What is quantitative topology and what quality of a topological space does it aim to quantify?
Thank you for all the answers in advance
This is a great question.
The beginning of an answer to this question can be found in
Real Analysis, Quantitative Topology, and Geometric Complexity
https://www.researchgate.net/publication/2096990_Real_Analysis_Quantitative_Topology_and_Geometric_Complexity
Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities
https://www.researchgate.net/publication/248780033_Finding_curves_on_general_spaces_through_quantitative_topology_with_applications_to_Sobolev_and_Poincare_inequalities
Typical problems tackled by quantitative topology are briefly described in
DIRECTIONS IN QUANTITATIVE TOPOLOGY
http://agingportfolio.org/projects/project/NSF-0504721
A detailed but narrow study of one aspect of quantitative topology is given in
Centralities in Simplicial Complexes
https://arxiv.org/pdf/1703.03641.pdf
Another aspect of quantitative topology is given in
Analysis and prediction of protein folding energy changes upon
mutation by element specific persistent homology
https://arxiv.org/pdf/1703.10966.pdf
Article Real Analysis, Quantitative Topology, and Geometric Complexity
Article Finding curves on general spaces through quantitative topolo...
This is a great question.
The beginning of an answer to this question can be found in
Real Analysis, Quantitative Topology, and Geometric Complexity
https://www.researchgate.net/publication/2096990_Real_Analysis_Quantitative_Topology_and_Geometric_Complexity
Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities
https://www.researchgate.net/publication/248780033_Finding_curves_on_general_spaces_through_quantitative_topology_with_applications_to_Sobolev_and_Poincare_inequalities
Typical problems tackled by quantitative topology are briefly described in
DIRECTIONS IN QUANTITATIVE TOPOLOGY
http://agingportfolio.org/projects/project/NSF-0504721
A detailed but narrow study of one aspect of quantitative topology is given in
Centralities in Simplicial Complexes
https://arxiv.org/pdf/1703.03641.pdf
Another aspect of quantitative topology is given in
Analysis and prediction of protein folding energy changes upon
mutation by element specific persistent homology
https://arxiv.org/pdf/1703.10966.pdf
Article Real Analysis, Quantitative Topology, and Geometric Complexity
Article Finding curves on general spaces through quantitative topolo...
Ok will try Muhammad , from application perspective
"Quantitative" aims at looking for regularity in the mapping or uniformity in the region (homeomorphism) . typical applications in graph theory . graphs are everywhere from stocks to Facebook
In terms of function continuity and comparison measures such as Lipschitz for "homotopics" (checking for changes )
Commonly : Manifold , T2 etc
Talking for topology is as talking for « shaping » (morphism // heteromorphic). From my point of view the term quantitative topology could be misleading in some cases. When make reference to 3-simplex as tetrahedron then the notion “quantitative” could be under question especially when talking for biological entities reflecting function (unit cell, drug screening etc.).
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