Abstract: In this talk we will talk about the relationship between hyperbolic cone-structures and their holonomy representations. Any hyperbolic structure on a given closed compact and orientable surface S induces in a very natural way a representation of the fundamental group \pi_1(S) in the Lie group PSL(2,R), which encodes geometric data about the structure. The reverse problem to recover a hyperbolic cone-structure from a given representation is more arduous and longer. Even worse it is not always possible.

Abstract: Rigidity of lattices in semisimple Lie groups has been widely studied so far. One of the most celebrated theorem is Mostow rigidity, which states the following. Assume n bigger or equal than 3. If two complete hyperbolic n-manifolds of finite volume have isomorphic fundamental groups then they are isometric. Equivalently their fundamental groups are conjugated in the group of isometries of the hyperbolic n-space. There are several ways to prove this theorem.

Quasimaps provide a compactification of the space of maps from smooth curves to a GIT quotient; Ciocan-Fontanine and Kim used them to prove beautiful wall-crossing formulae, comparing the resulting invariants with ordinary Gromov-Witten ones. In joint work with Navid Nabijou, we introduce the notion of genus zero relative quasimaps to a toric target X with respect to a smooth (but not necessarily toric) hyperplane section Y, extending work of Gathmann to this setting.

The dielectric constant is defined by an experiment with an infinite slab of dielectric between two infinite oppositely charged plates.  This talk describes the surprisingly delicate theorem that the physicists formula for the electrostatic potential of this idealized experiment is the limit, as the size tends to infinity, of finite size experiments.  Joint work with L, Ridgway SCOTT.