An introduction to persistent homology

[@Abacuscalcus]

Helpful intro, nice to see where this is being discussed in academia.

[@aatrn1]

Glad it was helpful!

[@dyllanusher1379]

Thanks for the great video!

[@aatrn1]

Thanks Dyllan, glad you liked it!

[@nksupreme1]

thanks for the talk. What software do you use to create the example about the sublevel filtration at 10:55?

[@HenryAdamsMath]

Hi NK, my former graduate student created that example clip. If you shoot me an email asking this same question, I can email it to the former student, in case they have any pointers for you!

[@nksupreme1]

@@HenryAdamsMath Alright, I will. Thanks!

[@nrk260717]

I am most interested in the part you skipped that talked about one, two and three dimensional betti numbers and how to calculate it ? But overall it was informative. Thank You for the talk. :)

[@anex004]

Here is a good intro for the first part: kzbin.info/www/bejne/nmHFn3pnjMapl8k

[@dyllanusher1379]

Hi Nitesh, en.wikipedia.org/wiki/Betti_number

[@matteopascale7187]

See Computing Persistent Homology by Zomorodian and Carlsson

[@andreasbeschorner1215]

Greetings. Any concepts for working with non metrizable topological spaces -- there are quite a lot of non pathological and non trivial ones and I imagine that, when interpreting data as topologies other than forming simplices, you might rather soon run into those.

[@HenryAdamsMath]

HI Andreas, thank you for the interesting question! To do persistent homology you need a filtration of a space, and many filtrations (such as the Cech or Vietoris-Rips filtrations) arise from a metric. But instead of growing balls according to a metric, you could replace the Cech complex complex with the nerve of any ordered collection of neighborhoods around points in your topological space. So perhaps this is of interest to you. I should also say that sublevelset persistent homology of a real-valued function f : X -> R does not require the domain space X to be metrizable. Folks like Vanessa Robins, me and my collaborators at Colorado State, and Benjamin Schweinhart and Robert MacPherson have studied how persistent homology relates to fractal dimension. So that involves wilder spaces, but we still typically have a metric. Folks in shape theory study wild spaces like subsets of R^n that have nonzero homology in dimensions >= n, but I'm not sure if their spaces are typically metrizable or not. The Vietoris-Rips complex of any finite metric space is itself metrizable. But in the limit, as you sample more and more points, the Vietoris-Rips complex of an infinite metric space need not be metrizable. Indeed, any simplicial complex which is not locally finite (some vertex is in an infinite number of simplices) is not metrizable. We use this as partial motiviation for introducing the Vietoris-Rips metric thickenings, which are always metric spaces, in www.math.colostate.edu/~adams/research/MetricReconstructionViaOptimalTransport.pdf.

[@andreasbeschorner1215]

@@HenryAdamsMath Hi Henry, thanks for the quick reply. Let me know if I understand it correctly: Instead of the sublevel approach you suggest using things as nerves or fibre products based on intersections with some subset for instance instead? And instead of "raising the level" I could for instance a) enlarge the subset for the intersection or b) consider a product of more fibres?

[@HenryAdamsMath]

@@andreasbeschorner1215 Let X be a set of points (vertices) in a metric space Y. Often we take X=Y, but not necessarily. For each x in X, let B(x;r) be the ball in Y of radius r about center point x. Then the Cech complex C(X,Y;r) is the nerve simplicial complex of these balls B(x,r) as x varies over all points in X. The vertex set of C(X,Y:r) is X, and this simplicial complex gets larger and larger as we increase r. Let's say now that you don't want Y to be a metric space. Let X be a set of points (vertices) in a topological space Y. You could take X=Y, but this is not necessary. Suppose that for each x in X, we have an increasing sequence of (perhaps open) neighborhoods N_1(x) \subset N_2(x) \subset N_3(x) \subset ... in Y that each contain the point x. Then for each integer i, one could consider the nerve simplicial complex N(X,Y;i) that is the nerve of the neighborhoods N_i(x) as x varies over all points in X. The vertex set of N(X,Y;i) is X, and this simplicial complex gets larger and larger as we increase i. Of course, there is no need to restrict i to vary over a discrete set. I didn't understand your comment about fiber products or products of more fibers --- I'm sure I'm misinterpreting what you have in mind!

[@NoNTr1v1aL]

Please recommend some good books on this topic at the graduate level.

[@aatrn1]

Certainly! Please see the list www.math.colostate.edu/~adams/advising/appliedTopologyBooks/ Let us know if you have more suggested books to add to this list!

[@NoNTr1v1aL]

@@aatrn1 Thank you for inspiring me!

[@vjysri2756]

Wow....is there a software or code I can use ?

[@HenryAdamsMath]

Hi Vjy, Yes, by now there is lots of good persistent homology code. See for example www.math.colostate.edu/~adams/advising/appliedTopologySoftware/ for a (incomplete) list of potential options!

[@ninnyhammerable]

go rams