This talk will introduce work in the area of Geometric Group Theory; no prior background in this area will be assumed. The commensurator of a subgroup H of a group G may be seen as a coarse approximation of the normalizer of H. We consider the situation where H is free abelian and G acts properly on a CAT(0) space, that is, a simply connected space of metric non-positive curvature. The structure of the normalizer of H and its action on the space are well understood in this context. However, the commensurator is more mysterious and it contains subtle information about the action which is not seen by the normalizer. For various classes of CAT(0) spaces we obtain structural results about the commensurator and its relation to the normalizer. In this talk, first I will give background on the commensurator and on CAT(0) spaces and groups, and then I will discuss various geometric tools and constructions used in our approach. This is joint work with Jingyin Huang.

In the early 1990s Ribet observed that the classical mod l multiplicity one results for modular curves, which are a consequence of the q-expansion principle, fail to generalize to Shimura curves. Specifically he found examples of Galois representations which occur with multiplicity 2 in the mod l cohomology of a Shimura curve with discriminant pq and level 1.

I will describe a new approach to proving multiplicity statements for Shimura curves, using the Taylor-Wiles-Kisin patching method (which was shown by Diamond to give an alternate proof of multiplicity one in certain cases), as well as specific computations of local Galois deformation rings done by Shotton. This allows us to re-interpret and generalize Ribet's result. I will prove a mod l "multiplicity 2^k" statement in the minimal level case, where k is a number depending only on local Galois theoretic data. This proof also yields additional information the Hecke module structure of the cohomology of a Shimura curve, which among other things has applications to the study of congruence modules.

If an algebraic curve over a field of characteristic 0 admits a finite
map to the projective line ramified only over three points, then it must
be definable over some number field. This fact has a famous converse due
to Belyi: any curve over a number field admits such a finite map over
its field of definition.

Similarly, if an algebraic curve over a field of characteristic p>0
admits a finite *tamely ramified* map to the projective line ramified
only over three points, then it must be definable over some finite
field. We prove the converse: any curve over a finite field admits such
a finite map over its field of definition.

A construction of Saidi shows that this reduces to the existence of a
single tamely ramified map. This is easy to establish over an infinite
field of odd characteristic, and only slightly harder (using
Poonen-style probabilistic techniques) over a finite field of odd
characteristic. To handle the case of a finite field of characteristic
2, we use a construction of Sugiyama-Yasuda that they used to establish
existence of tame morphisms over an algebraically closed field of
characteristic 2.

Joint work with Daniel Litt (Georgia) and Jakub Witaszek (Michigan).