
Elementary Applied Topology (2014) - adamnemecek
https://www.math.upenn.edu/~ghrist/notes.html
======
del_operator
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)

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rwilson4
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...

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nikofeyn
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
> 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.

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ajudson
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)

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improbable22
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?

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

~~~
soVeryTired
But now: persistent homology. Same question as above.

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kaitai
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...](https://www.ima.umn.edu/materials/2017-2018/SW8.13-15.18/27458/Bryn-
Keller_Two-Parameter-Persistence-for-Virtual-Ligand-Screening.pdf)). 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/](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.

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llamaz
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...](https://media.springernature.com/lw785/springer-
static/image/art%3A10.1186%2Fs13104-018-3482-7/MediaObjects/13104_2018_3482_Fig3_HTML.png)
\- from the open-access paper
[https://link.springer.com/article/10.1186/s13104-018-3482-7](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"

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

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efm
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
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

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kqr2
[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.

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romwell
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.

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bigmit37
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.

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nikofeyn
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
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).

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bigmit37
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 ?

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magoghm
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.

~~~
nikofeyn
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/](https://www.amazon.com/dp/1724510126/)

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valw
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!

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

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varlock
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.

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aggerdom
This typesetting is fantastic. Love the style.

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enriquto
> 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
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.

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31073
This sounds like the title to a Community episode.

