Research & Projects

We perform a narrative review of TDA, discussing its basic principles and advantages over traditional methods, which often struggle to capture the intricate patterns and relationships within coronary plaques. TDA can evaluate coronary atherosclerosis in novel ways using variations in the cover and filtration functions, highlighting new ways to characterize calcified and non-calcified plaques. It also highlights the potential of TDA in enhancing the understanding and management of this complex disease. Despite the challenges and future directions that need to be addressed, TDA shows potential as a valuable tool in assessing coronary atherosclerosis, specifically in non-calcified plaque detection, possibly surpassing traditional techniques in capturing the disease's intricate patterns. 

We review the quantification of topological uncertainty in AI applications for medical imaging, highlighting the importance of understanding spatial, structural, and shape-related characteristics in imaging data. Further, we explore various types of uncertainties, such as measurement and model uncertainties, and evaluate current methodologies for managing these uncertainties to improve diagnostic accuracy and treatment planning. Our review emphasizes the need for standardized frameworks and robust methods to ensure the reliable and ethical deployment of AI in medical settings, suggesting avenues for future research to enhance AI's clinical utility.  

Generative Adversarial Networks (GANs) have gained prominence in medical imaging due to their ability to generate realistic images. Traditional GANs, however, often fail to capture intricate topological features such as holes and connectivity components in real images. This study applies TopoGAN, a recently developed model tailored for medical imaging. TopoGAN dynamically learns and incorporates topological features like connectedness and loops, addressing a real-world medical data augmentation problem.

This project develops a theoretical and algorithmic framework for understanding how persistence diagrams of sliding-window time series behave under small computational and temporal perturbations . We analyze and compare the elder and local rule for pairing connected components in sublevel set filtrations, establishing explicit 1-Wasserstein stability bounds when switching between rules. We further prove tight stability guarantees under 1-shifts of the window and show that, for white noise, updating the local-rule persistence diagram has expected constant time complexity. To address worst-case update costs, we introduce a novel successor rule that limits the number of affected persistence pairs under any 1-shift to at most six, enabling logarithmic-time updates via balanced search trees. Our results are formulated within an optimal transport framework, clarifying how Wasserstein distances quantify structural changes in persistence diagrams. Together, these contributions provide both theoretical guarantees and practical tools for efficient streaming topological data analysis.

This workshop will explore the integration of Topological Data Analysis (TDA) techniques with current computational methods to advance medical data analysis, focusing on enhancing performance, generalizability, and explainability. By combining TDA with other computational approaches, such as deep learning, the event aims to tackle complex medical data challenges across various domains including disease diagnosis and personalized medicine. The workshop will serve as an interdisciplinary forum, bringing together professionals from mathematics, engineering, computer science, and medicine to discuss novel applications, share insights, and outline future directions in the field. 

This mini-tutorial covers the intersection of topological data analysis (TDA), machine learning, and uncertainty in medical data analysis. As biomedical datasets grow in complexity, TDA offers advanced tools to improve data analysis and visualization, especially when combined with deep learning. Participants will explore TDA's application to various medical data types, highlighting its role in enhancing analysis and generalizability, and the goal is to engage professionals from different fields to demonstrate TDA's value in medical research.

Research on topological deep learning: developing topology-biased GNN architectures by leveraging low-dimensional topoligical theory and applying discrete Morse theory to graph diffusion models; collaborating with Mayo Clinic on persistent homology for cardiovascular imaging.

• Leveraging Morse theory for diffusion on non-Euclidean objects (LENS2026, WAVC)

• Cobordism Theory in Topological Data Analysis (Networks in SCIENCE Conference, Berkeley)

• Graph Neural Networks & Applications in Quality (Gilead, Internal)

• Persistent Homology & Cycle Matching - Applications to Clustering (UT Austin, Junior Topology)

• Graph Neural Networks: A Comparitive Analysis (UT Austin)

• Topological Graph Layer: Topology in NN's (UT Austin, Junior Applied Mathematics Seminar)

• Introduction to Topological Data Analysis: Unvieling the Shape of Data (Austin Women in Data Science)

• Appications of Topology (UT Ausitn, Junior Topology Seminar)

• Hippocampal Spatial Rep, exhibit Hyperbolic Geometry that Expands with Experience (Computational Neuroscience Seminar, UT Austin)

• A Topological Paradigm for Hippocampal Spatial Representation using Persistent Homology (Computational Neuroscience Seminar, UT Austin)

• Topology in Neuroscience (UT Austin, Junior Topology)