Elementary Applied Topology (2014) | Hacker News Reader

Elementary Applied Topology (2014)

math.upenn.edu

215 pointsadamnemecek40 comments

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

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...

I am also interested in where the lambda calculus is referenced, I do not see it in the index.

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.

> 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.

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

This paper is from 2010, and as a sibling poster points out, it's just a clustering algorithm. I think you need to consider the 'dog that didn't bark' here: if the method turned out to be useful, where are the follow-up papers and the industrial applications?

Thanks. I skimmed the linked "Mapper" article they cite as their method, and it looks about as topological as t-SNE, i.e. in the sense of caring about local nearness but not global distance.

But did I miss any heavier stuff? Is this all people mean when they talk about topological data analysis?

The other famous TDA technique is persistent homology, this paper is a good intro https://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf

But now: persistent homology. Same question as above.

Progress has been slower than expected, in some ways. But I think that persistent homology and topological methods like Mapper are slowly allowing interesting research.

My favorite recent application: two-parameter persistent homology for drug discovery (link to pdf of talk: https://www.ima.umn.edu/materials/2017-2018/SW8.13-15.18/274...). Frankly I find two-parameter persistence very hard to interpret, though I hope to spend some time on it this summer. But it's undeniable that the work described in the link is an application that can't be reduced to clustering.

I also feel that describing Mapper as 'just a clustering tool' is not really accurate; I'm getting good results from using Mapper for feature discovery in some specific domains where clustering has truly failed because traits lie on continua in a way that renders usual clustering muddy and useless.

Edited to add: here's another interesting paper, about subgroups of type 2 diabetes patients: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4780757/ They use Mapper to find 'clusters' and then go back to more traditional statistics to test significance. I don't have the data so I don't know if clustering alone would have worked.

I'm currently doing a thesis using TDA. Excuse my shameless opportunism, but the topic I've come up with is using figure h [1] as a channel selection mechanism to select EEG channels [2]. I plan to then use a Gaussian process (similar to an SVM) to detect the time when a seizure occurs.

From your position of expertise, can you see any major problems off the top of your head?

[1] https://media.springernature.com/lw785/springer-static/image... - from the open-access paper https://link.springer.com/article/10.1186/s13104-018-3482-7

[2] quote: "Step III... we applied the Vietoris–Rips filtration to determine which particular sensors (thus, which areas of the brain) are more “involved”∖“significant” concerning the spreading of epileptic seizures"

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.

just look at his research papers.

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

Ghrist's youtube channel might be a good place to start: https://www.youtube.com/channel/UC5N5pRddyicAX1QJyJjIIdg

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

https://www.math.upenn.edu/~ghrist/preprints/ATSN.pdf

At the bottom of the original link.

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.

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.

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.

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).

Thank you. I am actually a self- learner and have only take calculus, linear algebra, probability, statistics. Currently learning bayesian statistics and reading “first course in abstract algebra” by John B Fraleigh.

I am guessing I should do Real Analysis and then the Munkres book to get an idea of topology ?

Right now, I'm self-studying analysis with Stephen Abbott's "Understanding Analysis". I strongly recommend it. It has clear explanations that help to develop the intuition about each concept and what I think are well chosen exercises.

there is a book by jay cummings called real analysis: a long-form mathematics textbook that i highly recommend checking out. used with a more "advanced" book, it could be a nice complement.

Real Analysis: A Long-Form Mathematics Textbook https://www.amazon.com/dp/1724510126/

I personally really enjoy Gamelin and Green's Topology, and (if you need it) the first chapter goes through topology of metric spaces. The rest of the book is well written; I couldn't quite find myself to like Munkres.

agreed. i had it in mind the person was a math major. although i think topology is approachable without analysis with some dedication, it certainly helps provide context for the abstraction.

i listed another nice book in another comment, but metric spaces: iteration and application by victor bryant is a pretty fun introduction to metric spaces and analysis at a very approachable level.

I know Munkres is the go-to standard for topology, but I think that Essential Topology by Martin D. Crossley is a better book. It's only 200 pages, packed full of great examples, and covers all the (as the title says) essential topics in undergrad topology (topological spaces, continuity, connectivity, compactness, Hausdorff property, homeomorphisms, homotopy, Euler number, homotopy groups, simplicial homology, etc.)

+1 on munkres. Also recommend Algebraic Topology by Hatcher as a followup (free on his website) and Introduction to Smooth Manifolds by Lee.

Here is the first two sentences of the first chapter, first section.

> A topological n-manifold is a space M locally homeomorphic to R^n. That is, there is a cover U = {U_{alpha}} of M by open sets along with maps \phi_{\alpha} : U_{\alpha} -> R^n that are continuous bijections onto their images with continuous inverses.

The book has an appendix, labeled 'background' which lays out more precisely the formal prerequisites from point-set topology and algebra which you should review. The algebra is surely going to be a bigger obstacle than differential equations.

for an expository book like this, i suggest having the pdf on one screen, and google/wikipedia on another.

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

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!

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

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.

This typesetting is fantastic. Love the style.

> 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.

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.

This sounds like the title to a Community episode.

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