Pawel Dlotko: Applied, computational topology
In this talk I will present various applications of computational topology. We will start by using persistence as a descriptor of porous material in analyzing databases of hypothetical zeolites and metal-organic-frameworks. Later we will discuss the idea of using persistence as a descriptor of tree and graph structures. We will consider an example of static (and moving) neuron trees and roots of plants. At the end, if time permits, I will speak about discrete approach to Maxwell’s equations and the way topology is used over there.
Anthea Monod: Constructing Probability Distributions for Barcodes
The concept of sufficiency is important in mathematical statistics, since it allows for the dimension reduction of a given sample of data via a functional summary without the loss of information. Using tropical algebra, we construct an embedding of the barcode space into R^n. We prove that this mapping is a sufficient statistic for barcodes, and moreover, admits an inner product structure in R^n, which provides the formulation of likelihood models for observations of barcodes. Since our mapping is injective and Lipschitz, we have the regularity conditions needed to model distributions of barcodes within the exponential family — an important class of distributions, which comprises the normal, Poisson, chi-squared, Bernoulli, exponential, and gamma distributions, among others. This is joint work with Sara K. Verovsek.
Magnus Bakke Botnan: Multidimensional Persistence
We know that there exists no barcode-like structure theorem for multi-D persistence. In particular, such persistence modules do not admit a `complete’ and `discrete’ invariant. But what does this really mean? By means of representation theory of quivers I will illustrate the di erent complexities of isomorphism classes for quivers of diff erent representation type. In particular, ordinary persistence is of ‘ i nite type', circular-valued persistence is `tame' and multi-D persistence is `wild'. Moreover, there exists software that allows you to decompose a multi-D persistence module into indecomposable summands. While such summands need not be `interval summands', I will still argue that they carry useful topological information for the user. Some examples of multi-parameter clustering will be provided. Joint work with Ulrich Bauer (TU Munich) and Johan Steen (NTNU Trondheim).
Bei Wang: Structural Inference of Point Clouds
Given potentially high-dimensional point cloud samples, can we infer the structures of the underlying space? In this talk, I will discuss a few theoretical and algorithmic developments in extracting structural features in point cloud data via topological data analysis. In particular, I will discuss scenarios when the point clouds are potentially noisy, and/or the underlying space contains singularities.