Cycle of seminars organized jointly by E. Angelini, A. Bernardi, L. Chiantini, M. Mella, G. Ottaviani.

To receive information on this cycle of seminars, contact the organizers by email or subscribe to the mailing list geometria.

Seminars typically take place every three weeks, between Bologna and Firenze .

Due to Covid-19 restrictions, seminars are currently taking place both in presence and online. Write to elena.angelini@unisi.it to get the online access codes.

Title: Maximum likelihood estimation for tensor normal models via castling transforms.

Abstract: Maximum likelihood estimation is a method for estimating the parameters of a statistical model. In recent years, there has been much interest in tensor normal models, which are Gaussian statistical models where the covariance matrix has Kronecker product structure. In this talk, we discuss sample size thresholds for maximum likelihood estimation for tensor normal models: that is, we determine the minimal number of samples such that, almost surely, (1) the likelihood function is bounded from above, (2) maximum likelihood estimates (MLEs) exist, and (3) MLEs exist uniquely. We obtain a complete answer for both real and complex models as well as some interesting structural consequences. Along the way, we will see a recently discovered connection between maximum likelihood estimation and stability questions in geometric invariant theory, castling transforms, and results on stabilizers in general positions. This is joint work with Harm Derksen and Visu Makam.

Title: Asymptotics of degrees and ED degrees of Segre products

Abstract: Two fundamental invariants attached to a projective variety are its classical algebraic degree and its Euclidean Distance degree (ED degree). In this talk, we present some results concerning the asymptotic behavior of these invariants for various Segre products and their duals. Friedland and Ottaviani found a beautiful formula expressing the number of singular vector tuples of a general tensor. From the formula, one derives the stabilization of the ED degree of Segre varieties, as soon as one of the factors has large enough dimension. We give an alternative viewpoint on this stabilization. Finally, we discuss the stabilization of the degree of the dual variety of the product between a projective variety and a smooth hyperquadric. This is joint work with Giorgio Ottaviani and Luca Sodomaco.

Title: Singular t-ples of tensors and their geometry

Abstract: 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. Their number is the EDdegree of the Segre variety. 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. In a following talk Emanuele Ventura will lecture about the asymptotic behaviour of EDdegree and other more advanced properties.

The slides of the talk are available at the following link .

or, in presence, room A218 Povo 1, Università di Trento

Title: The geometry of symmetric qubit states

Abstract: Symmetric multi-qubit quantum states are important in quantum information. A lot of well-known states, such as W, GHZ or Dicke states are symmetric. They appear in a large number of quantum protocols and algorithms. They are also good candidates to propose highly entangled states. In this talk I'll present how the question of their classification and the research of highly entangled states is tackled in the quantum information literature and what classical algebraic geometry has to say about it. I'll also propose some experimental work to connect the two pictures.

The registration of the talk is available at the following link .

This talk is in the framework of the Quantum Information, Algebra and Geometry Workgroup .