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Topological Data Analysis, CPSC 531F Page, Spring 2025: References and Resources
This page gives some references and resources
for TDA (Topological Data Analysis), for
CPSC 531F, Winter 2024-25, Term 2.
Materials below may have errors!
Errors will be corrected either here or in class.
If you are not attending my classes: use these materials at your
own risk!
This entire webpage is currently (January 2025) UNDER CONSTRUCTION.
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References |
UNDER CONSTRUCTION!!
We may use a number of references.
I believe that all these references are free of charge, at least if
you have a UBC library card or a CWL with library privileges.
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Linear Algebra and Applications:
I may refer to my older topics course, namely my
my Spring 2021 course on Applications of Linear
Algebra, especially
the reference page for this course,
especially the textbook
Matrix Analysis, by Horn and Johnson, (1985 edition, online
from the UBC Library)
which covers most of standard linear algebraic needed.
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Algebraic Topology and Differential Geometry:
There are a lot of excellent references for algebraic topology and
differential geometry available either free of charge, or at least
free to download with UBC library privileges
(you'll find dozens of such textbooks at
UBC library main webpage,
by typing "Algebraic Topology," and when you get the result
clicking "Full Text Online" and "Book/eBook"...)
Most textbooks on algebraic topology begin by discussing the
fundamental group, because it is easier to define and as important
as (co)homology groups. Below are a few that I'll
refer to this term.
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Singular Homology Theory, by W.S. Massey.
This textbook begins with cubical singular homology, which has
numerous pedagogical advantages over simplicial homology (singular
and regular). For example, since
the product a cube and an interval is again a cube,
the proof that two homotopic spaces have the same homology
groups is cleaner
(otherwise you have to triangulate the product of a simplex and an
interval).
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Algebraic Topology, by
Allen Hatcher
(see also
Hatcher's webpage on this textbook).
This textbook has many examples. Chapter 2, Section 1 (is independent
of Chapter 1 on homotopy groups and) gives you all the basics:
Δ-complexes, simplicial
complexes, simplicial homology, and its agreement with
singular homology (the usual, triangular version).
(You will need to triangulate the product of two simplicies,
but this is not too annoying.
On the bright side, the suspension of a
Δ-complex or a simplicial complex is immediately another one.)
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Differential Forms in Algebraic Topology, by Bott and Tu.
The easiest (co)homology theory to construct
comes from differential forms, and differential forms
have many applications
in physics.
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Although we may not use this, I admire the approach of
Fundamentals of Algebraic Topology, by Weintraub.
This textbook discusses homology by starting from the
Eilenberg-Steenrod axioms (Chapter 3), and then figures things out
from there.
(The point is that no one directly uses singular homology to compute
homology groups of topological spaces, except in a few simple
cases. Why not stress this from the start?)
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Algebra:
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List another reference here:
describe this reference.
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List another reference here:
describe this reference.
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TDA Resources |
Here are some resources and articles on TDA.
Most of these have been pointed out to me by others, including Tong Ling,
the participants of a
January 2023 Banff Workshop on Applied Hodge Theory), and others
(whom I'm happy to acknowledge if they wish).
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Expository Articles:
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Older Research Articles:
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Topological Persistence and Simplification, by Edelsbrunner, Letscher,
and Zomorodian, a 2002 article which, I believe, is where the idea
of "persistent homology" was originated;
here is the
preliminary FOCS 2000 version.
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Persistent
Barcodes for Shapes, by Carlsson, Zomorodian, Collins, and Guibas
is the 2004 article that, I believe, first proved the "Barcode
Theorem" as a special case of (graded) modules over a PID.
Algorithms were discussed in
Computing Persistent Homology, by Zomorodian and Carlsson.
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Recent Research Articles:
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Give a recent research article here.
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Books:
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Data Sets:
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AATRN:
The Applied Algebraic Topology Research
Network offers free membership, has various seminars,
and their very own
YouTube channel.
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