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Fabian Roll: A Unified View on the Functorial Nerve Theorem and its Variations

Time: Tue 2022-04-26 10.15

Location: KTH, 3721, Lindstedtsvägen 25, and Zoom

Video link: Meeting ID: 659 3743 5667

Lecturer: Fabian Roll (TU München)

Abstract

The history of the nerve construction reaches back at least to Alexandrov (1928) and the early days of algebraic topology. Nowadays, the nerve theorem, as well as the aspect of functoriality, play a crucial role in topological data analysis. In this area, nerves are the main tool to replace a point cloud with a combinatorial model that is suitable for computations.

In this talk, I will survey certain aspects of the long history of nerves and nerve theorems, mention applications to combinatorics, and draw the connection between the nerve theorem and persistent homology. Moreover, I will sketch a proof of the nerve theorem for covers by closed convex sets in Euclidean space that uses relatively elementary techniques. I will also set up the framework that allows us to discuss the aspect of functoriality and show two ways in which the presented nerve theorem can be turned functorial. Finally, I will state a "unified'' nerve theorem that establishes an equivalence between a covered space and its nerve under a variety of assumptions. If time permits, I will present some counterexamples that show the tightness of these assumptions and I will sketch the proof strategy, which uses standard techniques from modern homotopy theory such as model categories.

This is joint work with Ulrich Bauer, Michael Kerber, and Alexander Rolle.