• Speaker: Facundo Memoli, The Ohio State University
• Title: Some rigidity results for metric spaces via persistent homology
• Date: 05/03/2021
• Time: 2:00 PM - 3:00 PM
• Place: Online (virtual meeting)
• Contact: 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 Vietoris-Rips filtrations. Of particular interest is a relationship which we established linking persistence barcodes to Gromov's filling radius.

• Speaker: Yury Belousov, Higher School of Economics (HSE)
• Title: On the question of genericity of hyperbolic knots
• Date: 05/04/2021
• Time: 2:50 PM - 3:40 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Matthew Edward Hedden (heddenma@msu.edu)

(based on a joint work with A. Malyutin) Thurston’s famous classification theorem, of 1978, states that a non-toric non-satellite knot is hyperbolic, that is, its complement admits a complete hyperbolic metric of finite volume. Until recently there was the conjecture (known as Adams conjecture) saying that the percentage of hyperbolic knots amongst all the prime knots of n or fewer crossings approaches 100 as n approaches infinity. In 2017 Malyutin showed that this statement contradicts several other plausible conjectures. Finally, in 2019 Adams conjecture was found to be false. In this talk we are going to discuss the key ingredients of its disproof

• Speaker: Michael Kwok-Po Ng, University of Hong Kong
• Title: Low Rank Tensor Completion with Poisson Observations
• Date: 05/06/2021
• Time: 4:30 AM - 5:30 AM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Olga Turanova ()

Poisson observations for videos are important models in video processing and computer vision. In this talk, we study the third-order tensor completion problem with Poisson observations. The main aim is to recover a tensor based on a small number of its Poisson observation entries. An existing matrix-based method may be applied to this problem via the matricized version of the tensor. However, this method does not leverage on the global low-rankness of a tensor and may be substantially suboptimal. We employ a transformed tensor nuclear norm ball constraint and a bounded constraint of each entry, where the transformed tensor nuclear norm is used to get a lower transformed multi-rank tensor with suitable unitary transformation matrices. We show that the upper bound of the error of the estimator of the proposed model is less than that of the existing matrix-based method. Numerical experiments on synthetic data and real-world datasets are presented to demonstrate the effectiveness of our proposed model compared with existing tensor completion methods.

• Speaker: Ashleigh Thomas, Georgia Tech
• Title: The shape of worm behavior
• Date: 05/17/2021
• Time: 2:00 PM - 3:00 PM
• Place: Online (virtual meeting)
• Contact: 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 widely-studied 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 trade-off 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.

• Speaker: Katharine Turner, Australian National University
• Title: Generalisations of the Rips Filtration for quasi-metric spaces and asymmetric functions with corresponding stability results
• Date: 05/24/2021
• Time: 7:00 PM - 8:00 PM
• Place: Online (virtual meeting)
• Contact: 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 Gromov-Hausdorff 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 Gromov-Hasdorff distance called the correspondence distortion distance. These different constructions involve ordered-tuple homology, symmetric functions of the distance function, strongly connected components and poset topology.

• Speaker: Gerlind Plonka-Hoch , University of Göttingen
• Title: Recovery of sparse signals from their Fourier coefficients
• Date: 06/03/2021
• Time: 2:30 PM - 3:30 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Olga Turanova (turanova@msu.edu)

In this talk, we study a new recovery procedure for non-harmonic signals, or more generally for extended exponential sums y(t), which are determined by a finite number of parameters. For the reconstruction we employ a finite set of classical Fourier coefficients of y(t). Our new recovery method is based on the observation that the Fourier coefficients of y(t) possess a special rational structure. We apply the recently proposed AAA algorithm by Nakatsukasa et al. (2018) to recover this rational structure in a stable way. If a sufficiently large set of Fourier coefficients of y(t) is available, then our recovery method automatically detects the correct number of terms M of the exponential sums y(t) and reconstructs all unknown parameters of the signal model. Our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony's method. These results have been obtained jointly with Markus Petz and Nadiia Derevianko.

• Speaker: Rui Wang, Michigan State University
• Title: Persistent Spectral Graphs
• Date: 06/14/2021
• Time: 2:00 PM - 3:00 PM
• Place: Online (virtual meeting)
• Contact: 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 low-dimensional multiscale paradigm for revealing topological persistence and extracting geometric shapes from high-dimensional 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 non-harmonic spectra of PLMs give rise to additional geometric analysis of the shape of the data. We develop an open-source software package, called highly efficient robust multidimensional evolutionary spectra (HERMES) as well, to enable broad applications of PSGs in biology, engineering, and technology.

• Speaker: Veronica Ciocanel, Duke University
• Title: "Modeling and topological data analysis for biological ring channels"
• Date: 06/21/2021
• Time: 2:00 PM - 3:00 PM
• Place: Online (virtual meeting)
• Contact: 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 agent-based modeling framework to simulate interactions between actin filaments and motor proteins inside cells. We also develop tools based on topological data analysis to understand time-series 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.

• Speaker: Lori Ziegelmeier, Macalester College
• Title: Capturing Dynamics of Time-Varying Data via Topology
• Date: 06/28/2021
• Time: 2:00 PM - 3:00 PM
• Place: Online (virtual meeting)
• Contact: 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 time-varying 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 2-dimensional image that displays the (non-persistent 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 time-varying metric spaces: a crocker stack. Crocker stacks are convenient for visualization, amenable to machine learning, and satisfy a stability property.

• Speaker: Qi (Rose) Yu , University of California, San Diego
• Title: University of California, San Diego
• Date: 07/15/2021
• Time: 2:30 PM - 3:30 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Olga Turanova (turanova@msu.edu)

Applications such as climate science and transportation require learning complex dynamics from large-scale spatiotemporal data. Existing machine learning frameworks are still insufficient to learn spatiotemporal dynamics as they often fail to exploit the underlying physics principles. Representation theory can be used to describe and exploit the symmetry of the dynamical system. We will show how to design neural networks that are equivariant to various symmetries for learning spatiotemporal dynamics. Our methods demonstrate significant improvement in prediction accuracy, generalization, and sample efficiency in forecasting turbulent flows and predicting real-world trajectories. This is joint work with Robin Walters, Rui Wang, and Jinxi Li.

• Speaker: Mengsen Zhang, UNC Chapel Hill
• Title: Transitions and their topological signatures in social and brain dynamics
• Date: 07/26/2021
• Time: 2:00 PM - 3:00 PM
• Place: Online (virtual meeting)
• Contact: 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 dynamical-system context in which TDA is applied in the hope of eliciting new conversations across the boundary of empirical science, dynamical systems modeling, and TDA.

• Speaker: Wotao Yin, UCLA
• Title: Learning to Optimize: Fixed-Point Network
• Date: 07/29/2021
• Time: 2:30 PM - 3:30 PM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Olga Turanova (turanova@msu.edu)

Many applications require repeatedly solving a certain optimization problem, each time with new but similar data. “Learning to optimize” or L2O is an approach to develop algorithms that solve these similar problems very efficiently. L2O-generated algorithms have achieved significant success in signal processing and inverse-problem applications. This talk introduces the motivation for L2O and gives a quick overview of different types of L2O approaches for continuous optimization. Then, we will introduce Fixed Point Networks (FPNs), which incorporate fixed-point iterations into deep neural networks and provide abilities such as physics-based inversion, data-driven regularization, encoding hard constraints, and infinite depth. The FPNs are easy to train with a new Jacobian-free back propagation (JFB) scheme.