N3.12

In this talk, linear algebra for persistence modules will be introduced, together with a generalization of persistent homology. This theory permits us to handle the Mayer-Vietoris spectral sequence for persistence modules, and solve any extension problems that might arise. The result of this approach is a distributive algorithm for computing persistent homology. That is, one can break down the underlying data into different covering subsets, compute the persistent homology for each cover, and join everything together. This approach has the added advantage that one can recover extra geometrical information related to the barcodes. This addresses the common complaint that persistent homology barcodes are 'too blind' to the geometry of the data.

The triangular inequality is central in Mathematics. What would happen if we reverse it? We only obtain trivial spaces. However, if we mix it with an order structure, we obtain interesting spaces. We'll present linear antimetrics, prove a "masking theorem", and then look at a corollary which tells us about the "twin paradox" in special relativity; time is antimetric!

We can see the simplest setting of persistence from a functional point of view: given a fixed finite simplicial complex, we have the barcode function which, given a filter function over this complex, returns the corresponding persistent diagram. The bottleneck distance induces a topology on the space of persistence diagrams, and makes the barcode function a continuous map: this is a consequence of the stability Theorem. In this presentation, I will present ongoing work that seeks to deepen our understanding of the analytic properties of the barcode function, in particular whether it can be said to be smooth. Namely, if we smoothly vary the filter function, do we get smooth changes in the resulting persistent diagram? I will introduce a notion of differentiability/smoothness for barcode valued maps, and then explain why the barcode function is smooth (but not everywhere) with respect to the choice of filter function. I will finally explain why these notions are of interest in practical optimisation/learning situations. 

The amount and complexity of biological data has increased rapidly in recent years with the availability of improved biological tools. When applying persistent homology to large data sets, many of the currently available algorithms however fail due to computational complexity preventing many interesting biological applications. De Silva and Carlsson (2004) introduced the so called Witness Complex that reduces computational complexity by building simplicial complexes on a small subset of landmark points selected from the original data set. The landmark points are chosen from the data either at random or using the so called maxmin algorithm. These approaches are not ideal as the random selection tends to favour dense areas of the point cloud while the maxmin algorithm often selects outliers as landmarks. Both of these problems need to be addressed in order to make the method more applicable to biological data. We study new ways of selecting landmarks from a large data set that are robust to outliers. We further examine the effects of the different subselection methods on the persistent homology of the data.

Knots and their groups are a traditional topic of geometric topology. In this talk, I will explain how aspects of the subject can be approached as a homotopy theorist, rephrasing old results and leading to new ones. Part of this reports on joint work with Tyler Lawson.

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.

Hilbert's fifth problem asks informally what is the difference between Lie groups and topological groups. In 1950s this problem was solved by Andrew Gleason, Deane Montgomery, Leo Zippin and Hidehiko Yamabe concluding that every locally compact topological group is "essentially" a Lie group. In this talk we will show the complete proof of this theorem.

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.

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.