One of the central objects I am studying is the filtration of the (restricted) second configuration space of metric graphs. This space may be interpreted as a space of arrangements of n robots moving on a finite-length rail. I utilize the tools from TDA to investigate how the topological features of configuration spaces evolve as parameters vary. 

I have completed three projects to date, two of which are directly related to my thesis. In my thesis, I studied the bifiltrations of restricted second configuration spaces of finite metric trees. These metric graphs are the geometric realizations of weighted graphs. The first paper, Persistent Homology of Configuration Spaces of Trees, is currently under revision. The second paper (joint work with Murad Özaydın) will be submitted soon. Both papers are available on arXiv: 

• Persistent Homology of Configuration Spaces of Trees arXiv: 2310.05303, October 2023.

• Notes on Pointwise Finite-Dimensional 2-Parameter Persistence Modules arXiv: 2404.13877, April 2024. 

The third project addresses the Gromov-Hausdorff distance for directed topological spaces (equipped with a specific metric). It provides a tool for measuring the deviation between two directed topological spaces from being isometric. This is joint work with Lisbeth Fajstrup, Brittany Terese Fasy, Lydia Mezrag, Tatum Rask, Francesca Tombari, and Živa Urbančič. 

• Gromov--Hausdorff Distance for Directed Spaces arXiv: 2408.14394, August 2024.

My goal in research is to obtain a comprehensive understanding of multiparameter persistence modules. Compared to 1-parameter TDA, which is well-established with a robust theory of persistence modules, multiparameter TDA is still developing and faces challenges, particularly due to the absence of the multiparameter analogs of barcodes and a structure theorem for multiparameter persistence modules. 

The techniques I use involve multiparameter persistence modules, representations over partially ordered sets (posets), homological algebra, category theory, and discrete Morse theory.