N3.12

Configuration spaces of points in Euclidean space or on a manifold are well studied in algebraic topology. But what if the points have some positive thickness? This is a natural setting from the point of view of physics, since this the energy landscape of a hard-spheres system. Such systems are observed experimentally to go through phase transitions, but little is known mathematically.

In this talk, I will focus on two special cases where we have started to learn some things about the homology: (1) hard disks in an infinite strip, and (2) hard squares in a square or rectangle. We will discuss some theorems and conjectures, and also some computational results. We suggest definitions for "homological solid, liquid, and gas" regimes based on what we have learned so far.

This is joint work with Hannah Alpert, Ulrich Bauer, Robert MacPherson, and Kelly Spendlove.

The goal of topological data analysis is to apply tools form algebraic topology to reveal geometric structures hidden within high dimensional data. Mapper is among its most widely and successfully applied tools providing, a framework for the geometric analysis of point cloud data. Given a number of input parameters, the Mapper algorithm constructs a graph, giving rise to a visual representation of the structure of the data.  The Mapper graph is a topological representation, where the placement of individual vertices and edges is not important, while geometric features such as loops and flares are revealed.

However, Mappers method is rather ad hoc, and would therefore benefit from a formal approach governing how to make the necessary choices. In this talk I will present joint work with Francisco Belchì, Jacek Brodzki, and Mahesan Niranjan. We study how sensitive to perturbations of the data the graph returned by the Mapper algorithm is given a particular tuning of parameters and how this depend on the choice of those parameters. Treating Mapper as a clustering generalisation, we develop a notion of instability of Mapper and study how it is affected by the choices. In particular, we obtain concrete reasons for high values of Mapper instability and experimentally demonstrate how Mapper instability can be used to determine good Mapper outputs.

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.

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.