Research

My research interest is in topological data analysis(TDA), applied topology, homological algebra, and category theory. I’d like to develop theoretical groundwork for both existing and potential applications of mathematics in other fields.

Persistence Measures and Continuous Persistence Landscapes (link to paper)

Persistence diagram(PD) is a topological descriptor for data. I consider the limiting behavior of PDs as the size of data increases and define the limiting object to be the persistence measure(PM). I generalize the vectorization via persistence landscapes for PD to PM, called the continuous persistence landscape (CPL). I showed this vectorization is bijective, and from the CPL I can construct a measure on the upper left plane that recovers the PM.

Figure 1: Hiraoka et al. [1] showed for a dataset from material science, the points in the PD form clusterings. Due to the size of data, the multiplicities of points in the PD can reach the magnitude of 10^20. I’m interested in the asymptotic behavior of the PD as the number of points tends to infinity. By viewing the limiting object as a measure, I define the persistence measure in an axiomatic way. Then I construct a vectorization map from the set of persistence measures to functional space by extending the definition of persistence landscapes.

Wasserstein Stability for Barcodes and Persistence Landscapes (link to paper)

The TDA community often uses the p-Wasserstein distance as the metric on PDs. However, the p-Wasserstein distance requires a choice of metric between 2 points on the diagram, and there is no canonical choice. I defined a new metric on the set of interval modules based on their rank functions. This leads to a different Wasserstein distance on PDs compared to the conventional choices of ground metrics, while being topologically equivalent. Here are some cool plots comparing the metric balls from my choice (the red, weirdly shaped) to the conventional 1-norm and Manhattan norm.

The unconventional shape comes from that the metric uses not dimension invariants but the rank invariants, which form a complete set of invariants. Additionally, this choice of ground metric leads to a stronger stability result for PLs. I showed that PLs are L^1-stable w.r.t 1-Wasserstein distance on PDs. It also has a sharp bound and Lipschitz constant 1/2. And I showed that mapping from filtered chain complexes to PDs is 1-Wasserstein stable.

Topological Feature Tracking in Climate Systems: Analogous Barcodes and Vineyards for Bipersistence Modules

Joint work with Niny Arcila Maya, Yariana Diaz, Wenwen Li. This work is an extension from the MRC Climate Science at the Interface Between Topological Data Analysis and Dynamical Systems Theory.

In meteorology and climate science, a weather regime is a characteristic pattern of weather associated with repetitive atmospheric circulation that lasts for an extended period of time. Climate data can be obtained from dynamical systems, and nontrivial homology classes in the trajectory can reveal significant information about the attractor and any stabilizing loops. Similarly, persistent homology can indicate key trends in time series, potentially coming from a chaotic dynamical system. More specifically, Strommen et al. [2] have studied climate data using density-Rips filtrations. At selected density thresholds, they compute Rips barcodes. However, connecting information about persistent topological features across density thresholds is complicated in the multiparameter setting. We explore 2 methods to relate barcodes in a bipersistence module: analogous barcodes by Yoon et al. [3] and persistence vineyards by Cohen-Steiner et al. [4].

[1] Hiraoka et al. Hierarchical Structures of Amorphous Solids Characterized By Persistent Homology. PNAS 2016.

[2] Strommen, K., Chantry, M., Dorrington, J. et al. A topological perspective on weather regimes. Clim Dyn 60, 1415–1445 (2023).

[3] Yoon, H.R., Ghrist, R. & Giusti, C. Persistent extensions and analogous bars: data-induced relations between persistence barcodes. J Appl. and Comput. Topology 7, 571–617 (2023).

[4] Cohen-Steiner, D., Edelsbrunner, H, & Morozov, D. Vines and vineyards by updating persistence in linear time. In Proceedings of the Twenty-Second Annual Symposium on Computational Geometry, SCG ’06, page 119–126, New York, NY, USA, 2006. Association for Computing Machinery.