Loeb Lecture: Grassmannian as a Set of Symmetric Orthogonal Matrices

ABSTRACT: We will discuss a model of the Grassmannian where points are represented as symmetric orthogonal matrices. We call it the involution model because such matrices are involution matrices. Our initial interest in this simple model was   prompted by its utility in manifold optimization, as it yields algorithms both numerically stable and truly geometrical. But we soon realized that it is useful beyond numerics. We showed that the involution model is unique in that it gives a minimal equivariant and isometric embedding into Euclidean space, and also embeds as an adjoint orbit in a Lie group. Furthermore, the model generalizes to complex, quaternionic, and symplectic Grassmannians, as well as to arbitrary flag manifolds. We used the involution model to derive simple expressions for nearly every kind of curvature on the Grassmannian, so simple that calculating Riemann or Weyl curvature may now be a high-school activity. We also used it to resolve a conjecture about the degree of the Grassmannian as an affine variety, find new "Grassmannian generators" for classical groups, show that almost all common manifold optimization problems are NP-hard, characterize eigenvector distributions of various random matrix ensembles, and pack products of geodesic two-spheres in a Grassmannian. Time permitting, we will discuss how such models may be extended to other manifolds, a topic we call “manifold representation” in analogy with group representation.

This talk contains joint work Yihan Guo, Zehua Lai, Xiang Lu, Tianyun Tang, Thomas Wang, and most of all, Ke Ye.
 

Department Host: Ari Stern