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Description

Algebraic topology has been a key part of mathematics for at least a hundred years. Its main strength lies in its ability to detect shape of spaces. To provide qualitative description of spaces, for instance, how connected they are, various algebraic invariants have been developed, with simplicial cohomology being one of the most prominent.
In recent years there has been a very strong impetus in developing direct applications of algebraic topology, which resulted in the creation of a new sub-field of applied algebraic topology. The subject is young, is developing very fast and offers numerous opportunities for research problems. The recent preoccupation with Big Data is probably the most important motivation, and it has led to rapid creation of new techniques capable of dealing with finite discrete spaces of huge complexity and dimensionality.
This course will introduce the students to the key ideas of applied algebraic topology. Our main motivations will be applications to network theory and data analysis. This course is aimed at two kinds of students: pure mathematicians interested in applications, and applied mathematicians wishing to learn this new exciting theory straddling both disciplines. Additional reading will be suggested to support the course and to fill in any gaps in prerequisites.
I will provide regular problem sheets. A special feature of the course will be additional exercises in Matlab. I will not assume any prior experience with Matlab, and the course will not depend on Matlab expertise, but it will be a good idea for those interested in applications to follow one of many online Matlab courses to learn the basics. Specific recommendations will be given.
The course will introduce where appropriate specific examples of real-life data sets and networks; we will explain the problems, main difficulties, and solutions.

Semester

Spring 2017 (Monday, January 23 to Friday, March 31)

Timetable

  • Fri 12:05 - 12:55

Prerequisites

Basic topology and some algebraic topology, especially differential complexes and homology, cohomology. Some experience with Matlab would be useful, but guidance will be given to students who have not used this package before. An interest in crossing subject boundaries and a sense of intellectual adventure.

Syllabus

  1. Topological spaces and topological invariants
  2. Approximating topological spaces through simplicial complexes
  3. Persistent cohomology and Betti numbers
  4. Stability and periodic motion detection through persistent cohomology
  5. Feature detection with persistent cohomology: barcodes.
  6. Elementary ideas from simplicial Morse theory
  7. Applications of Morse theory: mapper and similar tools.
  8. Topology of sensor networks, coverage problems.
  9. Topological data analysis
  10. Shape classification and statistics

Lecturer


Jacek Brodzki
Email j.brodzki@soton.ac.uk
Phone (023) 80593648
Interests K-theory and Operator Algebras
vcard
Photo of Jacek Brodzki
Profile: I am fascinated by interactions between analysis and geometry. My current work revolves around problems in noncommutative geometry arising from the Baum-Connes conjecture, including characterisations of property A, exactness of groups and their consequences. An important part of my research is the study of geometry and topology of large data sets using the geometric, topological, and analytical techniques developed in the context of metric spaces and geometry of groups.


Students


Photo of Jack Aiston
Jack Aiston
(Newcastle)
Photo of Kiko Belchi
Kiko Belchi
(Southampton)
Photo of David Chillingworth
David Chillingworth
(Southampton)
Photo of Lorenzo De Biase
Lorenzo De Biase
(Cardiff)
Photo of Ai Guan
Ai Guan
(Lancaster)
Photo of Ewan Johnstone
Ewan Johnstone
(Liverpool)
Photo of Zoltan Kocsis
Zoltan Kocsis
(Manchester)
Photo of Rami Kraft
Rami Kraft
(Loughborough)
Photo of Jane Lyle
Jane Lyle
(Surrey)
Photo of Pedro Peres
Pedro Peres
(Exeter)
Photo of Gregory Roberts
Gregory Roberts
(Liverpool)
Photo of Lee Steinberg
Lee Steinberg
(Southampton)


Bibliography


Topology and dataCarlsson
Persistent homology---a surveyEdelsbrunner and Harer
Topological persistence and simplificationEdelsbrunner, Letscher and Zomorodian
Algebraic TopologyHatcher
Elementary Applied TopologyR. Ghrist
Homological sensor networksde Silva and Ghrist
Persistent Homology Transform for Modeling Shapes and SurfacesK. Turner, S. Mukherjee


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Assessment



The assessment for this course will be via a single take-home paper in April with 2 weeks to complete and submit online. There will be 4 questions and you will need 50% to pass.

End of semester exam for MAGIC087 Applied Algebraic Topology

Files:Exam paper
Released: Monday 24 April 2017 (63.4 days ago)
Deadline: Sunday 7 May 2017 (49.4 days ago)


Recorded Lectures


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