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Probabilistic intersection theory in Riemannian homogeneous spaces

Paul Breiding, Peter Bürgisser, Antonio Lerario, Léo Mathis

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.

Probabilistic intersection theory in Riemannian homogeneous spaces

Abstract

Let be a Riemannian homogeneous space, where is a compact Lie group with closed subgroup . Classical intersection theory states that the de Rham cohomology ring of describes the signed count of intersection points of submanifolds of in general position, when the codimensions add up to . We introduce the probabilistic intersection ring , whose multiplication describes the unsigned count of intersection points, when the are randomly moved by independent uniformly random elements of . The probabilistic intersection ring 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 of . The latter was introduced by the authors in [Adv. Math. 402, 2022]. There is a close connection to valuations of convex bodies: can be interpreted as a subspace of the space of translation invariant, even, continuous valuations on , 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.

Paper Structure

This paper contains 48 sections, 79 theorems, 438 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Algebras of zonoids
  3. The probabilistic intersection ring
  4. The class of a submanifold in the probabilistic intersection ring
  5. Probabilistic intersection problems
  6. The probabilistic intersection ring and the cohomology ring
  7. The probabilistic intersection ring of the complex projective space
  8. Probabilistic Schubert Calculus
  9. Basics on zonoids
  10. Centered, noncentered and virtual zonoids
  11. Zonoids and random vectors
  12. Pairing of zonoids
  13. Zonoids as measures
  14. Topologies and continuity
  15. Zonoid algebras
  16. ...and 33 more sections

Key Result

Lemma 2.5

Suppose $\varphi\colon V\to W$ is a linear map of Euclidean spaces with adjoint map $\varphi^T\colon W \to V$, i.e., $\langle \varphi (v), w \rangle = \langle v, \varphi^T (w)\rangle$ for all $v\in V$ and $w\in W$. Then, $\langle \varphi_* (K), L \rangle = \langle K, (\varphi^T)_* (L)\rangle$ for al

Figures (3)

  • Figure 1: The Vitale zonoids of a random vector $\xi$ with two states.
  • Figure 2: The disjoints sets $A_1,A_2,B_1,B_2,C\subset [k]$ from the proof of \ref{['prop:computealphak']}. In the proof, these sets come from partition $J=A_1\cup A_2\cup B_1\cup B_2$ and $T= B_1\cup A_2$ and $T' = B_2\cup A_2$.
  • Figure 3: The Young diagram of the partition $(4,3,1)$ contained in a $3\times 4$ rectangle. The outer corners are $(1,4),(2,3),(3,1)$.

Theorems & Definitions (201)

  • Remark 1.1
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • Remark 2.7
  • Proposition 2.8
  • ...and 191 more