2018年4月5日 10時30分 ～ 2018年4月5日 15時00分
Arthur Soulié (Strasbourg)
Talk 1, 10:30-12:00
Title: Homological stability with twisted coefficients and functor homology
Abstract: In 2017, Randal-Williams and Wahl set a general framework to prove homological stability with twisted coefficients for different families of groups, including automorphism groups of free groups, mapping class groups of orientable and non-orientable surfaces or mapping class groups of 3-manifolds. The twisted coefficients they use satisfy a “polynomial” condition which I will explain.
Moreover, when homological stability is satisfied, computing the colimit of the homology groups gives the stable value. I will present a general method set by Djament and Vespa to tackle this problem. This includes presenting a general result of splitting of the stable homology with twisted coefficients thanks to functor homology I extended from the framework of Djament and Vespa for the situation of mapping class group of surfaces.
Talk 2, 13:30-15:00
Title: Representations of braid groups and Long’s induction
Abstract: Linear representations of braid groups is a rich subject which appears in diverse contexts in mathematics. Even if, at the present time, a complete classification of these representations is probably out of reach, any new result which allows us to gain a better understanding of them is a useful contribution. In 1994, in a joint work with Moody, Long gave a method to construct from a linear representation of the braid group on n+1 strands, a new linear representation of the braid group on n strands. Moreover, the construction complicates in a sense the initial representation. For example, applying it to a one dimensional representation, the construction gives a mild variant of the unreduced Burau representation.
I will present this construction and give an overview of the classical families of representations of braid groups and present the original Long-Moody construction.