Computing Equivariant Cohomology for SL(n,Z)

Computing Equivariant Cohomology for SL(n,Z)

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Mark McConnell , Princeton University and CCR
Fine Hall 214

In a series of papers, Ash, Gunnells and McConnell have studied cohomology groups for congruence subgroups of SL(4,Z), and have verified experimentally that Hecke eigenclasses x for these cohomology groups seem to have attached Galois representations. The talk will have three parts.  (1) Very briefly, I will state the recent theorem of Peter Scholze which proves the existence of these Galois representations when x is torsion.  (2) I will describe how we perform our topological computations.  This uses a very concrete cell complex, the _well-rounded retract_, which lives naturally in the space of n-dimensional lattices.  (3) Our previous papers only used cohomology with constant coefficients.  Last summer, I wrote a Sage package to handle any coefficient module.  I will review the equivariant cohomology spectral sequence and how the Sage code computes it.