title:

"Hecke Operators for Arithmetic Groups via Cell Complexes"
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abstract:
The topology of locally symmetric spaces has many connections to number theory and automorphic forms.  We will present conjectures which attach Galois representations to Hecke eigenclasses in the cohomology of locally symmetric spaces for SL(n).  When n = 2, the spaces are quotients of the upper half-plane, and the conjectures extend classical results for modular forms.  We explain the well-rounded retract, a finite cell complex that lets us compute the cohomology of these spaces along with the Hecke action.  We describe recent work of Ash, Gunnells and McConnell giving the first concrete construction of cuspidal cohomology classes for GL(4).