Synopsis for Computational Algebraic Topology

Prerequisites:

The course will provide a self-contained, rapid introduction to (co)homology theory for simplicial sets. However, some familiarity with concepts from topology and homological algebra will be of help. It should be noted that MFoCS students may offer both this course and the Section A course ‘Algebraic Topology’ in Michaelmas Term for examination. Those offering only this course and with no familiarity with these topics would be well advised to at least sit in on the Section A course ‘Algebraic Topology’.

Overview:

Ideas and tools from algebraic topology have become more and more important in computational and applied areas of mathematics. This course will provide an introduction to the main concepts from algebraic topology, and explore areas of applications in data analysis and computing at the graduate level.

Learning outcomes:

Students should gain a working knowledge of homology and coho- mology of simplicial sets and improve their geometric intuition. Furthermore, they should gain an awareness of a variety of application in rather different, research active fields of applications.

Lecturers:

Prof Ulrike Tillmann (1-10), Prof Samson Abramsky (11-16)

Synopsis:

The course has two parts. The first will introduce students to the basic concepts and results from algebraic topology. In the second part applied topics are introduced and explored.
Core: Homology and cohomology (with Z/2Z coefficients) of simplicial sets; simplicial manifolds and Poincar´e duality (4 lectures); Alexander duality; simplicial Morse theory. (3 lectures)
Topic A: Persistent homology, barcodes and stability, applications to data analysis. (3 lectures)
Topic B: Applications to distributed computing. (3 lectures)
Topic C: Sheaf cohomology and applications to network flows. (3 lectures)

Learning support:

There will be four problem sheets and classes covering the core material. To support the topics lectures, students may be advised to read more broadly or work on additional problems set by the lecturers.

H. Edelsbrunner and J.L. Harer, Computational Topology - An Introduction, AMS (2010).
T. Kaczynski, K. Mischaikow, M. Mrozek, Computational Homology, Springer (2004).
See also, U. Tillmann, Lecture notes for CAT, 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.

Topic B:
M. Herlihy, S. Rajsbaum, Algebraic topology and distributed computing a primer, Computer science today (1995), 203–217.
M. Herlihy, S. Rajsbaum, Algebraic spans, Mathematical Structures in Computer Science 10 (2000), 549 – 573.

Topic C:
R. Ghrist, V. de Silva, Homological sensor networks, Notic. Amer. Math. Soc 54 (2007), 10–17.
R. Ghrist, Y. Hiraoka, Applications of sheaf cohomology and exact sequences to network coding, preprint 2011.
R. Ghrist, A. Muhammad, Coverage and hole-detection in sensor networks via homology, in Proceedings of the 4th international symposium on Information processing in sensor networks (2005), IEEE Press, 34–es.