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Reference on Persistent Homology
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I will be teaching a course on algebraic topology for MSc students and this semester, unlike previous ones where I used to begin with the fundamental group, I would like to start with ideas of singular homology as in Vick's book.
I am quite new to the ideas of persistent homology and have not done a single computations in this field. But, I like to learn on the subject. More is that I like to lead the course that I will be teaching so that towards the end, I can give some taste of persistent homology to students. But, I am not sure if there is any well written set of lecture notes on the material, or should we dive into the literature and start with some papers!?! The course involves of $3/2times 30$ hours of lectures.
Do you think this is possible or should I use some simplicial approaches instead? or you think it is more suitable to give this as a task to students to start as a project and discover the ideas for themsevles?!?!
I would be very grateful for any advise in terms of addressing to main references on the subject. I also would be grateful if you can give me some advise on the history of the subject; for instance when people decided to use homology to study biological problems and whether or not the main stream researchers in biology or data analysis really consider these kind of tools?!
reference-request co.combinatorics at.algebraic-topology
asked yesterday
user51223
1,207616
add a comment |Â
up vote
15
down vote
favorite
I will be teaching a course on algebraic topology for MSc students and this semester, unlike previous ones where I used to begin with the fundamental group, I would like to start with ideas of singular homology as in Vick's book.
I am quite new to the ideas of persistent homology and have not done a single computations in this field. But, I like to learn on the subject. More is that I like to lead the course that I will be teaching so that towards the end, I can give some taste of persistent homology to students. But, I am not sure if there is any well written set of lecture notes on the material, or should we dive into the literature and start with some papers!?! The course involves of $3/2times 30$ hours of lectures.
Do you think this is possible or should I use some simplicial approaches instead? or you think it is more suitable to give this as a task to students to start as a project and discover the ideas for themsevles?!?!
I would be very grateful for any advise in terms of addressing to main references on the subject. I also would be grateful if you can give me some advise on the history of the subject; for instance when people decided to use homology to study biological problems and whether or not the main stream researchers in biology or data analysis really consider these kind of tools?!
reference-request co.combinatorics at.algebraic-topology
asked yesterday
user51223
1,207616
add a comment |Â
up vote
15
down vote
favorite
up vote
15
down vote
favorite
I will be teaching a course on algebraic topology for MSc students and this semester, unlike previous ones where I used to begin with the fundamental group, I would like to start with ideas of singular homology as in Vick's book.
I am quite new to the ideas of persistent homology and have not done a single computations in this field. But, I like to learn on the subject. More is that I like to lead the course that I will be teaching so that towards the end, I can give some taste of persistent homology to students. But, I am not sure if there is any well written set of lecture notes on the material, or should we dive into the literature and start with some papers!?! The course involves of $3/2times 30$ hours of lectures.
Do you think this is possible or should I use some simplicial approaches instead? or you think it is more suitable to give this as a task to students to start as a project and discover the ideas for themsevles?!?!
I would be very grateful for any advise in terms of addressing to main references on the subject. I also would be grateful if you can give me some advise on the history of the subject; for instance when people decided to use homology to study biological problems and whether or not the main stream researchers in biology or data analysis really consider these kind of tools?!
reference-request co.combinatorics at.algebraic-topology
asked yesterday
user51223
1,207616
I will be teaching a course on algebraic topology for MSc students and this semester, unlike previous ones where I used to begin with the fundamental group, I would like to start with ideas of singular homology as in Vick's book.
I am quite new to the ideas of persistent homology and have not done a single computations in this field. But, I like to learn on the subject. More is that I like to lead the course that I will be teaching so that towards the end, I can give some taste of persistent homology to students. But, I am not sure if there is any well written set of lecture notes on the material, or should we dive into the literature and start with some papers!?! The course involves of $3/2times 30$ hours of lectures.
Do you think this is possible or should I use some simplicial approaches instead? or you think it is more suitable to give this as a task to students to start as a project and discover the ideas for themsevles?!?!
I would be very grateful for any advise in terms of addressing to main references on the subject. I also would be grateful if you can give me some advise on the history of the subject; for instance when people decided to use homology to study biological problems and whether or not the main stream researchers in biology or data analysis really consider these kind of tools?!
reference-request co.combinatorics at.algebraic-topology
reference-request co.combinatorics at.algebraic-topology
asked yesterday
user51223
1,207616
asked yesterday
user51223
1,207616
asked yesterday
user51223
1,207616
asked yesterday
user51223
1,207616
asked yesterday
user51223
1,207616
1,207616
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5 Answers
5
active
oldest
votes
up vote
9
down vote
Edelsbrunner and Harer's book seems good.
