Mathematics Colloquium

COLLOQUIUM WILL BE HELD IN 201 E. BRIDGE with Tea at 3:30 pm in 108 E. Bridge

The Langlands conjectures posit a deep connection between the cohomology of arithmetic locally symmetric spaces and Galois theory. In the case that the locally symmetric space is a Shimura variety (the simplest example being a modular curve) there has been enormous progress on the conjectures over the last 40+ years. However when the locally symmetric space is not a Shimura variety (for instance a hyperbolic 3-manifold) all the techniques appeared to break down and completely new phenomena emerge. Over the last 5+ years there has been surprisingly (at least to me) fast progress in this more general setting. Working with modular curves and arithmetic hyperbolic 3-manifolds as examples, in this talk I will briefly describe what is known in the case of modular curves, explain the new phenomena that arise in the more general setting and indicate some of what is now known. (These advances are due to many people, and I will try to give specific credit during my talk.)