Applied Algebraic Geometry 2020-2021
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.
Fourth Talk:
Tuesday January 26, 2021, at 15:30, via Zoom
Michael Walter (University of Amsterdam) will speak about:
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.
Third Talk:
Thursday December 10, 2020, at 15:00, via Zoom
Emanuele Ventura (Mathematisches Institut, Universität Bern) will speak about:
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.
Second Talk:
Thursday November 12, 2020, at 15:00, via Zoom
Giorgio Ottaviani (Università di Firenze) will speak about:
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 .
First Talk:
Wednesday October 28, 2020, at 11:30, via Zoom
or, in presence, room A218 Povo 1, Università di Trento
Frédéric A. Holwec (Belfort-Montbéliar) will speak about:
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 .