topology@ub

17 November 2020, 9:30

Abstract: A homotopy automorphism is an endomorphism which is also a weak equivalence. The space of homotopy automorphisms of a topological space X forms a topological monoid whose classifying space classifies all X-fibrations up to equivalence. Homotopy automorphisms of a closed oriented manifold M that restrict to the identity on a certain embedded closed disk, so called relative homotopy automorphisms, extend to relative homotopy automorphisms of any connected sum M#N. In particular we may study homological stability for relative homotopy automorphisms of iterated connected sums of closed oriented manifolds. In this series of talks I will discuss these topics from the point of view of rational homotopy theory. This is the first of two lectures on this topic.

24 November 2020, 9:30

[ Recording of the talk ]

1 December 2020, 9:30

Abstract: Consider a point cloud X in Rⁿ. On X one can build a filtered complex VR(X), known as the Vietoris-Rips complex. The persistent homology PH(X), computed from this filtered complex, contains a lot of interesting topological information. However, with ever-growing datasets, X and VR(X) become computationally expensive or intractable. This can be tackled by subdividing X, which leads to the following natural question: Given a cover U of X, and given some information about the local persistent homologies over each set in U, is it possible to recover PH(X) or at least some information about it? We will study this question using the Mayer-Vietoris spectral sequence, which is a powerful tool from algebraic topology, that encodes the relationship between the local complexes in its differentials. It provides a step-by-step procedure that allows parallelisation of most of the computations. If time permits, we will also briefly explain how the spectral sequence can be seen as an invariant in its own right. For this, we show that if local information is perturbed a bit, one recovers interleavings on the first pages of the spectral sequence. As a corollary, one can obtain a strong approximate Persistent Multinerve Theorem that extends the approximate version of the Nerve Theorem found by Govc and Scraba.

15 December 2020, 16:00

Abstract: Until recently, the role of applied topology in machine learning has largely been descriptive, with topological invariants used as summaries of shape data. However, a new wave of research aims to use applied topology in a prescriptive way, to simplify data or enforce desired structure. One happy discovery is that persistent homology, a well-studied invariant in applied topology, is just differentiable enough to make gradient descent optimization possible. This crucial feature of persistent homology is hampered by a computational bottleneck: the complexity of computing persistence diagrams is an unfeasible O(n³) in the worst case. To address this issue, we introduce a host of techniques from the machine learning and applied topology literatures: momentum, stochastic pooling, Čech complexes, persistence stability, etc. The resulting pipeline provides robust topological optima in a fraction of the time of the traditional method. We conclude with an application to cell image segmentation from the ISBI12 data set (coming from a symposium held in 2012 in Barcelona).