Applied Algebraic Geometry

2022-2023

"Nothing is pleasant that is not spiced with variety."

(Francis Bacon)

Tensors and secant varieties

The study of secant varieties of projective varieties is a classical problem in algebraic geometry, which presents considerable importance also for applied mathematics from a modern point of view. Indeed, the particular cases of the Segre variety, the Veronese variety and the Grassmannian lead to parametrizations respectively of the space of tensors, of the subspace of symmetric tensors and of the subspace of skew-symmetric tensors. In particular, their secant varieties provide stratifications of the spaces of tensors by the determination of tensor rank.

In general, tensors can be viewed as multi-dimensional arrays of numbers and present many utilities in several scientific areas, like computational complexity, quantum computing and computer science, or even more applied, like signal processing, bioinformatics and spectroscopy. Specifically, tensor decomposition has increased its relevance not only among researchers in algebraic geometry, but even in more applied communities.

Program

The structure of this program is based on a series of meetings among the entire academic year, with the aim of offering not only an important background about the subject, but also an opportunity to receive updates to the current situation of the research activities related to tensor decomposition, secant varieties and other topics in algebraic geometry. Moreover, it represents an occasion to share and exchange ideas with experts in specific research areas and also with young researchers at the beginning of their academic career. 

The meetings are usually planned to take place every three weeks between Dipartimento di Matematica e Informatica "Ulisse Dini" of University of Florence and Dipartimento di Matematica of University of Bologna. The duration is typically one hour.

There are two main activities. One concern a cycle of seminars held by several researchers in algebraic geometry. The other concerns a reading course on quantum computing. 

The participant group involves members affiliated to several institutions, which are Marche Polytechnic University, Polytechnic University of Milan, Polytechnic University of Turin, University of Bologna, University of Ferrara, University of Florence, University of Milan, University of Modena and Reggio-Emilia, University of Siena, University of Trento.

Organizers

Seminars

The idea of providing a cycle of meeting related to specific subjects related to tensors and secant varieties has been repeated every academic year from 2014, when the first edition took place through the program Geometria Algebrica Numerica. Among the several editions, the program has seen the participation of several professors and young researchers. In addition to the main members of the research groups by each institution, which continue to hold seminars regularly, there has also been participation of many other experts in algebraic geometry from other institutions in the world, which have occasionally offered to present their contribute and their more recent results.

Reading course

One of the more recent applications of theory of tensors regards the approach to quantum information and quantum computation. In particular, secant varieties and tensor product of complex Hilbert spaces turned out to be very important in dealing with quantum entanglement. 

The reading course is named Algebraic Geometry and Quantum Computing and it focuses on the analysis of some argument appearing in quantum mechanics from a point of view of algebraic geometry, with the main aim of creating a connection between both the languages which are used by physical researchers and mathematicians and formalizing the related mathematical theory.

Information

For any question and information about the cycle of seminars, to be added to the mailing list or to get the link to partecipate to the eventual online connection for the seminars, please write to cosimo.flavi@unifi.it.

Last meeting

Algebraic Geometry and Quantum Computing

Lecture  7 – Why Tensor Networks? 

Alessandra Bernardi

Università degli Studi di Trento (Trento, Italy)

The ground state of a Hamiltonian is a most suitable state to play with if one is interested in seeing quantum mechanics effects. These kinds of states naturally appear in a special "region" of the tensor space, namely on/very close to Tensor Network varieties. I would like to illustrate these connections and, in dependence on the time, to show one of the few examples where all the construction is exactly possibile, namely the so-called AKLT model.

Alma Mater Studiorum – Università di Bologna

Dipartimento di Matematica
Piazza di Porta San Donato 5 40127 Bologna

Tuesday, June 20, 2023
Time: 15:30

Room: Enriques