Probabilistic intersection theory in Riemannian homogeneous spaces

TL;DR

This paper introduces a probabilistic version of intersection theory on Riemannian homogeneous spaces, defining a new algebraic structure that captures intersection counts under random transformations, with explicit examples and applications.

Contribution

It develops the probabilistic intersection ring as a new algebraic framework, connecting it to valuations and extending classical intersection theory to a probabilistic setting.

Findings

Defined the probabilistic intersection ring as a Banach algebra.

Computed the ring structure for spheres and projective spaces.

Established a probabilistic intersection formula for complex projective space.

Abstract

Let $M=G/H$ be a Riemannian homogeneous space, where $G$ is a compact Lie group with closed subgroup $H$. Classical intersection theory states that the de Rham cohomology ring of $M$ describes the signed count of intersection points of submanifolds $Y_1, \ldots, Y_s$ of $M$ in general position, when the codimensions add up to $\dim M$. We introduce the probabilistic intersection ring $\mathrm{H}_{\mathbb E}(M)$, whose multiplication describes the unsigned count of intersection points, when the $Y_i$ are randomly moved by independent uniformly random elements of $G$. The probabilistic intersection ring $\mathrm{H}_{\mathbb E}(M)$ has the structure of a graded commutative and associative real Banach algebra. It is defined as a quotient of the ring of Grassmann zonoids of a fixed cotangent space $V$ of $M$. The latter was introduced by the authors in [Adv. Math. 402, 2022]. There is a close connection to valuations of convex bodies: $\mathrm{H}_{\mathbb E}(M)$ can be interpreted as a subspace of the space of translation invariant, even, continuous valuations on $V$, whose multiplication coincides with Alesker's multiplication for smooth valuations. We describe the ring structure of the probabilistic intersection ring for spheres, real projective space and complex projective space, relying on Fu [J. Diff. Geo. 72(3), 2006] for the latter case. From this, we derive an interesting probabilistic intersection formula in complex projective space. Finally, we initiate the investigation of the probabilistic intersection ring for real Grassmannians, outlining the construction of a probabilistic version of Schubert Calculus.