Edelsbrunner, Herbert; Harer, John L., Computational topology. An introduction, Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4925-5/hbk). xii, 241Â p. (2010). ZBL1193.55001.
answered yesterday
Igor Rivin
77.3k8109290
add a comment |Â
up vote
6
down vote
Maybe the following papers will be useful:
https://www.cambridge.org/core/journals/acta-numerica/article/topological-pattern-recognition-for-point-cloud-data/BB0DA0F0EBD79809C563AF80B555A23C (Topological pattern recognition for point cloud data, by Gunnar Carlsson).
https://escholarship.org/uc/item/2h33d90r (Persistent Homology: Theory and Practice, by Herbert Edelsbrunner and Dimitry Morozov).
answered yesterday
Zurab Silagadze
10.5k2368
add a comment |Â
up vote
6
down vote
The book by Steve Oudot is an alternative: Steve Y. Oudot. Persistence Theory: From Quiver Representations to Data Analysis (Mathematical Surveys and Monographs).
There is also a relatively new tutorial by Paweł Dłotko: Computational and applied topology, tutorial.
This introduction by Fugacci and others may also help you: Persistent homology: a step-by-step introduction for newcomers.
See also this question on studying persistent homology.
answered 22 hours ago
shadow
1612
New contributor
shadow is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
6
down vote
Since this area is developing rather quickly, there is a dearth of canonical references that would satisfy basic pedagogical requirements. If I were teaching a course on this material right now, I would probably use Oudot's nice book if the students had sufficient background, and the foundational paper of Zomorodian-Carlsson if they did not.
I haven't read Jose's recent article mentioned in Joe's answer, but here is what I remember of the good old days (with apologies to all the important stuff that got missed).
1992: Frosini introduces "size functions", which we would today consider equivalent to 0-dimensional persistent homology.
1995: Mischaikow + Mrozek publish a computer-assisted proof of chaos in the Lorenz equations; a key step involves computing Conley indices, which are relative homology classes. This produces considerable interest in machine computation of homology groups of spaces from finite approximations (eg large cell complexes).
1999: Robbins publishes this paper emphasizing that functoriality helps approximate the homology of an underlying space from Cech complexes of finite samples; meanwhile Kaczynski, Mischaikow and Mrozek publish their book on efficient homology computation via simple homotopy type reductions of cell complexes.
2002: Edelsbrunner, Letscher and Zomorodian introduce persistence from a computational geometry viewpoint; as written, their algorithm works only for subcomplexes of spheres and only with mod-2 coefficients.
2005: This was a big year.
Zomorodian and Carlsson reinterpret persistence of a filtration via the representation theory of graded modules over graded pid's, thus giving an algorithm for all finite cell complexes over arbitrary field coefficients; they also introduce the barcode, which is a perfect combinatorial invariant of certain tame persistence modules.
Edelsbrunner, Cohen-Steiner and Harer show that the map $$text[functions X to R] to text[barcodes]$$ obtained by looking at sublevel set homology of nice functions on triangulable spaces is 1-Lipschitz when the codomain is endowed with a certain metric called the bottleneck distance. This is the first avatar of the celebrated stability theorem.
2007: de Silva and Ghrist use persistence to give a slick solution to the coverage problem for sensor networks.
2008: Niyogi, Smale and Weinberger publish a paper solving the homology inference problem for compact Riemannian submanifolds of Euclidean space from finite uniform samples. Carlsson, with Singh and Sexton, starts Ayasdi, putting his money where his math is.
2009: Carlsson and Zomorodian use quiver representation theory to point out that getting finite barcodes for multiparameter persistence modules is impossible, highlighting dimension 2 as the new frontier for theoretical work in the field.
2010: Carlsson and de Silva, by now fully immersed in the quiver-rep zone, introduce zigzag persistence. The first software package for computing persistence (Plex, by Adams, de Silva, Vejdemo-Johansson,...) materializes.
2011: Nicolau, Levine and Carlsson discover a new type of breast cancer using 0-dimensional persistence on an old, and purportedly well-mined, tumor dataset.
