# Seminars:

The Centre is running a virtual seminar during the pandemic on Wednesdays, 11am to 12:30pm via Zoom. The Meeting-ID is 390 769 7241. Please contact Pawel Dlotko to obtain the passcode.

If you would like to give a talk as part of our seminar series please contact Pawel Dlotko.

- 04/11/2020

Niklas Hellmer (Dioscuri TDA, Warsaw): Nonembeddability of Persistence Diagrams with p>2 Wasserstein Metric.

**Abstract:**To apply kernel methods on persistence diagrams, they are mapped into a Hilbert space. The question is how much information we lose by doing so. I present recent work by Alexander Wagner that shows that persistence diagrams with p-Wasserstein metric do not admit a coarse embedding into a Hilbert space for p>2. - 11/11/2020

**No seminar due to public holiday!** - 18/11/2020

Pawel Dlotko (Dioscuri TDA, Warsaw): Mild introduction to TDA

**Abstract:**In this talk I will provide a general introduction to methods of TDA - concentrated mostly on persistent homology and mapper algorithm. I will present the basic theory, highlight the main application. But, I will mostly encourage questions and discussion. - 25/11/2020

John Harvey (Swansea University): Estimation of dimension

**Abstract:**I will survey various established and new techniques to estimate the dimension of a submanifold of Euclidean space from a sampled point cloud, how we might investigate underlying assumptions of the estimators, and how they relate to testing for manifoldness - 02/12/2020

Artem Dudko: On computability and computational complexity of Julia sets.

**Abstract:**In this talk I will give an introduction to computability and computational complexity of sets with the focus on Julia sets of polynomials. For a polynomial p(z) its Julia set is the set of points on a complex plane near which the iterations of p(z) behave chaotically. Roughly speaking, a set is called computable if there is an algorithm which can produce an approximation of this set (by a finite set of pixels) with any given precision. Computational complexity measures how hard it is to produce such approximations. In the talk I will show that already in the class of Julia sets of quadratic polynomials there are various interesting computational phenomena, including:

- large classes of functions having Julia sets of low (polynomial) computational complexity;

- functions with non-computable Julia sets;

- large classes of functions having Julia sets of arbitrarily high computational complexity. - 09/12/2020

Austin Lawson (UNC Greensboro): Persistence Curves: A canonical framework for summarizing persistence diagrams

**Abstract:**As Topological Data Analysis (TDA) grows in popularity so too does the need for topological methods compatible with modern machine learning algorithms. The use of machine learning algorithms directly on the space of persistence diagrams is challenging. For that reason, transforming these diagrams in a way that is compatible with machine learning is an important topic currently researched in TDA. A promising avenue in this vein involves mapping a persistence diagram to a real-valued summary function or vector that maintains the topological information contained in the diagram. Inspired by persistent Betti numbers, we discover a canonical way to generate these mappings, which we call the Persistence Curve Framework. In this talk, we will (1) define the framework and show that subsumes several other recent summary functions including the persistence landscapes; (2) explore theoretical results including stability; (3) demonstrate some recent applications using the PC framework including texture analysis, time series classification, and skin lesion analysis; and finally (4) we will discuss the future of applications and theoretical development of the framework. - 10/12/2020, 11 am
**Note the change of date (Thursday)!**

Rodrigo Azevedo Moreira de Silva: Proteins from Discrete Gauss-Bonnet Perspective

**Abstract:**Proteins are fundamental blocks of life, and their amino-acid sequence and geometric structure define their functions. However, the intrinsic rules governing their structure are still largely unknown. The emerging field of Topological Data Analysis has been contributing to the understanding of protein-protein interaction, protein classification, and structure. We will show our recent advances in the application of Euler Characteristic and local curvature analysis applied to large proteins and their molecular dynamics, and the insight about the role of high dimensional simplicies for modeling such systems. The technical difficulties found in processing the big data required for the modeling, as well as perspectives of the application of the discrete formulation of the Gauss-Bonnet theorem, will also be discussed. - 16/12/2020

Michael Grinfeld (University of Strathclyde, Glasgow): Topological Data Analysis techniques: unity in variety

**Abstract:**Given present-day vast computational resources, 21st century has seen a rapid development of techniques of topological data analysis. In addition to persistent homology, these techniques include Minkowski functionals and visibility graphs. I will review the Minkwoski functionals (and tensors) approach and the idea of a visibility graph, and will open for discussion two questions that seem to me to be central in this area: (a) which properties of the data are best captured by a which particular technique and (b) how are the different techniques connected, i.e. whether findings from one framework can be translated directly into results obtained by another. The thoughts on (a) and (b) will be very preliminary!