Title. Connectedness theorems in algebraic geometry

Abstract. Beginning with Bertini's connectedness theorem, and continuing through the "higher connectedness theorems" in SGA 2, there are many applications of connectedness, simple connectedness, etc. of algebraic varieties. I will explain a new connectedness theorem first appearing in the thesis of Cristian Minnocheri together with applications: a new proof of the Fulton-Hansen theorem, irreducibility of spaces of stable maps (Minoccheri, Dennis Tseng), and irreducibility of spaces parameterizing Veronese e-uple r-dimensional subvarieties of sufficiently general degree-d hypersurfaces (a question of Joe Harris).