2012 Chazal, de Silva, Glisse and Oudot unleash this beastly reworking of the stablity theorem. Gone are various assumptions about tameness and sub-levelsets. They show that bottleneck distance between barcodes arises from a certain "interleaving distance" on the persistence modules. This opens the door for more algebraic and categorical interpretations of persistence, eg Bubenik-Scott.
2013: Mischaikow and I retool the simple homotopy-based reductions to work for filtered cell complexes, thus producing the first efficient preprocessor for the Zomorodian-Carlsson algorithm along with a fast (at the time!) software package Perseus.
2015 Lesnick publishes a comprehensive study of the interleaving distance in the context of multiparameter persistence modules.
2018 MacPherson and Patel concoct bisheaves to attack multi-parameter persistence geometrically for fibers of maps to triangulable manifolds.
Good luck with your course!
answered 8 hours ago
Vidit Nanda
9,01123898
2
this is really great! the historical order and the mention of discovery of new breat cancer are really attracting! more than I could have wished for really!!!!
– user51223
6 hours ago
add a comment |Â
up vote
5
down vote
This paper was just released on the arXiv this morning:
"A Brief History of Persistence."
Jose A. Perea. 2018.
arXiv abstract.
"Persistent homology is currently one of the more widely known tools from
computational topology and topological data analysis. We present in this note a
brief survey on the evolution of the subject. The goal is to highlight the main
ideas, starting from the subject's computational inception more than 20 years
ago, to the more modern categorical and representation-theoretic point of view."
    Â
answered 22 hours ago
Joseph O'Rourke
83.2k15219683
thanks. I just saw it today!
– user51223
19 hours ago
add a comment |Â
5 Answers
5
active
oldest
votes
5 Answers
5
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
9
down vote
Edelsbrunner and Harer's book seems good.
Edelsbrunner, Herbert; Harer, John L., Computational topology. An introduction, Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4925-5/hbk). xii, 241Â p. (2010). ZBL1193.55001.
answered yesterday
Igor Rivin
77.3k8109290
add a comment |Â
up vote
9
down vote
Edelsbrunner and Harer's book seems good.
Edelsbrunner, Herbert; Harer, John L., Computational topology. An introduction, Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4925-5/hbk). xii, 241Â p. (2010). ZBL1193.55001.
answered yesterday
Igor Rivin
77.3k8109290
add a comment |Â
up vote
9
down vote
up vote
9
down vote
Edelsbrunner and Harer's book seems good.
Edelsbrunner, Herbert; Harer, John L., Computational topology. An introduction, Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4925-5/hbk). xii, 241Â p. (2010). ZBL1193.55001.
answered yesterday
Igor Rivin
77.3k8109290
Edelsbrunner and Harer's book seems good.
Edelsbrunner, Herbert; Harer, John L., Computational topology. An introduction, Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4925-5/hbk). xii, 241Â p. (2010). ZBL1193.55001.
answered yesterday
Igor Rivin
77.3k8109290
answered yesterday
Igor Rivin
77.3k8109290
answered yesterday
Igor Rivin
77.3k8109290
answered yesterday
Igor Rivin
77.3k8109290
77.3k8109290
add a comment |Â
add a comment |Â
up vote
6
down vote
Maybe the following papers will be useful:
https://www.cambridge.org/core/journals/acta-numerica/article/topological-pattern-recognition-for-point-cloud-data/BB0DA0F0EBD79809C563AF80B555A23C (Topological pattern recognition for point cloud data, by Gunnar Carlsson).
https://escholarship.org/uc/item/2h33d90r (Persistent Homology: Theory and Practice, by Herbert Edelsbrunner and Dimitry Morozov).
answered yesterday
Zurab Silagadze
10.5k2368
add a comment |Â
up vote
6
down vote
Maybe the following papers will be useful:
https://www.cambridge.org/core/journals/acta-numerica/article/topological-pattern-recognition-for-point-cloud-data/BB0DA0F0EBD79809C563AF80B555A23C (Topological pattern recognition for point cloud data, by Gunnar Carlsson).
https://escholarship.org/uc/item/2h33d90r (Persistent Homology: Theory and Practice, by Herbert Edelsbrunner and Dimitry Morozov).
answered yesterday
Zurab Silagadze
10.5k2368
add a comment |Â
up vote
6
down vote
up vote
6
down vote
Maybe the following papers will be useful:
https://www.cambridge.org/core/journals/acta-numerica/article/topological-pattern-recognition-for-point-cloud-data/BB0DA0F0EBD79809C563AF80B555A23C (Topological pattern recognition for point cloud data, by Gunnar Carlsson).
