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 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.

Exploring novel adaptations of cobordisms to demonstrate effectiveness for revealing underlying topological structure of high dimensional datasets under various choices of clustering, data projection functions (kernel SVM, UMAP, etc.)

Developing software to apply TDA techniques to real-world datasets from various domains.

• 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)