$osp(1,2)$, the Bannai-Ito Algebra and Generalizations

Abstract:

The Bannai-Ito algebra and its relation to the Bannai-Ito polynomials will be reviewed.  Emphasis will be put on some of the connections with the Lie superalgebra osp(1,2)osp(1,2).  Centralizing elements of osp(1,2)osp(1,2) associated to involutions will be identified and used to realize the Bannai-Ito algebra as the centralizer of osp(1,2)osp(1,2) in its co-product embedding in the three-fold direct product of this algebra with itself. The centralizing elements will be called upon to obtain a higher rank generalization and a hyperoctahedral version of the Bannai-Ito algebra.

About the speaker: Luc Vinet is Aisenstadt Professor of Physics at the Université de Montréal and the Director of the CRM (Centre de Recherches Mathématiques) , a position he held previously from 1993 to 1999. Born in Montréal, he holds a doctorate (3rd cycle) from the Université Pierre et Marie Curie (Paris) and a PhD from the Université de Montréal, both in theoretical physics. After two years as a research associate at MIT, he was appointed as assistant professor in the Physics Department at the Université de Montréal in the early 1980s and promoted to full professorship in 1992. His research interests in theoretical and mathematical physics include: exactly solvable problems, symmetries, algebraic

structures, special functions and quantum information, topics in which he written over 250 papers. In 1999, Luc Vinet joined the ranks of McGill University where he held the position of Vice-Principal (Academic) and Provost. From 2005 to 2010, he was the Rector of the Université de Montréal. Among numerous honours, he was awarded the Armand-Frappier Prize of the Government of Québec in 2009 and the 2012 CAP-CRM Prize in Theoretical and Mathematical Physics. He also holds an honorary doctorate from the Université Claude-Bernard (Lyon). He is a Fellow of the American Mathematical Society and was appointed Officer of the National Order of Quebec in 2017.

This is sample text

There are many situations where one imposes certain conditions on a function and its Fourier transform and then wants to optimize a certain quantity. I will describe how these types of Fourier optimization problems can arise in the context of the explicit formula, which relates the primes to the zeros of the Riemann zeta-function. These ideas lead to the strongest known estimates in the classical problem of bounding the maximum gap between consecutive primes assuming the Riemann hypothesis. Our answer depends on the size of the constant in the Brun-Titchmarsh inequality. Using the explicit formula in the other direction, we can also use Fourier optimization to prove the strongest known conditional estimates for the number of zeta zeros in an interval on the critical line. This is based on joint works with E. Carneiro, V. Chandee, and K. Soundararajan.

There are many situations where one imposes certain conditions on a function and its Fourier transform and then wants to optimize a certain quantity. I will describe how these types of Fourier optimization problems can arise in the context of the explicit formula, which relates the primes to the zeros of the Riemann zeta-function. These ideas lead to the strongest known estimates in the classical problem of bounding the maximum gap between consecutive primes assuming the Riemann hypothesis. Our answer depends on the size of the constant in the Brun-Titchmarsh inequality. Using the explicit formula in the other direction, we can also use Fourier optimization to prove the strongest known conditional estimates for the number of zeta zeros in an interval on the critical line. This is based on joint works with E. Carneiro, V. Chandee, and K. Soundararajan.