Abstracts

• Jacques Darné - Filtrations on the Mapping Class Group of the punctured sphere.

Abstract: The Mapping Class Group of the n-punctured sphere (fixing a basepoint) classically identifies with the quotient B_n^* of the Artin braid group by its center. This group acts faithfully on the fundamental group of the the n-punctured sphere, which is free on n-1 generators. From this action, one can define a filtration of B_n^* by subgroups B_n^* ⊃ A_1 ⊃ A_2 ⊃ ··· which are the analogues of the Johnson kernels in this situation. It turns out that this filtration is exactly the lower central series of the corresponding pure mapping class group P_n^*, and that we understand fairly well the associated Lie algebra. In this talk, I intend to explain these results and give you a taste of the techniques involved in their proofs.

• Ramanujan Santharoubane - Applications of quantum representations of mapping class groups.

Abstract: Representations of mapping class groups arising from Witten Reshetikhin–Turaev TQFT have exotic properties compared to representations coming from classical topology. I will review these properties and show what interesting results about mapping class groups and surface groups we can derive from these objects.

• Nathan Broaddus - The mapping class group of the connect sums of S^2 x S^1.

Abstract: Let M_n be the connect sum of n copies of S^2 x S^1. By a theorem of Laudenbach, the mapping class group, Mod(M_n), of M_n is an extension of the outer automorphism group Out(F_n) of the free group F_n by the finite abelian subgroup Twist(M_n) generated by Dehn twists about embedded 2-spheres. We give a new proof of Laudenbach’s result utilizing the set of trivializations of the tangent bundle TM_n and further show that the extension Twist(M_n) --> Mod(M_n) --> Out(F_n) splits. This is joint work with T. Brendle and A. Putman.