https://escholarship.org/uc/item/2h33d90r (Persistent Homology: Theory and Practice, by Herbert Edelsbrunner and Dimitry Morozov).
answered yesterday
Zurab Silagadze
10.5k2368
Maybe the following papers will be useful:
https://www.cambridge.org/core/journals/acta-numerica/article/topological-pattern-recognition-for-point-cloud-data/BB0DA0F0EBD79809C563AF80B555A23C (Topological pattern recognition for point cloud data, by Gunnar Carlsson).
https://escholarship.org/uc/item/2h33d90r (Persistent Homology: Theory and Practice, by Herbert Edelsbrunner and Dimitry Morozov).
answered yesterday
Zurab Silagadze
10.5k2368
answered yesterday
Zurab Silagadze
10.5k2368
answered yesterday
Zurab Silagadze
10.5k2368
answered yesterday
Zurab Silagadze
10.5k2368
10.5k2368
add a comment |Â
add a comment |Â
up vote
6
down vote
The book by Steve Oudot is an alternative: Steve Y. Oudot. Persistence Theory: From Quiver Representations to Data Analysis (Mathematical Surveys and Monographs).
There is also a relatively new tutorial by Paweł Dłotko: Computational and applied topology, tutorial.
This introduction by Fugacci and others may also help you: Persistent homology: a step-by-step introduction for newcomers.
See also this question on studying persistent homology.
answered 22 hours ago
shadow
1612
New contributor
shadow is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
6
down vote
The book by Steve Oudot is an alternative: Steve Y. Oudot. Persistence Theory: From Quiver Representations to Data Analysis (Mathematical Surveys and Monographs).
There is also a relatively new tutorial by Paweł Dłotko: Computational and applied topology, tutorial.
This introduction by Fugacci and others may also help you: Persistent homology: a step-by-step introduction for newcomers.
See also this question on studying persistent homology.
answered 22 hours ago
shadow
1612
New contributor
shadow is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
6
down vote
up vote
6
down vote
The book by Steve Oudot is an alternative: Steve Y. Oudot. Persistence Theory: From Quiver Representations to Data Analysis (Mathematical Surveys and Monographs).
There is also a relatively new tutorial by Paweł Dłotko: Computational and applied topology, tutorial.
This introduction by Fugacci and others may also help you: Persistent homology: a step-by-step introduction for newcomers.
See also this question on studying persistent homology.
answered 22 hours ago
shadow
1612
New contributor
shadow is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
The book by Steve Oudot is an alternative: Steve Y. Oudot. Persistence Theory: From Quiver Representations to Data Analysis (Mathematical Surveys and Monographs).
There is also a relatively new tutorial by Paweł Dłotko: Computational and applied topology, tutorial.
This introduction by Fugacci and others may also help you: Persistent homology: a step-by-step introduction for newcomers.
See also this question on studying persistent homology.
answered 22 hours ago
shadow
1612
New contributor
shadow is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 22 hours ago
shadow
1612
New contributor
shadow is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 22 hours ago
shadow
1612
answered 22 hours ago
shadow
1612
1612
New contributor
shadow is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
shadow is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
shadow is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
add a comment |Â
up vote
6
down vote
Since this area is developing rather quickly, there is a dearth of canonical references that would satisfy basic pedagogical requirements. If I were teaching a course on this material right now, I would probably use Oudot's nice book if the students had sufficient background, and the foundational paper of Zomorodian-Carlsson if they did not.
I haven't read Jose's recent article mentioned in Joe's answer, but here is what I remember of the good old days (with apologies to all the important stuff that got missed).
1992: Frosini introduces "size functions", which we would today consider equivalent to 0-dimensional persistent homology.
1995: Mischaikow + Mrozek publish a computer-assisted proof of chaos in the Lorenz equations; a key step involves computing Conley indices, which are relative homology classes. This produces considerable interest in machine computation of homology groups of spaces from finite approximations (eg large cell complexes).
1999: Robbins publishes this paper emphasizing that functoriality helps approximate the homology of an underlying space from Cech complexes of finite samples; meanwhile Kaczynski, Mischaikow and Mrozek publish their book on efficient homology computation via simple homotopy type reductions of cell complexes.
