Abstracts (August 7)

In this presentation, I will discuss persistent homology, a mathematical tool that characterizes the shape of data using topology. Mathematical foundations and applications to materials science will be presented. Our persistent homology-based software, HomCloud, will also be introduced.

The COVID-19 pandemic changed our lifestyles. It is expected that the changes in mental health also occurred because of these changes. In this study, we used the questionnaire responses that asked high school students in Tokyo about their depression before and during the COVID-19 pandemic and analyzed the group characteristics of changes in depression as topographical maps using energy landscape analysis (ELA), a method of multidimensional time-series data analysis. As a result, we visualized how the topographical maps of the depression changed in the COVID-19 pandemic and found that the COVID-19 pandemic has made the students less likely to become depressed. These results suggest that ELA is useful for the analysis of psychiatric questionnaires.

The formulation of higher-degree sheaf Laplacians on hypergraphs is hindered by the fundamental challenges of sparsity and orientational ambiguity. To address this, we propose a foundational methodology: symmetric simplicial lifting. This technique embeds a hypergraph into a richer structure, allowing for the systematic construction of Laplacians of arbitrary degree. We validate our framework with the Hypergraph Neural Sheaf Diffusion (HNSD) model, which leverages a degree-zero sheaf Laplacian to learn diffusion processes. The model achieves strong performance on key hypergraph benchmarks, demonstrating that our approach offers a principled pathway for higher-order signal analysis. This work is joint with Gahee Kim and Yong-Geun Oh.