Discrete Homotopy Theory and Applications

Explore discrete homotopy theory and its applications in this 49-minute lecture by Conrad Plaut. Delve into the evolution of this field, developed to understand generalized covering spaces of topological groups, and its extension to uniform and metric spaces. Learn how discrete chains and homotopies replace traditional paths, allowing for quantification of fundamental groups and topological features. Discover the simplicity of this approach compared to traditional methods, and its effectiveness in producing fundamental groups and regular covering spaces. Examine applications in generalized universal covering spaces, finiteness theorems in Riemannian geometry, topology of boundaries of CAT(0) spaces in geometric group theory, and spectra related to length spectra of compact Riemannian manifolds. Investigate concepts such as universal covers, critical loops, delta covers, phantom groups, uniform structures, and chain-connected spaces. Gain insights into the connections between discrete homotopy theory and graph theory, as well as potential applications in data science.

Introduction
Credits
The idea
The universal cover
Universal cover of a circle
Critical loop
Circle example
Delta covers
Covering spec spectrum
Phantom groups
Gramoff
Uniform Structures
Entourage Spaces
Inverse System
Spectre
History
Construction
Mark Length Spectrum
Fractals
Length Spectrum
Chain Connected Spaces
weakly chained Spaces
point one movable
point one moveable
point two inverse limit
taurus complexes
geometric group theory
general theorem
thank you
there are questions
I have a first question
Do you have connections to graph theory
Graph theory
Data science applications
Number tracking matrix