Abstract: 

Flag manifolds are a class of compact homogeneous spaces generalizing projective spaces, which arise naturally as boundaries of symmetric spaces. They attract interest from several communities, including algebraic geometry, representation theory, and the study of rigidity phenomena for discrete group actions.

In this talk, I will illustrate one of the most classical examples of such manifolds: the variety of complete flags of \(\mathbb{R}^3\). By definition, it is the space of all pointed projective lines in \(\mathbb{R}P^2\). It is homogeneous under the action of \(PGL(3,\mathbb{R})\) and admits an 8-fold covering by the 3-sphere. Using stereographic projection, we will see that intrinsic objects of the flag manifold, such as its Schubert cell decomposition, are closely related to familiar objects in the 3-sphere, notably Clifford tori and the Hopf fibration.

These visualizations, which arose at the beginning of an ongoing joint project with Thang Nguyen, were aimed at developing intuition for a completeness problem about compact manifolds locally modeled on the variety of complete flags of \(\mathbb{R}^n\), which I will briefly discuss.