Applied Algebraic Topology 7

 

23rd September 2016

Durham University

                                                   

 

10:30 – 11:00

CM 211

Coffee in Mathematical Sciences common room

11:00 – 11:45

CM 103

Ruben Sanchez-Garcia (University of Southampton)

Title: Geometry and Topology of Networks and Data (slides)

Abstract: This will be an overview talk on past and current work in applied graph theory and topology. I will describe previous work on network symmetry, and spectral clustering, ways of generalising network methods to higher dimensions via Laplacian operators, and an application to ranking in horse racing data.

11:45 – 14:15

Break for Lunch and Discussions

14:15 – 15:00

CM 103

Mark Grant (University of Aberdeen)

Title: The Poincaré-Hopf Theorem for line fields (revisited) (slides)

Abstract: A line field on a manifold is a section of the projectivized tangent bundle. These objects find applications in soft matter physics, where they may be used to model ordered media made up of rod-shaped molecules, such as nematic liquid crystals.

In this talk I will present an analogue of the Poincaré-Hopf theorem for line fields. For a line field with finitely many isolated topological defects on a closed manifold, this relates the sum of the local indices of the defects with the Euler characteristic of the manifold. A version of this result for orientable surfaces was known to H. Hopf. Our result extends Hopf’s result to higher dimensions, and corrects a result of L. Markus (published in the Annals of Mathematics in 1955).

 

This is joint work with D. Crowley.

15:15 – 16:00

CM 103

Yumi Boote (University of Manchester)

Title: Configuration spaces, the octonionic projective plane, and potential applications (slides)

Abstract:  For many years configuration spaces have attracted the attention of mathematicians, physicists, engineers, and scientists in other disciplines. For example, the elements of a configuration space of a topological space X may be interpreted as collections of data/particles.  In this talk, I shall focus on the space C_2(X) of unordered pairs of distinct particles on a cohomologically well-behaved X, and summarise my results on the geometry and integral cohomology rings of C_2(X) and two of its compactifications.  As an example, I shall discuss the case of the octonionic projective plane OP^2; it has been suggested by Hisham Sati that OP^2 could play a role in the M-theory of theoretical physics.

 

List of participants:

 

1.       Dirk Schuetz (Durham)

2.       Patrick Orson (Durham)

3.       Yumi Boote (Manchester)

4.       Vitaliy Kurlin (Liverpool)

5.       Mark Grant (Aberdeen)

6.       David Recio-Mitter (Aberdeen)

7.       Norbert Peyerimhoff (Durham)

8.       Gareth Williams (Open University)

9.       Daniel Ballesteros-Chavez (Durham)

10.   Mel Chen (Glasgow)

11.   Shiping Liu (Durham)

12.   Andrew Lobb (Durham)

13.   Ruben Sanchez-Garcia (Southampton)