Abstract. 
In Riemannian geometry, geodesics are curves which locally minimize
lengths. In general, it is a difficult and interesting question to
determine which geodesics of a manifold are in fact globally minimizing. In
settings of non-positive curvature (e.g., hyperbolic space), the
Cartan-Hadamard theorem says that all geodesics are minimizing, so the
presence of positive curvature (e.g., sphere) is necessary to destabilize
this minimization property. With Janek Wehr, we have used the point-of-view
of the particle technique to show that for random perturbations of
2-dimensional Euclidean space, enough positive curvature arises to
destabilize "generic" geodesics. I will present this work, as well as
discuss the extension to the more general setting of symmetric random
geometries. No background in geometry or probability will be required for
this talk, and it will be accessible to graduate students.