Research & Projects

My current research is on topological deep learning: developing topology-based GNN architectures by leveraging low-dimensional topoligical theory and applying discrete Morse theory to graph diffusion models (specifically as it relates to probabilistic forecasting and graph generation) and representation learning. I also work on harnessing topology for graph pooling methods.

You can find some of my previous work in bringing topology to medical image analysis below.

Graph neural networks (GNNs) are increasingly used in medical imaging and biological data analysis, where they model spatial relationships, multimodal interactions, and non-Euclidean features that conventional CNNs struggle to capture. Topology- and geometry-aware methods add multi-scale structure, higher-order relationships, and spatial priors, while hybrid graph-transformer and generative models address long-range dependencies and data scarcity. Clinical translation is still limited by inconsistent graph construction, weak external validation, computational cost, and poor interpretability. Progress will depend on biologically justified graph design, evaluation across diverse cohorts, and transparent validation. 

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.

• 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)