel stability criterion for time-varying filtrations, quantifying how perturbations in data sampling, noise, and manifold deformations propagate to persistence diagrams and barcodes. We establish Lipschitz-type bounds and multi-parameter stability results, enabling reliable interpretation of topological summaries in non-stationary environments. On the computational side, we design efficient algorithms for constructing and updating filtrations as manifolds evolve, leveraging incremental and parallel strategies to manage high-dimensional ambient spaces. We extend zigzag and vineyard filtrations to time-varying settings, enabling continuous tracking of topological features across successive time steps. Our methodology integrates adaptive sampling, curvature-aware neighborhood graphs, and landmark-based approximations to control computational complexity without sacrificing interpretability. We also address the challenge of scale separation by introducing multi-resolution encodings that adapt to local geometric changes, ensuring stability under both rapid and slow deformations. To demonstrate practical relevance, we apply the developed framework to a suite of real-world signals, including dynamic brain activity in functional MRI, time-series from physiological sensors, and evolving 3D shapes in motion capture data. We show that time-aware topological descriptors capture meaningful events such as phase transitions, onset of anomalies, and regime shifts that are often elusive to conventional feature engineering. The experiments compare standard persistent homology with time-resolved variants, highlighting improvements in robustness to noise, missing data, and sampling irregularities. We also explore interpretability by mapping persistent features to clinically or physically meaningful events, enabling actionable insights in domains like neuroscience and biomechanics. A theoretical contribution of the work is a cohesive framework linking dynamical systems perspectives with topological summaries, providing criteria for when topological changes reflect intrinsic geometric evolution versus observational artifacts. We identify regimes where TDA offers superior discriminative power for signal classification and anomaly detection, alongside limitations where local linear methods may outperform global topological summaries. The results are validated across synthetic benchmarks and diverse real-world datasets, demonstrating that time-varying TDA can serve as a robust, scalable, and interpretable tool for extracting lightweight yet informative descriptors from complex, evolving manifolds. The study concludes with guidelines for practitioners on parameter selection, stability checks, and interpretation of time-evolving topological features in applied signal analysis.

Project Overview

What This Project Is About

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The project explores how to study complex shapes, or manifolds, that change over time using topological ideas. It aims to find stable features of data as it evolves and to develop practical ways to compute these features from real-world signals like weather data, brain activity, or financial markets. The goal is to connect deep mathematical ideas with tools that can handle noisy, real data.

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The Problem It Addresses

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Many real-world signals live on shapes that shift over time, making it hard to capture their essential structure with traditional methods. Noise and rapid changes can obscure important patterns. This project tackles how to identify meaningful, persistent features that remain stable as the underlying manifold moves, and how to compute them efficiently for large datasets.

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Objectives of the Project

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1. Introduce time-varying manifolds and basic topological ideas in plain language.
2. Develop methods to extract stable features from evolving data.
3. Study how computation scales with data size and time steps.
4. Test approaches on real-world signal datasets.
5. Assess robustness to noise and sampling issues.

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What You Will Do Step by Step

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1) Learn the core concepts with simple examples. 2) Collect or obtain time-series datasets. 3) Apply lightweight computational tools to build representations of the evolving shape. 4) Identify and quantify stable features across time. 5) Compare results under different noise levels and sampling rates. 6) Interpret findings in the context of the chosen application area. 7) Document methods and discuss practical limitations.

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Expected Outcome

A clear, implementable workflow for analyzing time-varying manifolds, including a set of stable features and their interpretation. The project should provide guidance on when these methods work well and how to apply them to real-world signals, with insight into potential applications and limitations.