6 ms·

Elementary Applied Topology (2014)

215 points by adamnemecek 8y ago

romwell 8y ago
Note: all chapters available as free PDF. The book is not expensive either (it'll probably cost you more to print it yourself). Great text, great author. It's also a pleasure to see Ghrist give a talk.
peignoir 8y ago
I wished I could read / remember the math notations to understand it... anyone else in my case?
- efm 8y ago
  Ghrist's youtube channel might be a good place to start: https://www.youtube.com/channel/UC5N5pRddyicAX1QJyJjIIdg https://www.youtube.com/channel/UC5N5pRddyicAX1QJyJjIIdg
  - del_operator 8y ago
    Clicking this video I am fairly certain I’ve read a handwritten book by Ghrist on Applied Algebraic Topology and Sensor Networks. It’s short and fun
    - kqr2 8y ago
      https://www.math.upenn.edu/~ghrist/preprints/ATSN.pdf https://www.math.upenn.edu/~ghrist/preprints/ATSN.pdf At the bottom of the original link.
rwilson4 8y ago
I was skeptical about the “applied” part, but glancing through it, the examples are really interesting! Discussing magnetic fields in the same chapter as lambda calculus, very cool! Seems a bit light on details though...
- nikofeyn 8y ago
  why skeptical when he is a top applied mathematician? and he explains in the preface why it is seemingly light on details. it's because the details (the math) are elsewhere. the applications (as isolated entities) are also out there to a degree. what isn't out there is the bridge connecting the math to the applications, and so that is what his book sets out to do. > Discussing magnetic fields in the same chapter as lambda calculus where did you see this? i don't remember seeing it and couln't find it.
  - soVeryTired 8y ago
    > why skeptical when he is a top applied mathematician? I see a lot of mathematics in the book, but from skimming it, I can't see a 'killer application' as such. I'd like to see an example of a practical problem that can be solved using reasonably deep topolgical methods (beyond, say the Euler characteristic) that can't be solved in any other way. Ghrist was championing persistent homology for network analysis as such an application a few years ago, but I'm not sure if it ever progressed beyond 'toy' problems to give state-of-the-art results.
    - improbable22 8y ago
      From skimming, it looks to me like the examples are facing the other way: they're there to help you grasp the mathematics. Would also be curious to hear of practical applications of heavy-duty topology, if anyone can think of some.
      - nikofeyn 8y ago
        just look at his research papers.
    - ajudson 8y ago
      Here is a paper for you: "Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival" https://www.pnas.org/content/108/17/7265 https://www.pnas.org/content/108/17/7265
bigmit37 8y ago
Topology seems like a really cool field. Can I jump into this book without any topology knowledge? I have not taken differential equations yet and wonder is that is a prerequisite.
- myth_drannon 8y ago
  From Goodreads review: " This is not so much a math textbook as an extended infomercial for the modern field of topology, with applications and intuition stressed over formal results.As the author's intention appears to be to give a broad overview of the field and its potential uses which will encourage readers to pursue deeper study, ..." So looks like the answer is yes
- nikofeyn 8y ago
  you will understand very little of the math in this book without a substantial background in mathematics. however, and i think the author would agree, you should jump in anyway. you'll pick up pictures, applications, vocabulary, and some ideas, all of which could help pave the way as you learn the material more traditionally. the book sets goalposts so to speak, especially since it is written in a more expository manner with lots of example applications. i don't understand anywhere near everything in the book (much less than that probably), and i still get something from it. in terms of learning some topology and geometry, i recommend topology by james munkres and an introduction to manifolds by loring tu. both are accessible to junior and senior undergraduates and beginning graduate students.
  - throwawaymath 8y ago
    I strongly second the recommendation of Munkres' Topology. It's basically the gold standard. However, I think don't think it's accessible to junior/senior undergraduates unless they've taken a prior course in analysis. Technically Munkres' Topology has no formal prerequisites, but it's a rough go of it if you haven't had a rigorous exposure to continuity and metric spaces already. The second part of the book also assumes knowledge of elementary abstract algebra. If the person you're replying to is a junior or senior undergrad in a math major they might be alright (presumably they'll have taken analysis by then). If not, Topology is not what I'd recommend as the first place to learn about metrics and continuous functions (Calculus doesn't cut it for that).
31073 8y ago
This sounds like the title to a Community episode.
openloop 8y ago
This is good stuff. See also: https://en.m.wikipedia.org/wiki/Topological_data_analysis https://en.m.wikipedia.org/wiki/Topological_data_analysis
aggerdom 8y ago
This typesetting is fantastic. Love the style.
- enriquto 8y ago
  > This typesetting is fantastic. Love the style. I dislike this kind of typesetting, the sans-serif font is nearly unreadable, and the math displays are just slightly smaller than the regular text, with the same font. WHY? I would love to have access to the .tex source to compile it in a saner style. The drawings are excellent, though. I'm enjoying the ones on Morse theory.
  - jhanschoo 8y ago
    Not a fan either. It commits the same sin as the typical Computer Modern typically used in TeX but to a worse degree: far too light for easy reading. IIRC it's name is either Iwona or Kurier. For free fonts for typesetting in TeX, I would recommend one of the Times-based fonts. I personally typesetting typically in EB Garamond with a very new free Unicode math font called Garamond Math, but the setup is quite elaborate.
del_operator 8y ago
I’ve also been reading about persistence more in the Edelsbrunner and Harer Computational Toplogy text, but glancing at this book on page 104 I see good examples and intuitive build up. I found this overview I liked for persistence in ML https://arxiv.org/pdf/1811.00252.pdf https://arxiv.org/pdf/1811.00252.pdf
varlock 8y ago
Generalizing the topic, does anyone know if there is a map of subjects/branches in mathematics and books/blogs/etc giving examples of real-world applications? Personally, I know the (existence of the) maths behind some applications (e.g. JPEG -> DCT) because I have learned it while at uni, but in general, if I encounter a mathematical subject I struggle to find examples of its application.
valw 8y ago
I'd take the 'Elementary' qualifier with a grain of salt here :) - there are more elementary notions of topology (such as point-set topology, continuity, connectedness, compactness etc.) that can already get you pretty far in many practical situations. Thanks for posting this, added to my reading list!
- bschne 8y ago
  Do you have any pointers to good books / lecture videos / introductions to start with for these?

