Surface Braid Groups and Mapping Class Groups

Mini course by

Professor Paolo Bellingeri

Université de Caen

November 4 - 8, 2019

From November 4 to November 8, 2019, Professor Paolo Bellingeri from the University of Caen will teach a mini course on surface braid groups and mapping class groups at the Atlantic Algebra Centre. The lectures will take place at the St. John's campus of Memorial University.

Surface braid groups are a natural generalization of classical braid groups and of fundamental groups of surfaces. They were first defined by Zariski during the 1930's (although braid groups on the sphere had been considered much earlier by Hurwitz), and they were re-discovered during the 1960's in the study of mapping class groups and configuration spaces. These groups, introduced as an "algebraic" tool, turned out to be very difficult to understand. It is now common to use mapping class techniques to study the properties of surface braid groups.

In the last decade the interest in these groups grew notably, in particular due to their relations with knot theory and mapping class groups and, quite astonishingly, with robotics. The mini course will start with different definitions and group presentations for these groups; this first part will allow us to present several combinatorial properties, such as residual properties, central series, and related quotients, which will lead to some applications to finite type invariant theory as well as linear and "symmetric" representations for surface braid groups. We will then discuss the relation between surface braid groups and mapping class groups. We will end with an overview on classic and more recent applications to knot theory in 3-manifolds. Here and there we will present some open questions.

The lecturer of the mini course, Paolo Bellingeri, is a professor at the University of Caen and the director of the Federation of Normandy Laboratories of Mathematics. He also serves as the lead scientist of the research group "Algèbre Représentations et Topologie pour l'Informatique Quantique et Classique" (ARTIQ). He has published widely on the topic of the mini course and is one of the organizers of the conference series "WinterBraids" on this research area.

The precise schedule of the mini course is

Monday, November 4, 9:00 am - 10:50 am, EN 1004 (Engineering Building)

Tuesday, November 5, 10:30 am - 11:50 am, EN 1004 (Engineering Building)

Wednesday, November 6, 9:00 am - 10:50 am, EN 1004 (Engineering Building)

Thursday, November 7, 13:00 pm - 14:50 pm, EN 1004 (Engineering Building)

Friday, November 8, 9:00 am - 10:00 am, EN 1004 (Engineering Building)

The mini course is co-organzised with the AARMS Collaborative Research Group "Groups, Rings, Lie and Hopf algebras". We plan to provide partial support for members of this research group and for students from Atlantic Canada. For any questions about the mini course, please contact Yorck Sommerhäuser at sommerh@mun.ca.