17 nov 2020 -- 16:00
Geometry in Como - Online
The talk will take place on Microsoft Teams. The link is available below and is also on https://researchseminars.org/seminar/DifferentialAndAlgebraicGeometry.
Abstract.
The space of tensors considered in this talk is the tensor product of some real vector spaces of finite dimension $V_1,\ldots,V_d$. This space contains the Segre variety of decomposable (or rank one) tensors. There is a natural invariant metric on the space of tensors, called Frobenius metric.
In optimization setting one considers the (complex) critical points on the Segre variety of the distance function from a given tensor, they are called singular $t$-ples, among them there is the best rank one approximation.
In the symmetric setting, when $d=2$, these critical points are just the eigenvectors of a symmetric matrix.
The geometry of the critical points is appealing, since they lie in a linear space called critical space, which has dimension smaller than the number of critical points, in other words the critical points are linearly dependent, unless the matrix case. We expose some properties of singular $t$-ples.