Beijing Institute of Mathematical Sciences and Applications

Matthew Burfitt , Tyrone Cutler , Jing Yan Li , Jie Wu , Jia Wei Zhou

Taras Panov

Monday, November 13, 2023 2:30 PM - 3:30 PM

A3-4-101

Zoom 230 432 7880 (BIMSA)

Given a finite pseudo-metric space $(X,d_X)$, the Vietoris--Rips filtration is a sequence $\{R(X,t)\}_{t\ge0}$ of filtered flag simplicial complexes associated with~$X$. The simplicial homology of $R(X,t)$ is used to define the most basic persistence modules in data science, the persistent homology of~$X$. In toric topology, a finer homological invariant of a simplicial complex $K$ is considered, the bigraded homology of the moment-angle complex $\mathcal Z_K$ associated with~$K$. The moment-angle complex $\mathcal Z_K$ is a toric space patched from products of discs and circles parametrised by simplices in a simplicial complex~$K$. It has a bigraded cell decomposition and the corresponding bigraded homology groups $H_{-i,2j}(\mathcal Z_K)$ contain the simplicial homology groups $H_k(K)$ as a direct summand. Algebraically, the bigraded homology modules $H_{-i,2j}(\mathcal Z_K)$ are the bigraded components of the $\text{Tor}$-modules of the Stanley--Reisner ring $\mathbf k[K]$ and can be expressed via the Hochster decomposition as the sum of the reduced simplicial homology groups of all full subcomplexes $K_I$ of~$K$. The bigraded homology of the moment-angle complexes $\mathcal Z_{R(X,t)}$ associated with the Vietoris--Rips filtration $\{R(X,t)\}_{t\ge0}$ can be used to define bigraded persistent homology modules and bigraded barcodes of a point cloud (data set)~$X$. Simple examples show that bigraded persistent homology can distinguish between points clouds that are indistinguishable by the ordinary persistent homology. Double homology $\text{\it HH}_*(\mathcal Z_K)$ is the homology of the chain complex $\text{\it CH}_*(\mathcal Z_K)=(H_*(\mathcal Z_K),\partial')$ obtained by endowing the bigraded homology of~$\mathcal Z_K$ with the second differential~$\partial'$. The bigraded double homology modules are smaller than the bigraded homology modules, and therefore might be more computationally accessible. More importantly, persistent homology modules defined from the bigraded double homology of the Vietoris--Rips filtration have the stability property, which roughly says that the bigraded barcode is robust to changes in the input data. This is a joint work with Anthony Bahri, Ivan Limonchenko, Jongbaek Song and Donald Stanley.