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Results tagged with at.algebraic-topology
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user 83274
Homotopy theory, homological algebra, algebraic treatments of manifolds.
35
votes
1
answer
4k
views
Why is persistent cohomology so much faster than persistent homology
I refer to this paper: de Silva, Vin; Morozov, Dmitriy; Vejdemo-Johansson, Mikael. Dualities in persistent (co)homology. Inverse Problems 27 (2011), no. 12, 124003, 17 pp. (Journal link, arXiv link).
…
- 1,229
24
votes
2
answers
2k
views
Research directions in persistent homology
I am interested in what are the possible directions for new research in persistent homology (more of the mathematical theoretical aspects rather than the computer algorithm aspects).
So far from goog …
10
votes
1
answer
699
views
Persistent homology over the integers
Is it likely that in the future, there will be interest in computing persistent homology over the integers (or other PIDs)?
Currently, persistent homology is usually done over a field (like $\mathbb{ …
- 1,229
7
votes
1
answer
422
views
Correspondence between persistence module and graded module over $R[t]$
In the paper "Computing Persistent Homology" by Zomorodian and Carlsson, it is stated as Theorem 3.1 that:
The correspondence $\alpha$ defines an equivalence of categories between the category of …
- 1,229
6
votes
0
answers
232
views
Explicit formula for higher order Bockstein
The formula for the Bockstein $\beta:H_n(X;\mathbb{Z}/p\mathbb{Z})\to H_{n-1}(X;\mathbb{Z}/p\mathbb{Z})$ is $$\beta[c\otimes 1]=[\frac{1}{p}\partial c\otimes 1]$$
(McCleary page 456)
How about for th …
- 1,229
5
votes
2
answers
356
views
Is there an upper bound on the number of points in point cloud for which we compute the pers...
I am interested in the topic of persistent homology (topological data analysis). According to what I read, there is some roadblock in the analysis of "big data" using persistent homology as it is cons …
- 1,229
5
votes
0
answers
522
views
Strong Convergence vs Conditional Convergence for Spectral Sequences (Is there a simple expl...
I am curious if there is a relatively simple explanation of what is the difference between strong convergence and conditional convergence for Spectral Sequences?
(Hopefully a simpler explanation than …
- 1,229
4
votes
1
answer
194
views
Can Bockstein Spectral Sequence detect multiple summands of the same power, in homology?
I understand that the differential $d^k$ of the Bockstein S.S. (mod p) is nonzero iff the homology $H_*(X;\mathbb{Z})$ has summand of the form $\mathbb{Z}/p^k$.
How about for multiple summands in the …
- 1,229
4
votes
1
answer
574
views
Algorithm for computing fundamental group of simplicial complexes
For computing homology of a simplicial complex, there is the well-known reduction algorithm.
How about for fundamental group of simplicial complexes? Is there any (implementable) algorithm to compute …
- 1,229
3
votes
2
answers
339
views
Good, detailed references for "mod p lower central series"
I am looking for good, detailed references for "mod $p$ lower central series".
So far I only find papers such as (https://core.ac.uk/download/pdf/81193793.pdf, https://www.sciencedirect.com/science/a …
- 1,229
2
votes
1
answer
132
views
Persistent homology stability results (query about Lipschitz functions)
One of the beneficial properties of persistent homology is its stability results (so called robustness to noise).
Usually the referenced paper is this paper
titled "Lipschitz functions have $L_p$-st …
- 1,229