A path connected topological space is simply connected if the space of based paths is path connected. Equivalently, the fundamental group is zero or any connected covering space is trivial. However, these notions do not capture the correct notion in the world of algebraic geometry. For example, if $X$ is a Riemann surface then the Zariski topology (the usual topology in algebraic geometry) on $X$ is equivalent to the cofinite topology, so $X$ is simply connected.
In this talk, we will introduce a few definitions of simply connectedness in algebraic geometry - each corresponding to one of the equivalent definitions above. We will then compare these definitions and discuss how their consequences differ from their topological counterparts.