## Workshop on

## Geometry, Topology, and Computation

## June 12 - 14, 2019

## Mathematikon, Heidelberg University

This workshop is organized jointly between the

- Cluster of Excellence
**STRUCTURES: a unifying approach to emergent phenomena in the physical world, mathematics, and data**at Heidelberg University - Center
**MathSee: Mathematics in Sciences, Engineering, and Economics**at the Karlsruhe Institute of Technology

with further support from the joint research training group **RTG 2229 Asymptotic Invariants and Limits of Groups and Space**.

## Speaker

Pau Vilimelis Aceituno (MPI Leipzig)

Ulrich Bauer (TU Munich)

Paul Breiding (MPI Leipzig)

Jacek Brodzki (Southampton)

Matthew Burfitt (Southampton)

Herbert Edelsbrunner (IST Austria)

Jürgen Jost (MPI Leipzig)

Ingrid Membrillo-Solis (Southampton)

Katharina Ölsböck (IST Austria)

Miriam Pirashvili (Southampton)

## Organizer

Peter Albers

Roman Sauer

Anna Wienhard

## Registration

If you want to participate you are kindly requested to send an e-mail to Mrs Nicole Umlas (numlas "at" mathi.uni-heidelberg.de) not later than Monday, May 27, 2019.

## Schedule

## Titles and abstracts

**Aceituno** - The network in recurrent neural networks: Computational effects and emergence

*Abstract:* Recurrent neural networks (RNNs) are the state-of-the-art machine learning techniques to process time series, with various training algorithms that, despite their success, are still not fully understood and whose biological plausibility is heavily contested. In this work we take a more theoretical approach and focus on the relationship between network structure and information processing. To do so we focus on Reservoir Computing, a specific paradigm of RNNs where only the last layer is trained. By taking a geometric perspective on that training we connect the dynamics of the RNN to their machine learning performance, both in terms of memory – here the amount of inputs that the network can store – and in terms of frequency specialization – representing the type of signals that the RNN can process successfully. To impose the previously mentioned dynamics onto our RNNs we use notions from control theory – namely feedback loops and poles – and we formalize them through ideas from random matrix theory relating the structure of large networks to the eigenvalues of their adjacency matrices.

We conclude by referring to the biological plausibility of our results, relating the emergence of feedback loops of the appropriate length to synaptic time-dependent plasticity and the memory to previous experiments on context-dependent integration tasks.

**Bauer** - Persistence Diagrams as Diagrams

*Abstract:* We explore the perspective of viewing persistence diagrams, or persistence barcodes, as diagrams in the categorical sense. Specifically, we consider functors indexed over the reals and taking values in the category of matchings, which has sets as objects and partial bijections as morphisms.\

This yields a categorical structure on barcodes, turning the bottleneck distance into an interleaving distance, and allowing for a simple reformulation of the induced matching theorem, which has been used to prove the algebraic stability of persistence barcodes.

We will also discuss an explicit construction of a barcode for a pointwise finite-dimensional persistence module that doesn’t require a decomposition into indecomposable interval summands, and that is actually functorial on monomorphisms of persistence modules (along with a dual construction, which is functorial on epimorphisms).

This is joint work with Michael Lesnick (Albany).

**Breiding** - Monte Carlo Homology

*Abstract:* Persistent homology is a method for computing the homology of a topological space from a finite point sample. The underlying idea is as follows: for varying t, put a ball of radius t around each point and compute the homology of the union of the balls. Under the assumption that the finite point sample was drawn from the uniform distribution on a manifold, a theorem by Niyogi, Smale and Weinberge tells us how to fix t to get the correct homology with high probability. In this talk, I want to discuss that one can use the Niyogi-Smale-Weinberger Theorem for computing the homology of algebraic varieties. The algebraic structure opens the path to sampling from the uniform distribution, estimating the volume and computing the reach.

**Brodzki** - Geometry, Topology, and the structure of data

*Abstract:* Modern data is astonishing in its variety, and is far removed from anything that could be held in spreadsheet or analysed using traditional methods. Through intense research effort, topology has emerged as a source of novel methodology to provide insight into the structure of very complex, high dimensional data. It provided us with tools like persistent homology, which are used to compute numerical topological characteristics of the data. More recently, these methods have been augmented by geometric insights, which are valuable in capturing the structure of and relationships between complex shapes.

