Past Topological Data Analysis Seminar

14 June 2019
15:00
Jeffrey Giansiracusa
Abstract

One of the key properties of 1-parameter persistent homology is that its output can entirely encoded in a purely combinatorial way via persistence diagrams or barcodes.  However, many applications of topological data analysis naturally present themselves with more than 1 parameter. Multiparameter persistence suggests itself as the natural invariant to use, but the problem here is that the moduli space of multiparameter persistence diagrams has a much more complicated structure and we lack a combinatorial diagrammatic description.  An alternative approach was suggested by work of Giansiracusa-Moon-Lazar, where they investigated calculating a series of 1-parameter persistence diagrams as the other parameter is varied. In this talk I will discuss work in progress to produce a refinement of their perspective, making use the Algebraic Stability Theorem for persistent homology and work of Bauer-Lesnick on induced matchings.

  • Topological Data Analysis Seminar
7 June 2019
15:00
Ilya Chevyrev
Abstract

In this talk I will introduce the concept of the path signature and motivate its recent use in analysis of time-ordered data. I will then describe a new feature map for barcodes in persistent homology by first realizing each barcode as a path in a vector space, and then computing its signature which takes values in the tensor algebra over that vector space. The composition of these two operations — barcode to path, path to tensor series — results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness.

  • Topological Data Analysis Seminar
24 May 2019
15:30
Oliver Vipond
Abstract

I will give an introduction to the asymptotic behaviour of random geometric complexes. In the specific case of a simplicial complex realised as the Cech complex of a point process sampled from a closed Riemannian manifold, we will explore conditions which guarantee the homology of the Cech complex coincides with the homology of the underlying manifold. We will see techniques which were originally developed to study random geometric graphs, which together with ideas from Morse Theory establish homological connectivity thresholds.

  • Topological Data Analysis Seminar
24 May 2019
15:00
Wojciech Reise
Abstract

I will give an overview of audio identification methods on spectral representations of songs. I will outline the persistent homology-based approaches that I propose and their shortcomings. I hope that the review of previous work will help spark a discussion on new possible representations and filtrations.

  • Topological Data Analysis Seminar
10 May 2019
15:00
Abstract

The class of sheaf Laplacians can be characterized as the convex closure of a certain set of sparse semidefinite matrices. From this viewpoint, the study of sheaf Laplacians becomes a question of linear algebra on sparse matrices. I will discuss the applications of this perspective to the problems of approximating, sparsifying, and learning sheaves.

  • Topological Data Analysis Seminar
3 May 2019
15:00
Abstract

This talk will cover the connections of persistence with the topology of random structures. This includes an overview of various results from stochastic topology as well as the role persistence ideas  play in the analysis. This will include results on the maximally persistent classes and minimum spanning acycles/generalised trees.

  • Topological Data Analysis Seminar

Pages