Orders in finite-dimensional algebras over number fi give rise to interesting locally symmetric spaces and algebraic varieties. Hilbert modular varieties or arithmetically defined hyperbolic 3-manifolds, compact ones as well as noncompact ones, are familiar examples. In this talk we discuss various cases related to the general linear group $GL(2)$ over orders in division algebras defined over some number field. Geometry, arithmetic, and the theory of automorphic forms are interwoven in a most fruitful way in this work. Finally we indicate a construction of non-vanishing square-integrable cohomology classes for such arithmetically defined groups.[-]

We will describe two projects. The first which is joint with Avner Ash and Paul Gunnells, concerns arithmetic subgroups $\Gamma$ of $G = SL_4(Z)$. We compute the cohomology of $\Gamma \setminus G/K$, focusing on the cuspidal degree $H^5$. We compute a range of Hecke operators on this cohomology. We fi Galois representations that appear to be attached to the Hecke eigenclasses, based on the operators we have computed. We have done this for both non-torsion and torsion classes. The second project, which is joint with Bob MacPherson, is an algorithm for computing the Hecke operators on the cohomology $H^d$ of $\Gamma$ in $SL_n(Z)$ for all $n$ and all $d$.[-]

We investigate the geometry of Galois deformation rings in the defect zero setting but when the Taylor-Wiles hypothesis does not hold. In particular, we consider the question of whether or not the map from the local deformation ring to the global deformation ring is a local complete intersection map and the role the Taylor-Wiles hypothesis plays in this question. We exhibit an example in the context of classical weight two modular forms where this does not hold and shows that a resulting Tor algebra acts on the cohomology of a modular orbifold. This is joint work in progress with Preston Wake.[-]

In this talk, we consider the limit multiplicity question (and some variants): how many automorphic forms of fixed infinity-type and level N are there as N grows? The question is well-understood when the archimedean representation is a discrete series, and we focus on non-tempered cohomological representations on unitary groups. Using an inductive argument which relies on the stabilization of the trace formula and the endoscopic classification, we give asymptotic counts of multiplicities, and prove the Sarnak-Xue conjecture at split level for (almost!) all cohomological representations of unitary groups. Additionally, for some representations, we derive an average Sato-Tate result in which the measure is the one predicted by functoriality. This is joint work with Rahul Dalal.[-]