215 points by adamnemecek 8y ago

Note: all chapters available as free PDF. The book is not expensive either (it'll probably cost you more to print it yourself). Great text, great author. It's also a pleasure to see Ghrist give a talk.

peignoir 8y ago

I wished I could read / remember the math notations to understand it... anyone else in my case?

efm 8y ago
Ghrist's youtube channel might be a good place to start: https://www.youtube.com/channel/UC5N5pRddyicAX1QJyJjIIdg https://www.youtube.com/channel/UC5N5pRddyicAX1QJyJjIIdg
- del_operator 8y ago
  Clicking this video I am fairly certain I’ve read a handwritten book by Ghrist on Applied Algebraic Topology and Sensor Networks. It’s short and fun
  - kqr2 8y ago
    https://www.math.upenn.edu/~ghrist/preprints/ATSN.pdf https://www.math.upenn.edu/~ghrist/preprints/ATSN.pdf At the bottom of the original link.

rwilson4 8y ago

I was skeptical about the “applied” part, but glancing through it, the examples are really interesting! Discussing magnetic fields in the same chapter as lambda calculus, very cool! Seems a bit light on details though...

nikofeyn 8y ago
why skeptical when he is a top applied mathematician? and he explains in the preface why it is seemingly light on details. it's because the details (the math) are elsewhere. the applications (as isolated entities) are also out there to a degree. what isn't out there is the bridge connecting the math to the applications, and so that is what his book sets out to do. > Discussing magnetic fields in the same chapter as lambda calculus where did you see this? i don't remember seeing it and couln't find it.
- soVeryTired 8y ago
  > why skeptical when he is a top applied mathematician? I see a lot of mathematics in the book, but from skimming it, I can't see a 'killer application' as such. I'd like to see an example of a practical problem that can be solved using reasonably deep topolgical methods (beyond, say the Euler characteristic) that can't be solved in any other way. Ghrist was championing persistent homology for network analysis as such an application a few years ago, but I'm not sure if it ever progressed beyond 'toy' problems to give state-of-the-art results.
  - improbable22 8y ago
    From skimming, it looks to me like the examples are facing the other way: they're there to help you grasp the mathematics. Would also be curious to hear of practical applications of heavy-duty topology, if anyone can think of some.
    - nikofeyn 8y ago
      just look at his research papers.
  - ajudson 8y ago
    Here is a paper for you: "Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival" https://www.pnas.org/content/108/17/7265 https://www.pnas.org/content/108/17/7265

bigmit37 8y ago

Topology seems like a really cool field. Can I jump into this book without any topology knowledge? I have not taken differential equations yet and wonder is that is a prerequisite.

