Talk_id  Date  Speaker  Title 
29072

Monday 5/3 2:00 PM

Facundo Memoli, The Ohio State University

Some rigidity results for metric spaces via persistent homology
 Facundo Memoli, The Ohio State University
 Some rigidity results for metric spaces via persistent homology
 05/03/2021
 2:00 PM  3:00 PM
 Online (virtual meeting)
 Shelley Kandola (kandola2@msu.edu)
Persistence barcodes provide computable signatures for datasets. These signatures absorb both geometric and topological information in a stable manner. One question that has not yet received too much attention is: how strong are these signatures? A related question is that of ascertaining their relationship to other more classical invariants such as curvature. In this talk I will describe some results about characterizing metric spaces via persistence barcodes arising from VietorisRips filtrations. Of particular interest is a relationship which we established linking persistence barcodes to Gromov's filling radius.

29075

Monday 5/17 2:00 PM

Ashleigh Thomas, Georgia Tech

The shape of worm behavior
 Ashleigh Thomas, Georgia Tech
 The shape of worm behavior
 05/17/2021
 2:00 PM  3:00 PM
 Online (virtual meeting)
 Shelley Kandola (kandola2@msu.edu)
We apply sliding window embeddings and persistent homology to video data of the behavior and locomotion of C. elegans, a worm that is a widelystudied model organism in biology. The goal is to produce a quantitative and interpretable summary of complex behavior to use as a measurement tool for future biological experiments.
We distinguish and classify videos from various environmental conditions, analyze the tradeoff between accuracy and interpretability, and use representative cycles of persistent homology to produce synthetic videos of stereotypical behaviors. The talk will end with new data that shows the significant variation that can occur in worm behavior and a discussion on adapting TDA methods to data that is sometimes far from periodic.

29082

Monday 5/24 7:00 PM

Katharine Turner, Australian National University

Generalisations of the Rips Filtration for quasimetric spaces and asymmetric functions with corresponding stability results
 Katharine Turner, Australian National University
 Generalisations of the Rips Filtration for quasimetric spaces and asymmetric functions with corresponding stability results
 05/24/2021
 7:00 PM  8:00 PM
 Online (virtual meeting)
 Shelley Kandola (kandola2@msu.edu)
Rips filtrations over a finite metric space and their corresponding persistent homology are prominent methods in Topological Data Analysis to summarize the ``shape'' of data. For finite metric space $X$ and distance $r$ the traditional Rips complex with parameter $r$ is the flag complex whose vertices are the points in $X$ and whose edges are $\{[x,y]: d(x,y)\leq r\}$. From considering how the homology of these complexes evolves as we increase $r$ we can create persistence modules (and their associated barcodes and persistence diagrams). Crucial to their use is the stability result that says if $X$ and $Y$ are finite metric space then the bottleneck distance between persistence modules constructed by the Rips filtration is bounded by $2d_{GH}(X,Y)$ (where $d_{GH}$ is the GromovHausdorff distance). Using the asymmetry we construct four different constructions analogous to the persistent homology of the Rips filtration and show they also are stable with respect to a natural generalisation of the GromovHasdorff distance called the correspondence distortion distance. These different constructions involve orderedtuple homology, symmetric functions of the distance function, strongly connected components and poset topology.

29084

Monday 6/14 2:00 PM

Rui Wang, Michigan State University

Persistent Spectral Graphs
 Rui Wang, Michigan State University
 Persistent Spectral Graphs
 06/14/2021
 2:00 PM  3:00 PM
 Online (virtual meeting)
 Shelley Kandola (kandola2@msu.edu)
Persistent homology (PH) is one of the most popular tools in topological data analysis (TDA), while graph theory has had a significant impact on data science. This work introduces persistent spectral graph (PSG) theory to create a unified lowdimensional multiscale paradigm for revealing topological persistence and extracting geometric shapes from highdimensional datasets. In PSG theory, families of persistent Laplacian matrices (PLMs) corresponding to various topological dimensions are constructed via a filtration to sample a given dataset at multiple scales. The harmonic spectra from the null spaces of PLMs offer the same topological invariants, namely persistent Betti numbers, at various dimensions as those provided by PH, while the nonharmonic spectra of PLMs give rise to additional geometric analysis of the shape of the data. We develop an opensource software package, called highly efficient robust multidimensional evolutionary spectra (HERMES) as well, to enable broad applications of PSGs in biology, engineering, and technology.

