To any group G we  associate the space Ch(G) of all characters on G. After defining this space and discussing its interesting properties, I’ll turn to discuss dynamics on such spaces. Our main result is that the action of any​ arithmetic group on the character space of its amenable/solvable radical is stiff, i.e., any probability measure which is stationary under random walks must be invariant. This generalizes a classical theorem of Furstenberg for dynamics on tori. Relying on works of Bader, Boutonnet, Houdayer, and Peterson, this stiffness result is used to deduce dichotomy statements (and ‘charmenability’) for (not necessarily semi-simple) higher rank arithmetic groups pertaining to their normal subgroups, dynamical systems, representation theory and more. The talk is based on a joint work with Uri Bader. 

Abstract: Given a smooth surface in Euclidean 3-space, a classical question in differential geometry asks whether the surface can be continuously deformed through a smooth, nontrivial family of isometric surfaces. If such a family exists and does not arise from rigid motions of R^3, then the surface is said to be flexible. An old conjecture asserts that flexible, smooth closed surfaces do not exist. In this talk, we survey this question and the general uniqueness problem for isometric immersions. We then present new examples of flexible, smooth immersed cylinders in R^3 which are neither flat nor minimal. We conclude with a discussion of potential approaches to the construction of flexible, smooth closed surfaces. These results are part of upcoming work with Andrew Sageman-Furnas.