Research & Projects

Graph neural networks (GNNs) are increasingly used for medical imaging analysis and biological data modeling, where the integration of radiomics, topology, geometry, and generative artificial intelligence (AI) may improve representation learning from medical images and related biomedical data. Across the reviewed literature, GNNs show particular value for modeling spatial relationships, multimodal interactions, graph-structured biological networks, and non-Euclidean imaging features that are difficult to capture using conventional convolutional architectures alone. Topology- and geometry-aware approaches further expand this capability by encoding multi-scale structure, higher-order relationships, curvature, geodesic organization, and equivariant spatial priors. Hybrid graph-transfomer models and generative graph methods represent emerging directions for modeling long-range dependencies, augmenting scarce datasets, supporting synthetic pretraining, and improving representation learning in low-label or heterogeneous biomedical settings. However, clinical translation remains limited by variability in graph construction, limited external validation, computational cost, scalability constraints, interpretability challenges, and uncertainty regarding the biological realism of synthetic data. Overall, this review highlights that GNN-based medical imaging analysis is most likely to advance when graph construction is biologically justified, model performance is evaluated across diverse clinical cohorts, and technical gains are paired with transparent validation, interpretability, and implementation strategies.  

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.

• Topology-Aware Graph Generation: Harnessing Discrete Morse Theory (ICERM; poster)

• Introduction to Morse & Cobordism Theory With Potential Applications (Virginia Tech)

• Graph Neural Networks and Neural Algorithmic Alignment (UT Austin)

• Discrete Morse Theory & Graph Diffusion (UT Austin)

• 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)