The Poincare-Hopf Theorem for line fields revisited Seminar
- Time:
- 14:00 - 15:00
- Date:
- 30 January 2017
- Venue:
- Room 1037, Building 58, Social Sciences Building, University of Southampton, Highfield, Southampton, SO17 1BJ
For more information regarding this seminar, please telephone Dr Stephen Theriault on 023 8059 2141 or email S.D.Theriault@southampton.ac.uk .
Event details
A line field is a smooth assignment of a tangent line to each point of a manifold (and therefore may be thought of as the projective analogue of a vector field). Such objects find applications in soft matter physics, where they are used to model ordered media made up of rod-shaped molecules, such as nematic liquid crystals. I will present an analogue of the Poincaré-Hopf Theorem for line fields with point singularities. For orientable surfaces, such a result appears in Hopf's 1956 lecture notes on Differential Geometry. In 1955 Markus presented such a result in all dimensions, but his statement is only valid in even dimensions greater than 2. In 1984 Jänich presented a Poincaré-Hopf Theorem for line fields with more complicated singularities, focussing on the complexities arising in the generalised setting. Our proof is a correction of Markus' proof, and is valid in all dimensions. This is joint work with Diarmuid Crowley.
Speaker information
Dr Mark Grant , University of Aberdeen. I am a lecturer in the Institute of Pure and Applied Mathematics at the University of Aberdeen, with interests in algebraic and differential topology and their applications. I am organizer of the Aberdeen topology seminar. I am one of the organizers of the UK research network Applied Algebraic Topology, supported by the LMS, the GMJ and the IMA. I am also a keen participant at MathOverflow, and have an Erdős number of 3.