Extra QLunch: Luis Velazques

Speaker: Luis Velazques, University of Zaragoza

Title: BABEL

Abstract: But the lord came to see the city and tower the people were building. The lord said, “If as one people speaking the same language they have begun to do this, then nothing they plan to do will be impossible for them”

The story of the Tower of Babel reflects the advantages of speaking a common language that allows the communication among different communities, and the confusion that may arise in the opposite situation, which led the Tower of Babel unfinished when the speeches were confounded so that the Babel citizens could no longer understand each other, and ended scattered around the world.

The good part of the message is that the Tower of Babel could reach the heaven finally if the citizens succeeded in unifying their languages again. This story represents metaphorically the actual situation of knowledge, whose high level of specialization has led to the lack of communication among different fields which have developed their particular language, hindering their mutual enrichment.     

This talk intends to illustrate the opportunities that unifying the languages of mathematical sciences and their applications may open, showing their mathematical payoffs. The paradigm chosen for this starts with a connection among several problems coming from apparently unrelated areas of mathematics and quantum theory. At the heart of these unexpected links is the notion of Schur function, one of the gems bequeathed to us by Issai Schur, a very original worker who left his mark all over mathematics.

The alluded results touch areas of harmonic analysis, orthogonal polynomial theory, random walks, as well as their quantum version, known as quantum walks.

REFERENCES

M.J. Cantero, F.A. Grünbaum, L. Moral, L. Velázquez,

Matrix valued Szegő polynomials and quantum random walks,

Comm. Pure Appl. Math. 63 (2010) 464-507.

F.A. Grünbaum, L. Velázquez, A.H. Werner, R.F. Werner,

Recurrence for discrete time unitary evolutions,

Commun. Math. Phys. 320 (2013) 543-569.

Bourgain, F.A. Grünbaum, L. Velázquez, J. Wilkening,

Quantum recurrence of a subspace and operator-valued Schur functions,

Commun. Math. Phys. 329 (2014) 1031-1067.

Cedzich, F.A. Grünbaum, L. Velázquez, A.H. Werner, R.F. Werner,

A quantum dynamical approach to matrix Khrushchev's formulas,

Comm. Pure Appl. Math. 69 (2016) 909-957.

F.A. Grünbaum, L. Velázquez,

A generalization of Schur functions: applications to Nevanlinna functions, orthogonal polynomials, random walks and unitary and open quantum walks,

Adv. Math. 326 (2018) 352-464.