In this talk, I will provide an overview of new techniques from topology and geometry and illustrate them on particular examples. One set of data was created through the study of CT scans of human lungs, and another addresses the problem of classification of three-dimensional shapes.

**Burfitt** - A numerical measure for the stability of Mapper

*Abstract: *Mapper is a very important tool in the implementation of topological data analysis and has been used with great success in many contexts. Despite this the Mapper method is rather ad hoc, varying a lot with respect changes in the input parameters. We develop a numerical measure for the stability of Mapper based on the exiting theory of clustering instability. This allows us to obtain concrete reasons for high values of Mapper instability and experimentally demonstrate how it can be applied to determine good Mapper outputs over variations of the Mapper parameters.

Our approach tackles directly the inherent instability of the choice of clustering procedure and requires very few assumption on the specifics of the data or chosen Mapper construction, making it applicable to any Mapper-type algorithm.

This is joint work with Francisco Belchì, Jacek Brodzki, and Mahesan Niranjan.

**Edelsbrunner** - Shape reconstruction and distortion in Bregman space

*Abstract:* How does the measure of dissimilarity affect the reconstruction of shape from point data, and how do we quantify the influence? We approach this question by extending popular Euclidean reconstruction algorithms to Bregman space. A particularly interesting Bregman divergence is the relative entropy, whose infinitesimal version defines the Fisher metric. We explain the connections and illustrate the findings with sample results of the implemented algorithms.

Joint work with Katharina Oelsboeck and Hubert Wagner.

**Jost** -** **Curvature and eigenvalues of graphs and hypergraphs and their applications in data analysis

*Abstract:* In recent years, concepts and ideas from Riemannian geometry, like curvature inequalities and eigenvalue estimates, have been transferred to graph theory to create new approaches for the identification of structural properties of graphs. Some of them also provide powerful new tools for geometric data analysis. This can also be extended to hypergraphs as occurring, for instance, in the modelling of chemical reaction networks.

**Membrillo-Solis** - The geometry and topology of configuration spaces of molecules

*Abstract: *A molecule is an example of an n-body system formed by the nuclei and the electrons of the constituent atoms. The space of all possible positions that the nuclei in a molecule can adopt is called the molecular configuration space. The study of these spaces is of particular interest in Chemistry since it can help to understand the relationship between the structural and chemical properties of molecules.

In this talk I will present a mathematical framework to study configuration spaces of molecules. I will show that in this framework, configuration spaces come equipped with a group of symmetries which give rise to both principal bundles and orbifolds. I will also show how data analytic methods, such as Procrustes analysis, local dimensionality reduction and persistent homology, can be used to explore the geometric and topological properties of these configuration spaces.

**Ölsböck** - The Hole System of Triangulated Shapes

*Abstract:* We are given a finite set of points sampled from an object, e.g. atoms of a protein, or points on a surface provided by a laser scanner. The goal is to reconstruct the shape of the object, and analyze and manipulate the hole system of the resulting model.

First, I will introduce the 'Alpha complex', which often gives a good reconstruction in the form of a triangulated shape. Allowing the scale parameter to vary, we get a nested sequence of such discrete shapes. The hole system of this sequence is best described with 'persistent homology' and best visualized with the persistence diagram.

In many cases, holes are important for the functionality of an object. Therefore, we are intersted in operations to manipulate the holes of the reconstructed shape. We define operations to open and close holes, while preserving dependences between them.

**Pirashvili** - Decomposability of certain multidimensional persistence modules

*Abstract:* The theory of persistent homology is closely linked to quiver representations. Persistence modules are representations of certain quivers. While one-dimensional persistent homology deals with representations of the A_n type quivers, which happen to be tame, multidimensional persistent homology is complicated due to the wild nature of the quivers whose representations we need to consider.

In this project, we looked at conditions placed on the representations which put restrictions on the representations not easily accounted for through the tools used in representation theory. We show how these may lead to a discrete invariant for the persistence modules.