Algebraic topology vs differential topology

Algebraic and Differential Topology in Data Analysis (ADTDA) 

The course will cover some recent applications of topology and differential geometry in data analysis. Tools of differential and algebraic topology are starting to impact the area of data sciences, where the mathematical apparatus thus far was dominated by the ideas from statistical learning, computational linear algebra and high-dimensional normed space theories. While the research community around ADT topics in data analysis is lively and fast-growing, the area is somewhat sparsely represented in campus syllabi.

Syllabus

1. Tools from algebraic topology Homotopy equivalence, Simplicial homology, Nerve lemma, Dowker’s theorem
Applications​: Cech complexes and their topology in robotics and neurophysiology. Netflix problem complexes.

2. Topological Approximations Vietoris-Rips and Čech complexes. Topology reconstruction from random samples: Niyogi-Smale-Weinberger. Topology reconstruction from dense samples: Hausmann-Latscher. Sketches: Merge trees; Reeb graphs.
Applications​: Mapper.

3. Topological Inference Persistent Homology: Algorithms; Stability. Biparametric persistence
Applications​: Image patches spaces. Textures and characterization of materials

4. Euler calculus  Integration with respect to Euler characteristics. Topological Signal Processing. Valuations; Hadwiger’s Theorem. Average persistence for Gaussian fields.
Applications​: Topological Sensor Networks. Shape reconstruction through Euler transform.

5. Aggregation Spaces with averaging. Arrow theorem and Topological Social Choice. Aggregation in CAT(0) spaces
Applications​: Consensus in phylogenetic analysis. Political polarization

6. Clustering Basic clustering tools. Kleinberg’s Impossibility theorem. Carlsson-Memoli functorial approach to clustering. 

7. Tools from differential topology Useful topological spaces: manifolds, subanalytic sets, simplicial complexes. Transversality, Sard’s theorem. Whitney’s embedding theorem.
Applications​: Dimensionality reduction; Embeddings

The course will rely mainly on the recent papers, and a few textbooks, like

R Ghrist, Elementary Applied Topology, Createspace, 2014
A Hatcher, Algebraic Topology, CUP, 2002
V Prasolov, Elements of Combinatorial and Differential Topology, AMS, 2006

To receive credit the students will be expected

• to take (a fraction of) class notes (and produce a LaTeX source), (sign up here) and
• either present a paper (from a list), or to run a computational project (from a set of provided topics).

The class will be conducted remotely: the lectures will be held via zoom (one time registration required) at the planned times, – 9:30 on Tuesdays and Thursdays. The first lecture is on March 24.

While most of materials and links will be posted here, we will be also using moodle for course-specific announcements. If you auditing the course, let me know  so I can add you to the list of users.

Weekly updates:

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• Idea
• Examples
• Entries in differential topology
• Related entries
• References

Differential topology is the subject devoted to the study of topological properties of differentiable manifolds, smooth manifolds and related differential geometric spaces such as stratifolds, orbifolds and more generally differentiable stacks.

Differential topology is also concerned with the problem of finding out which topological (or PL) manifolds allow a differentiable structure and the degree of nonuniqueness of that structure if they do (e.g. exotic smooth structures). It is also concerned with concrete constructions of (co)homology classes (e.g. characteristic classes) for differentiable manifolds and of differential refinements of cohomology theories.

Examples

Many considerations, and classification problems, depend crucially on dimension, and the case of high-dimensional manifolds (the notion of ‘high’ depends on the problem) is often very different from the situation in each of the low dimensions; thus there are specialists’ subjects like 33-(dimensional) topology and 44-topology. There are restrictions on an underlying topology which is allowed for some sorts of additional structures on a differentiable manifold.

For example, only some even-dimensional differentiable manifolds allow for symplectic structure and only some odd-dimensional one allow for a contact structure; in these cases moreover special constructions of topological invariants like Floer homology and symplectic field theory exist.

This yields the relatively young subjects of symplectic and contact topologies, with the first significant results coming from Gromov. Any (Hausdorff paracompact finite-dimensional) differentiable manifold allows for riemannian structure however; therefore there is no special subject of ‘riemannian topology’.

Entries in differential topology

References

Though some of the basic results, methods and conjectures of differential topology go back to Poincaré, Whitney, Morse and Pontrjagin, it became an independent field only in the late 1950s and early 1960s with the seminal works of Smale, Thom, Milnor and Hirsch. Soon after the initial effort on foundations, mainly in the American school, a strong activity started in Soviet Union (Albert Schwarz, A. S. Mishchenko, S. Novikov, V. A. Rokhlin, M. Gromov…).

Introductions and monographs:

• John Milnor, Differential topology, chapter 6 in T. L. Saaty (ed.) Lectures On Modern Mathematic II 1964 (pdf)

• John Milnor, Lectures on the h-cobordism theorem, 1965 (pdf)

• James R. Munkres, Elementary Differential Topology, Annals of Mathematics Studies 54 (1966), Princeton University Press (doi:10.1515/9781400882656).

• Andrew H. Wallace, Differential topology: first steps, Benjamin 1968.

• Victor Guillemin, Alan Pollack, Differential topology, Prentice-Hall 1974

• Morris Hirsch, Differential topology, Springer GTM 33 (1976) (doi:10.1007/978-1-4684-9449-5, gBooks)

• T. Bröcker, K. Jänich, C. B. Thomas, M. J. Thomas, Introduction to differentiable topology, 1982 (translated from German 1973 edition; ∃\exists also 1990 German 2nd edition)

• Raoul Bott, Loring Tu, Differential Forms in Algebraic Topology, Graduate Texts in Math. 82, Springer 1982. xiv+331 pp.