2002: Edelsbrunner, Letscher and Zomorodian introduce persistence from a computational geometry viewpoint; as written, their algorithm works only for subcomplexes of spheres and only with mod-2 coefficients.
2005: This was a big year.
Zomorodian and Carlsson reinterpret persistence of a filtration via the representation theory of graded modules over graded pid's, thus giving an algorithm for all finite cell complexes over arbitrary field coefficients; they also introduce the barcode, which is a perfect combinatorial invariant of certain tame persistence modules.
Edelsbrunner, Cohen-Steiner and Harer show that the map $$text[functions X to R] to text[barcodes]$$ obtained by looking at sublevel set homology of nice functions on triangulable spaces is 1-Lipschitz when the codomain is endowed with a certain metric called the bottleneck distance. This is the first avatar of the celebrated stability theorem.
2007: de Silva and Ghrist use persistence to give a slick solution to the coverage problem for sensor networks.
2008: Niyogi, Smale and Weinberger publish a paper solving the homology inference problem for compact Riemannian submanifolds of Euclidean space from finite uniform samples. Carlsson, with Singh and Sexton, starts Ayasdi, putting his money where his math is.
2009: Carlsson and Zomorodian use quiver representation theory to point out that getting finite barcodes for multiparameter persistence modules is impossible, highlighting dimension 2 as the new frontier for theoretical work in the field.
2010: Carlsson and de Silva, by now fully immersed in the quiver-rep zone, introduce zigzag persistence. The first software package for computing persistence (Plex, by Adams, de Silva, Vejdemo-Johansson,...) materializes.
2011: Nicolau, Levine and Carlsson discover a new type of breast cancer using 0-dimensional persistence on an old, and purportedly well-mined, tumor dataset.
2012 Chazal, de Silva, Glisse and Oudot unleash this beastly reworking of the stablity theorem. Gone are various assumptions about tameness and sub-levelsets. They show that bottleneck distance between barcodes arises from a certain "interleaving distance" on the persistence modules. This opens the door for more algebraic and categorical interpretations of persistence, eg Bubenik-Scott.
2013: Mischaikow and I retool the simple homotopy-based reductions to work for filtered cell complexes, thus producing the first efficient preprocessor for the Zomorodian-Carlsson algorithm along with a fast (at the time!) software package Perseus.
2015 Lesnick publishes a comprehensive study of the interleaving distance in the context of multiparameter persistence modules.
2018 MacPherson and Patel concoct bisheaves to attack multi-parameter persistence geometrically for fibers of maps to triangulable manifolds.
Good luck with your course!
answered 8 hours ago
Vidit Nanda
9,01123898
2
this is really great! the historical order and the mention of discovery of new breat cancer are really attracting! more than I could have wished for really!!!!
– user51223
6 hours ago
add a comment |Â
up vote
6
down vote
Since this area is developing rather quickly, there is a dearth of canonical references that would satisfy basic pedagogical requirements. If I were teaching a course on this material right now, I would probably use Oudot's nice book if the students had sufficient background, and the foundational paper of Zomorodian-Carlsson if they did not.
I haven't read Jose's recent article mentioned in Joe's answer, but here is what I remember of the good old days (with apologies to all the important stuff that got missed).
1992: Frosini introduces "size functions", which we would today consider equivalent to 0-dimensional persistent homology.
1995: Mischaikow + Mrozek publish a computer-assisted proof of chaos in the Lorenz equations; a key step involves computing Conley indices, which are relative homology classes. This produces considerable interest in machine computation of homology groups of spaces from finite approximations (eg large cell complexes).
1999: Robbins publishes this paper emphasizing that functoriality helps approximate the homology of an underlying space from Cech complexes of finite samples; meanwhile Kaczynski, Mischaikow and Mrozek publish their book on efficient homology computation via simple homotopy type reductions of cell complexes.
2002: Edelsbrunner, Letscher and Zomorodian introduce persistence from a computational geometry viewpoint; as written, their algorithm works only for subcomplexes of spheres and only with mod-2 coefficients.
2005: This was a big year.
Zomorodian and Carlsson reinterpret persistence of a filtration via the representation theory of graded modules over graded pid's, thus giving an algorithm for all finite cell complexes over arbitrary field coefficients; they also introduce the barcode, which is a perfect combinatorial invariant of certain tame persistence modules.
