# Gabriel Larotonda

## Conjugate points in the Grassmann manifold of a $C^*$-algebra

For a (real or complex) $C^*$-algebra $\mathcal A$, let $P^*=P^2=P\in\mathcal A$. Consider $Gr(P)$ the unitary orbit of $P$ in $\mathcal A$, which is one of the connected components of the Grassmanian manifold of $\mathcal A$. For a natural connection (which appears in different disguises) in $Gr(P)$, its geodesics are described by $\gamma(t)=e^{tZ}Pe^{-tZ}$ for $Z^*=-Z$ and $Z=ZP+PZ$. The exponential map of this connection is then $\mathrm{Exp}_P(Z)=e^ZPe^{-Z}$. In this talk we will describe conjugate points along geodesics in the Grassmannian, both in the metric sense (cut locus for the Finsler metric induced by the norm of $\mathcal A$) and in the tangent sense (the differential of the exponential map along $\gamma$ is not invertible). In the infinite dimensional setting, the distinction between monoconjugate and epiconjugate points will be relevant.

This is part of recent research in collaboration with Esteban Andruchow and Lazaro Recht (CONICET, Argentina).