Tamal Dey - A Framework for Topological Summarization

Explore a comprehensive framework for topological summarization in this 57-minute lecture by Tamal Dey, presented for the Applied Algebraic Topology Network. Delve into key concepts such as covers, nerves, and maps between covers, before progressing to more advanced topics like pullback covers and hierarchical covering complexes. Learn about the Multiscale Mapper and its persistence diagram, as well as the stability of Hierarchical Function-Ordered Simplicial Structures (HFOSS) against perturbations. Examine various cover examples and stability against function perturbations, with a focus on PL functions and combinatorial Multiscale Mapper. Gain insights into the metric viewpoint and Čech filtrations using du before concluding with a Q&A session. This talk provides a thorough exploration of topological summarization techniques and their applications in algebraic topology.

Intro
Covers and Nerves
Maps and covers
Maps between covers
Pullback covers
Hierarchical covering and complexes
Multiscale Mapper
Persistence diagram of MM
Interleaving between HFOSS
Stability against perturbation of HFOCs
Cover examples
Stability against function perturbation
MM for PL functions
Combinatorial MM
Metric viewpoint
Čech filtrations using du
Conclusion/Questions