Abstracts

Jurgen Jost, Geometry and Topology of Data

Many data are equipped with metric relations, that is, distances between data points. We capture these metric relations systematically through intersection patterns of distance balls. Such patterns are used in topological data analysis to identify persistent homology groups. But from the perspective of metric geometry, a more systematic theory can be developed that links these patterns to notions of generalized curvature and concepts like hyperconvexity. Eventually, we can transfer all geometric relations into a topological construction that records the intersection patterns of all configurations of distance balls.

The talk represents joint work with Parvaneh Joharinad.

Kevin Li, Topological complexity of hyperbolic groups.

The topological complexity TC(X) of a space X is an integer-valued homotopy invariant, measuring the complexity of the robot motion planning problem in X. By restricting ourselves to aspherical spaces, we obtain an interesting invariant of discrete groups. In a recent breakthrough, Dranishnikov has computed the topological complexity of hyperbolic groups. I will present a simplified proof of his result, which generalises easily to certain relatively hyperbolic groups.

Peter Bubenik, Path metrics and algebraic Wasserstein distances

Wasserstein distances provide quantitative comparisons of probability measures in optimal transport theory and of persistence diagrams in topological data analysis. I will describe an analogous framework for producing distances in algebraic settings. In the classical case, transportation plans are assigned a cost and the infimum of these costs results in a metric. In our algebraic case, we assign a cost to zigzags of morphisms and the infimum of these costs is a metric which we call the path metric. As an example, we obtain a path metric for generalized persistence modules indexed by a poset with a measure on its underlying set. From this path metric, we define a Wasserstein distance which generalizes the Wasserstein distance for persistence diagrams.

This is joint work with Jonathan Scott and Don Stanley.

Fedor Manin, Configuration spaces of disks in an infinite strip

The topology of the configuration space of points in the plane is well-understood. In statistical physics and topological robotics, one may instead want to understand the configuration space of thick particles in a constrained space. The topology of such spaces is quite complicated, and several models have been used to simplify it. One such model is that of n unit disks in an infinitely long strip of radius w, introduced by Alpert, Kahle, and MacPherson. After introducing the background, I will discuss joint work with Hannah Alpert studying the homology of this family of spaces and how it changes as we add more disks or widen the strip.

Stephan Mescher, Geodesic complexity of Riemannian manifolds

Geodesic complexity is motivated by Farber’s notion of topological complexity of a space, which gives a topological description of the motion planning problem in robotics. Motivated by this, D. Recio-Mitter recently introduced geodesic complexity as an isometry invariant of geodesic spaces which formalizes the notion of efficient robot motion planning mathematically, i.e. of motion planning along shorts paths. In my talk, I will present recent work with M. Stegemeyer, in which we study the geodesic complexity of complete Riemannian manifolds. Using structure results for cut loci from Riemannian geometry, we derive lower and upper bounds for geodesic complexity from the geometry of cut loci and illustrate our results by some examples.

Andrew Newman, Subcomplexes of random flag complexes

The coedge ideal of a graph is the monomial ideal generated by its nonedges; equivalently it's the Stanley--Reisner ideal of the graph's flag complex. Using Hochster's formula the algebraic structure of the resulting Stanley--Reisner ring can be interpreted from the topology of subcomplexes of the flag complex. After briefly discussing this connection between commutative algebra and topology, I will discuss recent results about coedge ideals of random graphs coming from results in stochastic topology. This is based on joint work with Anton Dochtermann.

D. Yogeshwaran, Poisson process approximation for critical points of random distance function.

We shall consider extremal critical points (along with scaled critical radius) of a random distance function and show a Poisson process approximation result for the same. This is obtained as a consequence of a general Poisson process approximation result for stabilizing functionals of Poisson processes that arise in stochastic geometry. The bounds are derived for the Kantorovich-Rubinstein distance between a point process and an appropriate Poisson point process. This is a joint work with Omer Bobrowski (Technion) and Matthias Schulte (Hamburg Institute of Technology).

Sergio Barbarossa. "Topological Signal Processing"

One of the main goals of machine learning is to find effective representations of the observed data and to extract relevant information. The goal of this talk is to establish a basic framework to represent signals over a topological space and to derive effective ways to process them to extract features useful for learning. We start introducing algorithms to infer the structure of the signal domain from the observed data, focusing on simplicial and cell complexes, and then we show how to find dictionaries enabling signal representations achieving a good trade-off between sparsity and accuracy. Then, we propose how to design filters operating over edge signals, highlighting the harmonic, solenoidal or irrotational components. Finally, we show how to exploit the established framework to design deep neural networks operating on flow signals.