Advances in Applied and Computational Topology
About this Title
Afra Zomorodian, The D. E. Shaw Group, New York, NY, Editor
Publication: Proceedings of Symposia in Applied Mathematics
Publication Year 2012: Volume 70
ISBNs: 978-0-8218-5327-6 (print); 978-0-8218-8997-8 (online)
What is the shape of data? How do we describe flows? Can we count by integrating? How do we plan with uncertainty? What is the most compact representation? These questions, while unrelated, become similar when recast into a computational setting. Our input is a set of finite, discrete, noisy samples that describes an abstract space. Our goal is to compute qualitative features of the unknown space. It turns out that topology is sufficiently tolerant to provide us with robust tools.
This volume is based on lectures delivered at the 2011 AMS Short Course on Computational Topology, held January 4–5, 2011 in New Orleans, Louisiana.
The aim of the volume is to provide a broad introduction to recent techniques from applied and computational topology. Afra Zomorodian focuses on topological data analysis via efficient construction of combinatorial structures and recent theories of persistence. Marian Mrozek analyzes asymptotic behavior of dynamical systems via efficient computation of cubical homology. Justin Curry, Robert Ghrist, and Michael Robinson present Euler Calculus, an integral calculus based on the Euler characteristic, and apply it to sensor and network data aggregation. Michael Erdmann explores the relationship of topology, planning, and probability with the strategy complex. Jeff Erickson surveys algorithms and hardness results for topological optimization problems.
Graduate students and research mathematicians interested in applied and computational topology.
Table of Contents
- Afra Zomorodian – Topological data analysis
- Marian Mrozek – Topological dynamics: Rigorous numerics via cubical homology
- Justin Curry, Robert Ghrist and Michael Robinson – Euler calculus with applications to signals and sensing
- Michael Erdmann – On the topology of discrete planning with uncertainty
- Jeff Erickson – Combinatorial optimization of cycles and bases