Classical intersection theory relies on the notion of generic position: intersection numbers are defined for sufficiently general configurations and encoded algebraically by cohomology. Over the real numbers, this paradigm often breaks down—intersection counts can vary with position or even vanish entirely.

In this talk, I present a probabilistic approach to real intersection theory, replacing “generic” configurations with “random” ones. A motivating example is a variation on a classical question: what is the expected number of real lines intersecting four random lines in real projective space? In joint work with P. Bürgisser, we showed that this expectation equals the volume of a specific convex body, uncovering an unexpected link between real algebraic geometry and convex geometry. Building on this insight, further joint work with Bürgisser, Breiding, and Mathis led to the construction of the probabilistic intersection ring—an algebraic framework in which probabilistic intersection computations take place.

The elements of this ring are equivalence classes of convex bodies, and its product is induced by the zonoid algebra, a convex-geometric analogue of the exterior algebra. As in classical intersection theory, products encode intersection data—here, expected intersection counts. This framework provides a foundation for probabilistic Schubert calculus and highlights deep connections with valuation theory and integral geometry.

This is based on joint works  with P. Bürgisser, P. Breiding and L. Mathis