• John Milnor, Topology from the differential viewpoint, Princeton University Press, 1997. (ISBN:9780691048338, pdf)

• Mladen Bestvina (notes by Adam Keenan), Differentiable Topology and Geometry, 2002 (pdf, pdf)

• C. T. C. Wall, Differential topology, Cambridge Studies in Advanced Mathematics 154, 2016

• Joel W. Robbin, Dietmar Salamon, Introduction to differential topology, 294 pp, webdraft 2018 pdf

• Riccardo Benedetti, Lectures on Differential Topology, Graduate Studies in Mathematics 218, AMS 2021 (arXiv:1907.10297, ISBN: 978-1-4704-6674-9)

Survey with connections to algebraic topology:

See also

• Wikipedia, Differential topology

Generalization to equivariant differential topology:

I would say, it depends on how much Differential Topology you are interested in. Generally speaking, Differential Topology makes use of Algebraic Topology at various places, but there are also books like Hirsch' that introduce Differential Topology without (almost) any references to Algebraic Topology. Having said that, topological theory built on differential forms needs background/experience in Algebraic Topology (and some Homological Algebra). In other words, for a proper study of Differential Topology, Algebraic Topology is a prerequisite.

Addendum (book recommendations):

1) For a general introduction to Geometry and Topology:

• Bredon "Topology and Geometry": I can wholeheartedly recommend it! First part covers all the necessary and important general topology, then moves on to Differentiable Manifolds, after which it goes to Algebraic Topology (fundamental groups, (co)homology,homotopy theory). I think it is a good first book because it is self-contained, has exercises and gives a taste of different basic parts of modern geometry and topology without leaving the impression that they are isolated from each other.

2) For Algebraic Topology:

• Hatcher "Algebraic Topology": personally, I find his book overrated and quite annoying. It made me hate algebraic topology in my undergraduate years! YMMV.

• Spanier "Algebraic Topology": probably not a good first book, a little outdated, but much less annoying than Hatcher's. There is a lot to learn from it!

• Switzer "Algebraic Topology": great second book for Algebraic Topology, covering various topics of modern topology.

• Rotman "Introduction to Algebraic Topology": good introduction to very basic Algebraic Topology.

• tom Dieck "Algebraic Topology": good introduction to Algebraic Topology, but it contains several typos, so it's probably not so great for beginners. On the bright side, that keeps you on your toes to make sure you are paying attention :P It covers a good amount of topics too.

• Greenberg and Harper "Algebraic Topology... a first course": "reader-friendly" introduction to basic Algebraic Topology, but some prior experience with the language of category might be helpful.

• Davis and Kirk "Lecture notes on algebraic topology": since they are lecture notes, the material is "compressed" without filling in between (which is nice in my opinion) and covers various important topics of Algebraic Topology. The notes can be downloaded from their homepage .

• May "A concise course in algebraic topology": in my opinion, not suitable for readers without prior experience with Algebraic Topology (unless they are very gifted students).

3) For Algebraic Topology with homotopical focus:

• Selick "Introduction To Homotopy Theory": it is suited for readers who already have some experience with the basic concepts of Algebraic Topology, Category theory and Homological Algebra. Although the first part deals with all these prerequisites, the material would get dense for readers without prior exposure to them.

• Gray "Homotopy Theory - An Introducton to Algebraic Topology": nice self-contained introductory exposition with exercises, suitable for beginners in Algebraic Topology (knowledge of general point-set topology is assumed)

• Fomenko, Fuchs "Homotopic Topology" - it is one of those books where you have to work your way through it (i.e. lots of stuff left to the reader as exercises).

• Aguilar, Gitler, Prieto "Algebraic Topology from a Homotopical Viewpoint": only requires solid understanding of basic general topology. Prior experience with Algebraic Topology and Category Theory might be helpful, but not necessary.

• Strom "Modern Classical Homotopy Theory": little gem that only assumes solid understanding of basic general topology. All the theory is presented as mini-problems to work through. What better way to learn something than to work it out on your own. Thus some experience with Algebraic Topology might be helpful, but not strictly necessary. And it is self-contained in the sense that it takes care of the necessary category theory.

• Whitehead "Elements of homotopy theory": requires a first course in algebraic topology.

• Warner "Topics in Topology and Homotopy Theory": not suitable for beginners, it is more of an encyclopedia (mere 900+ pages).

4) For Differential Topology:

• Hirsch' "Differential Topology": self-contained, in particualar requires no prior knowledge of Algebraic Topology;

• Milnor's "Topology from Differentiable Viewpoint": self-contained, in particular requires no prior knowledge of Algebraic Topology;

• Milnor's "Morse Theory": the classic book on Morse theory;

• Guillemin and Pollack's "Differential Topology": self-contained, the last chapter introduces some cohomology theory, thus it does not omit this important tool.

5) For Differential Algebraic Topology:

• Milnor and Stasheff's "Characteristic Classes": A first course in smooth manifolds might be helpful, but not necessary.

• Bott and Tu's "Differential Forms in Algebraic Topology": another classic text, the title is self-explanatory;

• Kreck's "Differential algebraic topology - from stratifolds to exotic spheres": advanced text.