Edelsbrunner, Cohen-Steiner and Harer show that the map $$text[functions X to R] to text[barcodes]$$ obtained by looking at sublevel set homology of nice functions on triangulable spaces is 1-Lipschitz when the codomain is endowed with a certain metric called the bottleneck distance. This is the first avatar of the celebrated stability theorem.
2007: de Silva and Ghrist use persistence to give a slick solution to the coverage problem for sensor networks.
2008: Niyogi, Smale and Weinberger publish a paper solving the homology inference problem for compact Riemannian submanifolds of Euclidean space from finite uniform samples. Carlsson, with Singh and Sexton, starts Ayasdi, putting his money where his math is.
2009: Carlsson and Zomorodian use quiver representation theory to point out that getting finite barcodes for multiparameter persistence modules is impossible, highlighting dimension 2 as the new frontier for theoretical work in the field.
2010: Carlsson and de Silva, by now fully immersed in the quiver-rep zone, introduce zigzag persistence. The first software package for computing persistence (Plex, by Adams, de Silva, Vejdemo-Johansson,...) materializes.
2011: Nicolau, Levine and Carlsson discover a new type of breast cancer using 0-dimensional persistence on an old, and purportedly well-mined, tumor dataset.
2012 Chazal, de Silva, Glisse and Oudot unleash this beastly reworking of the stablity theorem. Gone are various assumptions about tameness and sub-levelsets. They show that bottleneck distance between barcodes arises from a certain "interleaving distance" on the persistence modules. This opens the door for more algebraic and categorical interpretations of persistence, eg Bubenik-Scott.
2013: Mischaikow and I retool the simple homotopy-based reductions to work for filtered cell complexes, thus producing the first efficient preprocessor for the Zomorodian-Carlsson algorithm along with a fast (at the time!) software package Perseus.
2015 Lesnick publishes a comprehensive study of the interleaving distance in the context of multiparameter persistence modules.
2018 MacPherson and Patel concoct bisheaves to attack multi-parameter persistence geometrically for fibers of maps to triangulable manifolds.
Good luck with your course!
answered 8 hours ago
Vidit Nanda
9,01123898
2
this is really great! the historical order and the mention of discovery of new breat cancer are really attracting! more than I could have wished for really!!!!
– user51223
6 hours ago
add a comment |Â
up vote
6
down vote
up vote
6
down vote
Since this area is developing rather quickly, there is a dearth of canonical references that would satisfy basic pedagogical requirements. If I were teaching a course on this material right now, I would probably use Oudot's nice book if the students had sufficient background, and the foundational paper of Zomorodian-Carlsson if they did not.
I haven't read Jose's recent article mentioned in Joe's answer, but here is what I remember of the good old days (with apologies to all the important stuff that got missed).
1992: Frosini introduces "size functions", which we would today consider equivalent to 0-dimensional persistent homology.
1995: Mischaikow + Mrozek publish a computer-assisted proof of chaos in the Lorenz equations; a key step involves computing Conley indices, which are relative homology classes. This produces considerable interest in machine computation of homology groups of spaces from finite approximations (eg large cell complexes).
1999: Robbins publishes this paper emphasizing that functoriality helps approximate the homology of an underlying space from Cech complexes of finite samples; meanwhile Kaczynski, Mischaikow and Mrozek publish their book on efficient homology computation via simple homotopy type reductions of cell complexes.
2002: Edelsbrunner, Letscher and Zomorodian introduce persistence from a computational geometry viewpoint; as written, their algorithm works only for subcomplexes of spheres and only with mod-2 coefficients.
2005: This was a big year.
Zomorodian and Carlsson reinterpret persistence of a filtration via the representation theory of graded modules over graded pid's, thus giving an algorithm for all finite cell complexes over arbitrary field coefficients; they also introduce the barcode, which is a perfect combinatorial invariant of certain tame persistence modules.
Edelsbrunner, Cohen-Steiner and Harer show that the map $$text[functions X to R] to text[barcodes]$$ obtained by looking at sublevel set homology of nice functions on triangulable spaces is 1-Lipschitz when the codomain is endowed with a certain metric called the bottleneck distance. This is the first avatar of the celebrated stability theorem.
2007: de Silva and Ghrist use persistence to give a slick solution to the coverage problem for sensor networks.
2008: Niyogi, Smale and Weinberger publish a paper solving the homology inference problem for compact Riemannian submanifolds of Euclidean space from finite uniform samples. Carlsson, with Singh and Sexton, starts Ayasdi, putting his money where his math is.