myth_drannon 8y ago
From Goodreads review: " This is not so much a math textbook as an extended infomercial for the modern field of topology, with applications and intuition stressed over formal results.As the author's intention appears to be to give a broad overview of the field and its potential uses which will encourage readers to pursue deeper study, ..." So looks like the answer is yes
nikofeyn 8y ago
you will understand very little of the math in this book without a substantial background in mathematics. however, and i think the author would agree, you should jump in anyway. you'll pick up pictures, applications, vocabulary, and some ideas, all of which could help pave the way as you learn the material more traditionally. the book sets goalposts so to speak, especially since it is written in a more expository manner with lots of example applications. i don't understand anywhere near everything in the book (much less than that probably), and i still get something from it. in terms of learning some topology and geometry, i recommend topology by james munkres and an introduction to manifolds by loring tu. both are accessible to junior and senior undergraduates and beginning graduate students.
- throwawaymath 8y ago
  I strongly second the recommendation of Munkres' Topology. It's basically the gold standard. However, I think don't think it's accessible to junior/senior undergraduates unless they've taken a prior course in analysis. Technically Munkres' Topology has no formal prerequisites, but it's a rough go of it if you haven't had a rigorous exposure to continuity and metric spaces already. The second part of the book also assumes knowledge of elementary abstract algebra. If the person you're replying to is a junior or senior undergrad in a math major they might be alright (presumably they'll have taken analysis by then). If not, Topology is not what I'd recommend as the first place to learn about metrics and continuous functions (Calculus doesn't cut it for that).

31073 8y ago

This sounds like the title to a Community episode.

openloop 8y ago

This is good stuff. See also: https://en.m.wikipedia.org/wiki/Topological_data_analysis https://en.m.wikipedia.org/wiki/Topological_data_analysis

aggerdom 8y ago

This typesetting is fantastic. Love the style.

enriquto 8y ago
> This typesetting is fantastic. Love the style. I dislike this kind of typesetting, the sans-serif font is nearly unreadable, and the math displays are just slightly smaller than the regular text, with the same font. WHY? I would love to have access to the .tex source to compile it in a saner style. The drawings are excellent, though. I'm enjoying the ones on Morse theory.
- jhanschoo 8y ago
  Not a fan either. It commits the same sin as the typical Computer Modern typically used in TeX but to a worse degree: far too light for easy reading. IIRC it's name is either Iwona or Kurier. For free fonts for typesetting in TeX, I would recommend one of the Times-based fonts. I personally typesetting typically in EB Garamond with a very new free Unicode math font called Garamond Math, but the setup is quite elaborate.

del_operator 8y ago

I’ve also been reading about persistence more in the Edelsbrunner and Harer Computational Toplogy text, but glancing at this book on page 104 I see good examples and intuitive build up. I found this overview I liked for persistence in ML https://arxiv.org/pdf/1811.00252.pdf https://arxiv.org/pdf/1811.00252.pdf

varlock 8y ago

Generalizing the topic, does anyone know if there is a map of subjects/branches in mathematics and books/blogs/etc giving examples of real-world applications? Personally, I know the (existence of the) maths behind some applications (e.g. JPEG -> DCT) because I have learned it while at uni, but in general, if I encounter a mathematical subject I struggle to find examples of its application.

valw 8y ago

I'd take the 'Elementary' qualifier with a grain of salt here :) - there are more elementary notions of topology (such as point-set topology, continuity, connectedness, compactness etc.) that can already get you pretty far in many practical situations. Thanks for posting this, added to my reading list!

bschne 8y ago
Do you have any pointers to good books / lecture videos / introductions to start with for these?