Singular vector tuples of tensors and Kalman varieties

Kalman varieties of tensors are algebraic varieties consisting of tensors whose singular vector tuples lay on prescribed subvarieties. They were first studied by Ottaviani and Sturmfels in the context of matrices. I will talk about some families of Kalman varieties, extending previous work of Ottaviani and Shahidi to the partially symmetric context, highlighting the special role of isotropic vectors in the spectral theory of tensors. I will indicate how to describe the totally isotropic Kalman variety as a dual variety and how to obtain a generating function whose coefficients are the degrees of these varieties. This is based on a joint work with Shahidi and Sodomaco.