2009: Carlsson and Zomorodian use quiver representation theory to point out that getting finite barcodes for multiparameter persistence modules is impossible, highlighting dimension 2 as the new frontier for theoretical work in the field.
2010: Carlsson and de Silva, by now fully immersed in the quiver-rep zone, introduce zigzag persistence. The first software package for computing persistence (Plex, by Adams, de Silva, Vejdemo-Johansson,...) materializes.
2011: Nicolau, Levine and Carlsson discover a new type of breast cancer using 0-dimensional persistence on an old, and purportedly well-mined, tumor dataset.
2012 Chazal, de Silva, Glisse and Oudot unleash this beastly reworking of the stablity theorem. Gone are various assumptions about tameness and sub-levelsets. They show that bottleneck distance between barcodes arises from a certain "interleaving distance" on the persistence modules. This opens the door for more algebraic and categorical interpretations of persistence, eg Bubenik-Scott.
2013: Mischaikow and I retool the simple homotopy-based reductions to work for filtered cell complexes, thus producing the first efficient preprocessor for the Zomorodian-Carlsson algorithm along with a fast (at the time!) software package Perseus.
2015 Lesnick publishes a comprehensive study of the interleaving distance in the context of multiparameter persistence modules.
2018 MacPherson and Patel concoct bisheaves to attack multi-parameter persistence geometrically for fibers of maps to triangulable manifolds.
Good luck with your course!
answered 8 hours ago
Vidit Nanda
9,01123898
Since this area is developing rather quickly, there is a dearth of canonical references that would satisfy basic pedagogical requirements. If I were teaching a course on this material right now, I would probably use Oudot's nice book if the students had sufficient background, and the foundational paper of Zomorodian-Carlsson if they did not.
I haven't read Jose's recent article mentioned in Joe's answer, but here is what I remember of the good old days (with apologies to all the important stuff that got missed).
1992: Frosini introduces "size functions", which we would today consider equivalent to 0-dimensional persistent homology.
1995: Mischaikow + Mrozek publish a computer-assisted proof of chaos in the Lorenz equations; a key step involves computing Conley indices, which are relative homology classes. This produces considerable interest in machine computation of homology groups of spaces from finite approximations (eg large cell complexes).
1999: Robbins publishes this paper emphasizing that functoriality helps approximate the homology of an underlying space from Cech complexes of finite samples; meanwhile Kaczynski, Mischaikow and Mrozek publish their book on efficient homology computation via simple homotopy type reductions of cell complexes.
2002: Edelsbrunner, Letscher and Zomorodian introduce persistence from a computational geometry viewpoint; as written, their algorithm works only for subcomplexes of spheres and only with mod-2 coefficients.
2005: This was a big year.
Zomorodian and Carlsson reinterpret persistence of a filtration via the representation theory of graded modules over graded pid's, thus giving an algorithm for all finite cell complexes over arbitrary field coefficients; they also introduce the barcode, which is a perfect combinatorial invariant of certain tame persistence modules.
Edelsbrunner, Cohen-Steiner and Harer show that the map $$text[functions X to R] to text[barcodes]$$ obtained by looking at sublevel set homology of nice functions on triangulable spaces is 1-Lipschitz when the codomain is endowed with a certain metric called the bottleneck distance. This is the first avatar of the celebrated stability theorem.
2007: de Silva and Ghrist use persistence to give a slick solution to the coverage problem for sensor networks.
2008: Niyogi, Smale and Weinberger publish a paper solving the homology inference problem for compact Riemannian submanifolds of Euclidean space from finite uniform samples. Carlsson, with Singh and Sexton, starts Ayasdi, putting his money where his math is.
2009: Carlsson and Zomorodian use quiver representation theory to point out that getting finite barcodes for multiparameter persistence modules is impossible, highlighting dimension 2 as the new frontier for theoretical work in the field.
2010: Carlsson and de Silva, by now fully immersed in the quiver-rep zone, introduce zigzag persistence. The first software package for computing persistence (Plex, by Adams, de Silva, Vejdemo-Johansson,...) materializes.
2011: Nicolau, Levine and Carlsson discover a new type of breast cancer using 0-dimensional persistence on an old, and purportedly well-mined, tumor dataset.
