Braids: Knots, Garside Groups and Mapping Class Groups

   PI: Juan González-Meneses

Abstract: Braids groups are mathematical objects of particular relevance, as they appear as an important component in several fields of Mathematics, not only in Group theory but also in Knot Theory, Geometric Theory of surface automorphisms, Algebraic Geometry or even Cryptography. 
This project brings together a team of first level specialists in the theory of braids, and aims to go deeper in the study of these objects from three points of view: As topological objects, using the knowledge of braid groups to discover properties of knots and links, as happens with Jones of HOMFLYPT polynomials; As algebraic objects, using the structure discovered by Garside and others, which is shared by groups as relevant as Artin groups; And finally, as geometric objects, looking at braids as mapping classes (automorphisms of the punctured disc modulo isotopy), where the general theory of Nielsen-Thurston behaves in a very particular way, and can be used to understand topological and algebraic properties of braids. 
Each of these distinct points of view to study braids can be used to understand the others, in this way braids become a link relating algebraic, geometric and topological results.