In algebraic geometry, intersection theory aims at describing how algebraic manifolds intersect one another in an ambient space. Recent progress about self-intersections have been made by studying how an algebric manifold intersects its infenitesimal perturbations. These results open the way to a better understanding of derivate self-intersections.
For a more complete description of this result (in french), click here.
 Julien Grivaux. Derived geometry of the first formal neighborhood of a smooth analytic cycle, to appear.