2012 Chazal, de Silva, Glisse and Oudot unleash this beastly reworking of the stablity theorem. Gone are various assumptions about tameness and sub-levelsets. They show that bottleneck distance between barcodes arises from a certain "interleaving distance" on the persistence modules. This opens the door for more algebraic and categorical interpretations of persistence, eg Bubenik-Scott.
2013: Mischaikow and I retool the simple homotopy-based reductions to work for filtered cell complexes, thus producing the first efficient preprocessor for the Zomorodian-Carlsson algorithm along with a fast (at the time!) software package Perseus.
2015 Lesnick publishes a comprehensive study of the interleaving distance in the context of multiparameter persistence modules.
2018 MacPherson and Patel concoct bisheaves to attack multi-parameter persistence geometrically for fibers of maps to triangulable manifolds.
Good luck with your course!
answered 8 hours ago
Vidit Nanda
9,01123898
edited 5 hours ago
answered 8 hours ago
Vidit Nanda
9,01123898
answered 8 hours ago
Vidit Nanda
9,01123898
answered 8 hours ago
Vidit Nanda
9,01123898
9,01123898
2
this is really great! the historical order and the mention of discovery of new breat cancer are really attracting! more than I could have wished for really!!!!
– user51223
6 hours ago
add a comment |Â
2
this is really great! the historical order and the mention of discovery of new breat cancer are really attracting! more than I could have wished for really!!!!
– user51223
6 hours ago
2
2
this is really great! the historical order and the mention of discovery of new breat cancer are really attracting! more than I could have wished for really!!!!
– user51223
6 hours ago
this is really great! the historical order and the mention of discovery of new breat cancer are really attracting! more than I could have wished for really!!!!
– user51223
6 hours ago
add a comment |Â
up vote
5
down vote
This paper was just released on the arXiv this morning:
"A Brief History of Persistence."
Jose A. Perea. 2018.
arXiv abstract.
"Persistent homology is currently one of the more widely known tools from
computational topology and topological data analysis. We present in this note a
brief survey on the evolution of the subject. The goal is to highlight the main
ideas, starting from the subject's computational inception more than 20 years
ago, to the more modern categorical and representation-theoretic point of view."
    Â
answered 22 hours ago
Joseph O'Rourke
83.2k15219683
thanks. I just saw it today!
– user51223
19 hours ago
add a comment |Â
up vote
5
down vote
This paper was just released on the arXiv this morning:
"A Brief History of Persistence."
Jose A. Perea. 2018.
arXiv abstract.
"Persistent homology is currently one of the more widely known tools from
computational topology and topological data analysis. We present in this note a
brief survey on the evolution of the subject. The goal is to highlight the main
ideas, starting from the subject's computational inception more than 20 years
ago, to the more modern categorical and representation-theoretic point of view."
    Â
answered 22 hours ago
Joseph O'Rourke
83.2k15219683
thanks. I just saw it today!
– user51223
19 hours ago
add a comment |Â
up vote
5
down vote
up vote
5
down vote
This paper was just released on the arXiv this morning:
"A Brief History of Persistence."
Jose A. Perea. 2018.
arXiv abstract.
"Persistent homology is currently one of the more widely known tools from
computational topology and topological data analysis. We present in this note a
brief survey on the evolution of the subject. The goal is to highlight the main
ideas, starting from the subject's computational inception more than 20 years
ago, to the more modern categorical and representation-theoretic point of view."
    Â
answered 22 hours ago
Joseph O'Rourke
83.2k15219683
This paper was just released on the arXiv this morning:
"A Brief History of Persistence."
Jose A. Perea. 2018.
arXiv abstract.
"Persistent homology is currently one of the more widely known tools from
computational topology and topological data analysis. We present in this note a
brief survey on the evolution of the subject. The goal is to highlight the main
ideas, starting from the subject's computational inception more than 20 years
ago, to the more modern categorical and representation-theoretic point of view."
    Â
answered 22 hours ago
Joseph O'Rourke
83.2k15219683
edited 15 hours ago
answered 22 hours ago
Joseph O'Rourke
83.2k15219683
answered 22 hours ago
Joseph O'Rourke
83.2k15219683
answered 22 hours ago
Joseph O'Rourke
83.2k15219683
83.2k15219683
thanks. I just saw it today!
– user51223
19 hours ago
add a comment |Â
thanks. I just saw it today!
– user51223
19 hours ago
thanks. I just saw it today!
– user51223
19 hours ago
thanks. I just saw it today!
– user51223
19 hours ago
add a comment |Â
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