# C3.9 Computational Algebraic Topology - Material for the year 2019-2020

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Some familiarity with the main concepts from algebraic topology, homological algebra and category theory will be helpful.

16 lectures

### Assessment type:

- Mini Project

Ideas and tools from algebraic topology have become more and more important in computational and applied areas of mathematics. This course will provide at the masters level an introduction to the main concepts of (co)homology theory, and explore areas of applications in data analysis and in foundations of quantum mechanics and quantum information.

Students should gain a working knowledge of homology and cohomology of simplicial sets and sheaves, and improve their geometric intuition. Furthermore, they should gain an awareness of a variety of application in rather different, research active fields of applications with an emphasis on data analysis and contextuality.

The course has two parts. The first part will introduce students to the basic concepts and results of (co)homology, including sheaf cohomology. In the second part applied topics are introduced and explored.

*Core:* Homology and cohomology of chain complexes. Algorithmic computation of boundary maps (with a view of the classification theorm for finitely generated modules over a PID). Chain homotopy. Snake Lemma. Simplicial complexes. Other complexes (Delaunay, Cech). Mayer-Vietoris sequence. Poincare duality. Alexander duality. Acyclic carriers. Discrete Morse theory. (6 lectures)

*Topic A:* Persistent homology: barcodes and stability, applications todata analysis, generalisations. (4 lectures)

*Topic B:* Sheaf cohomology and applications to quantum non-locality and contextuality.Sheaf-theoretic representation of quantum non-locality and contextuality asobstructions to global sections. Cohomological characterizations and proofs of contextuality.(6 lectures)

H. Edelsbrunner and J.L. Harer, *Computational Topology - An Introduction*,AMS (2010).

See also, U. Tillmann, Lecture notes for CAT 2012, in http://people.maths.ox.ac.uk/tillmann/CAT.html

*Topic A:*

- G. Carlsson,
*Topology and data*, Bulletin A.M.S.46 (2009), 255-308. - H. Edelsbrunner, J.L. Harer,
*Persistent homology: A survey*, Contemporary Mathematics 452 A.M.S. (2008), 257-282. - S. Weinberger,
*What is ... Persistent Homology?*, Notices A.M.S. 58 (2011), 36-39. - P. Bubenik, J. Scott,
*Categorification of Persistent Homology*, Discrete Comput. Geom. (2014), 600-627.

*Topic B:*

- S. Abramsky and Adam Brandenburger, The Sheaf-Theoretic Structure Of Non-Locality and Contextuality. In
*New Journal of Physics*, 13(2011), 113036, 2011. - S. Abramsky and L. Hardy, Logical Bell Inequalities, {Phys. Rev. A} 85, 062114 (2012).
- S. Abramsky, S. Mansfield and R. Soares Barbosa, The Cohomology of Non-Locality and Contextuality, in
*Proceedings of Quantum Physics and Logic 2011*, Electronic Proceedings in Theoretical Computer Science, vol. 95, pages 1-15, 2012.