The linearity problem for braid groups

Haris Stylianakis

Friday 22nd January, 2016 16:00-17:00 Maths 326


We can define the braid group as the mapping class group of the punctured disc. An old problem on braid groups was whether is linear or not; that is, is there a faithful representation of braid groups on the general linear group with entries in the field of complex numbers? One candidate for such a representation is the Burau representation introduced by Werner Burau during the 1930s. For many years it was believed that the Burau representation is faithful, until 1999 in which Bigelow proved that if the number of strands is at least 5, then the Burau representation is not faithful. Eventually in 2000 Krammer came up with a new linear representation, and he proved that braid groups are linear. The same period Bigelow independently used the latter linear representation to give a different proof that braid groups are linear. In this talk we will define braid groups, and we will construct the Burau and Krammer's representations.

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