In recent years, topological and geometric data analysis (TGDA) has emerged as a new and promising field for processing, analyzing and understanding complex data. Indeed, geometry and topology form natural platforms for data analysis, with geometry describing the ''shape'' behind data; and topology characterizing / summarizing both the domain where data are sampled from, as well as functions and maps associated with them. In this talk, I will show how topological (and geometric ideas) can be used to analyze graph data, which occurs ubiquitously across science and engineering. Graphs could be geometric in nature, such as road networks in GIS, or relational and abstract. I will particularly focus on the reconstruction of hidden geometric graphs from noisy data, as well as graph matching and classification. I will discuss the motivating applications, algorithm development, and theoretical guarantees for these methods. Through these topics, I aim to illustrate the important role that topological and geometric ideas can play in data analysis.

 In this work we systematically introduce relations to topological data analysis (TDA) in the categories of sets, simplicial complexes and vector spaces to characterize and study the general dynamical behaviors in a consistent way. The proposed framework not only offers new insights to the classical TDA methodologies, but also motivates new approaches to interesting applications of TDA in dynamical metric spaces, dynamical coverings, etc. The associated algorithm which produces barcode invariants, and relations in more general categories will also be discussed.

In this work we present a thorough study of the theoretical properties and devise efficient algorithms for the persistent Laplacian, an extension of the standard combinatorial Laplacian to the setting of simplicial pairs: pairs of simplicial complexes related by an inclusion, which was recently introduced by Wang, Nguyen, and Wei. 

In analogy with the non-persistent case, we establish that the nullity of the q-th persistent Laplacian equals the q-th persistent Betti number of any given simplicial pair which provides an interesting connection between spectral graph theory and TDA. 

We further exhibit a novel relationship between the persistent Laplacian and the notion of Schur complement of a matrix. This relation permits us to uncover a link with the notion of effective resistance from network circuit theory and leads to a persistent version of the Cheeger inequality.

This relationship also leads to a novel and fundamentally different algorithm for computing the persistent Betti number for a pair of simplicial complexes which can be significantly more efficient than standard algorithms. 

Active matter systems, ranging from liquid crystals to populations of cells and animals, exhibit complex collective behavior characterized by pattern formation and dynamic phase transitions. However, quantitative analysis of these systems is challenging, especially for heterogeneous populations of varying sizes, and typically requires expertise in formulating problem-specific order parameters. I will describe an alternative approach, using a combination of topological data analysis and machine learning, to investigate emergent behaviors in self-organizing populations of interacting discrete agents.

Given a persistence diagram with n points, we give an algorithm that produces a sequence of n persistence diagrams converging in bottleneck distance to the input diagram, the ith of which has i distinct (weighted) points and is a 2-approximation to the closest persistence diagram with that many distinct points. For each approximation, we precompute the optimal matching between the ith and the (i+1)st. Perhaps surprisingly, the entire sequence of diagrams as well as the sequence of matchings can be represented in O(n) space. The main approach is to use a variation of the greedy permutation of the persistence diagram to give good Hausdorff approximations and assign weights to these subsets. We give a new algorithm to efficiently compute this permutation, despite the high implicit dimension of points in a persistence diagram due to the effect of the diagonal. The sketches are also structured to permit fast (linear time) approximations to the Hausdorff distance between diagrams -- a lower bound on the bottleneck distance. For approximating the bottleneck distance, sketches can also be used to compute a linear-size neighborhood graph directly, obviating the need for geometric data structures used in state-of-the-art methods for bottleneck computation.

Understanding the motion of small bodies at a fluid interface has relevance to a range of natural systems and technological applications. In this talk, we discuss two systems where capillarity and fluid inertia govern the dynamics of millimetric particles at a fluid interface.

In the first part, we present a study of superhydrophobic spheres impacting a quiescent water bath.  Under certain conditions particles may rebound completely from the interface - an outcome we characterize in detail through a synthesis of experiments, modeling, and direct numerical simulation.  In the second half, we introduce a system wherein millimetric disks trapped at a fluid interface are vertically oscillated and spontaneously self-propel.  Such "capillary surfers" interact with each other via their collective wavefield and self-assemble into a myriad of cooperative dynamic states.  Our experimental observations are well captured by a first theoretical model for their dynamics, laying the foundation for future investigations of this highly tunable active system.

Eva Antonopoulou

Surfactants in drop-on-demand inkjet printing

The rapid development of new applications for inkjet printing and increasing complexity of the inks has created a demand for in silico optimisation of the ink jetting performance. Surfactants are often added to aqueous inks to modify the surface tension. However, the time-scales for drop formation in inkjet printing are short compared to the time-scales of the surfactant diffusion resulting a non-uniform surfactant distribution along the interface leading to surface tension gradients. We present both experiments and numerical simulations of inkjet break-up and drop formation in the presence of surfactants investigating both the surfactant transport on the interface and the influence of Marangoni forces on break-up dynamics. The numerical simulations were conducted using a modified version of the Lagrangian finite element developed by our previous work by including the solution for the transport equation for the surfactants over the free surface. During the initial phase of a “pull-push-pull” drive waveform, surfactants are concentrated at the front of the main drop with the trailing ligament being almost surfactant free. The resulting Marangoni stresses act to delay and can even prevent the break-off of the main drop from the ligament. We also examine and present some initial results on the effects of surfactants on the shape oscillations  of the main drop. Although there is little change to the oscillation frequency, the presence of surfactants significantly increases the rate of decay due to the rigidification of the surface, by modifying the internal flow within the droplet and enhancing the viscous dissipation.

Hadrien Oliveri

An optic ray theory for nerve durotaxis

During the development of the nervous system, neurons extend bundles of axons that grow and meet other neurons to form the neuronal network. Robust guidance mechanisms are needed for these bundles to migrate and reach their functional target. Directional information depends on external cues such as chemical or mechanical gradients. Unlike chemotaxis that has been extensively studied, the role and mechanism of durotaxis, the directed response to variations in substrate rigidity, remain unclear. We model bundle migration and guidance by rigidity gradients by using the theory of morphoelastic rods. We show that at a rigidity interface, the motion of axon bundles follows a simple behavior analogous to optic ray theory and obeys Snell’s law for refraction and reflection. We use this powerful analogy to demonstrate that axons can be guided by the equivalent of optical lenses and fibers created by regions of different stiffnesses.