29085

Monday 6/21 2:00 PM

Veronica Ciocanel, Duke University

"Modeling and topological data analysis for biological ring channels"
 Veronica Ciocanel, Duke University
 "Modeling and topological data analysis for biological ring channels"
 06/21/2021
 2:00 PM  3:00 PM
 Online (virtual meeting)
 Shelley Kandola (kandola2@msu.edu)
Actin filaments are polymers that interact with motor proteins inside cells and play important roles in cell motility, shape, and development. Depending on its function, this dynamic network of interacting proteins reshapes and organizes in a variety of structures, including bundles, clusters, and contractile rings. Motivated by observations from the reproductive system of the roundworm C. elegans, we use an agentbased modeling framework to simulate interactions between actin filaments and motor proteins inside cells. We also develop tools based on topological data analysis to understand timeseries data extracted from these filamentous network interactions. We use these tools to compare the filament organization resulting from motors with different properties. We are currently interested in gaining insights into myosin motor regulation and the resulting actin architectures during cell cycle progression. This work also raises questions about how to assess the significance of topological features in topological summaries such as persistence diagrams.

29086

Monday 6/28 2:00 PM

Lori Ziegelmeier, Macalester College

Capturing Dynamics of TimeVarying Data via Topology
 Lori Ziegelmeier, Macalester College
 Capturing Dynamics of TimeVarying Data via Topology
 06/28/2021
 2:00 PM  3:00 PM
 Online (virtual meeting)
 Shelley Kandola (kandola2@msu.edu)
One approach to understanding complex data is to study its shape through the lens of algebraic topology. While the early development of topological data analysis focused primarily on static data, in recent years, theoretical and applied studies have turned to data that varies in time. A timevarying collection of metric spaces as formed, for example, by a moving school of fish or flock of birds, can contain a vast amount of information. There is often a need to simplify or summarize the dynamic behavior. One such method is a crocker plot, a 2dimensional image that displays the (nonpersistent but varying with scale) topological information at all times simultaneously. We use this method to perform exploratory data analysis and investigate parameter recovery via machine learning in the collective motion model of D’Orsogna et al. (2006). Then, we use it to choose between unbiased correlated random walk models of Nilsen et al. (2013) that describe motion tracking experiments on pea aphids. We then introduce a new tool to summarize timevarying metric spaces: a crocker stack. Crocker stacks are convenient for visualization, amenable to machine learning, and satisfy a stability property.

29088

Monday 7/26 2:00 PM

Mengsen Zhang, UNC Chapel Hill

Transitions and their topological signatures in social and brain dynamics
 Mengsen Zhang, UNC Chapel Hill
 Transitions and their topological signatures in social and brain dynamics
 07/26/2021
 2:00 PM  3:00 PM
 Online (virtual meeting)
 Shelley Kandola (kandola2@msu.edu)
Understanding biological and social interactions in a dynamical system framework amounts to identifying stable patterns of activity and tracking transitions between them. When the system’s dimension is low (few
interacting elements), one can often visually inspect the time series and write down the relevant variables and differential equations. Once we venture into higher dimensions, this traditional modeling approach becomes intractable: what constitutes a “stable pattern” or a “transition” in the empirical data depends on the spatiotemporal scales at which we look at the system. We show how persistent homology serves as a natural tool for characterizing such multiscale dynamics in rhythmic social interaction. Furthermore, a key goal of studying transitions is to understand the relations between distinct stable dynamic patterns – they
provide access to the global organization of the dynamical system. We show how existing topological data analysis (TDA) tools can be adapted to understand the network of transitions in brain dynamics. The talk focuses on the empirical and dynamicalsystem context in which TDA is applied in the hope of eliciting new
conversations across the boundary of empirical science, dynamical systems modeling, and TDA.
