Abstract: The generalised Hurwitz theorem states that exist only fifteen composition algebras over any given field: seven Hurwitz algebras (four division and three split), six para-Hurwitz algebras (three division and three split) and two Okubo algebras (one division and one split). Of these fifteen algebras only three are division composition algebras of dimension eight: the Octonions, which are unital; the para-Octonions that are para-unital and the Okubo algebra which is non-unital. In this talk we will show how all of them are suitable for the construction of the 16-dimensional Moufang plane and how the collineation groups of such a plane allows the realisation of all possible real forms of the Lie Groups G2 and F4 along with two real forms of E6.  
This work is in collaboration with Alessio Marrani and Francesco Zucconi. 

Abstract: The concept of the finite difference will be applied to the sequence of Vietoris’ numbers. Some algebraic properties will be explored, and the generating function will be presented. We will also extend the study to the quaternionic sequence with Vietoris’ numbers as its components. This is a joint work with Paula Catarino.

Abstract: The spectrum and the numerical range of a bounded operator is arguably two of the most important concepts associated with linear operators. Their properties are greatly influenced by the underlying field. For instance, the usual definition of spectrum cannot be used in the setting of infinite dimensional quaternionic Hilbert spaces, leading to the notion of S-spectrum. Regarding the numerical range, in complex Hilbert spaces, the Toeplitz-Hausdorff Theorem ensures that the numerical range is always convex. However, in the setting of quaternionic Hilbert spaces, this convexity property may no longer hold. Moreover, it is generally challenging to describe the numerical range for operators in quaternionic spaces, and its shape remains largely unpredictable. A notable exception occurs for normal operators, where more is known about both convexity and the structure of the numerical range. In this presentation, we focus on numerical range of bounded linear operators in quaternionic Hilbert spaces and explore its connection to the S-spectrum of these operators. Our goal is to extend several classical results from complex Hilbert space theory to the quaternionic case. In particular, we provide a detailed characterization of the numerical range for normal operators in quaternionic Hilbert spaces.
This is joint work with Cristina Diogo and Luís Carvalho.

Abstract: In this talk we will describe what is known about the metacommutation problem in the ring of Hurwitz integers, and describe some properties about the cycle structure of the metacommutation map, which is a permutation. In particular, we will talk about the sign of this permutation and the number of fixed points.

Abstract:  We will present recent results establishing analogues of Hilbert's Nullstellensatz over the quaternion algebra, which generalize a classical theorem of Jacobson.

Abstract: In this talk, we will initially survey the existing results on transposed Poisson structures and Poisson algebras transposed in the Heisenberg-Virasoro super algebra, in addition to introducing transposed Poisson structures in the Heisenberg-Virasoro ultra algebra. And finally, we will present the result we obtained that tells us that there are no non-trivial 1/2-ultra derivations in the Heisenberg-Virasoro ultra algebra and, as a consequence, we obtained that there are no non-trivial transposed Poisson ultra algebra structures defined on the Heisenberg-Virasoro ultra algebra.

Abstract: My participation aims to shed light on the octonion-valued Fourier transform (OFT) by establishing a generalized version of Miyachi's theorem for the OFT. This generalization extends two well-known uncertainty principles, namely Hardy's theorem and the Cowling-Price theorem, to the OFT domain.

Abstract: In this talk we introduce two new sequences: the gaussian repunit numbers and the quaternion repunit numbers. We establish some properties of these sequences, as well as recurrence relations, the Binet formula, and Catalan's, Cassini's, and d'Ocganes identities.

Abstract: The base elements of octonions generate a so-called octonion loop of 16 elements, analogously one can obtain the sedenion loop and so on. The map that can behave as an automorphism for some pair of elements and as an anti-automorphism for another pair of elements of some loop is called a half- automorphism. We will present the group of half-automorphisms of Cayley-Dickson loops.  

My talk is based on a joint work with Peter Plaumann and Maria de Lourdes Merlini Giuliani.

Abstract:  We use the classical Fueter mapping theorem to build a system of Clifford-Appell polynomials in the quaternionic setting. We will discuss various properties of such polynomials and study their algebraic behavior with respect to the well-known Cauchy-Kowalevski product. Then, based on this system, we study a family of quaternionic reproducing kernel Hilbert spaces in this framework. As particular examples, we will treat results in the case of Hardy and Fock spaces and present counterparts of creation, annihilation, shift, and backward shift operators. We will present some connections and applications of this approach to quaternionic polyanalytic functions and Schur analysis. If time allows, we will discuss extensions of these results in the Clifford case.  

Abstract: My “hypercomplex” view at two less well-known books of Paul Halmos (1916 - 2006) and Walter Rudin (1921 - 2010) aims to call attention to some almost overseen connections between holomorphic functions of one or several complex variables on one side and hyperholomorphic functions of one or several hypercomplex variables on the other side.  These books are Finite dimensional vector spaces (Halmos, 1942) resp. Function theory in polydiscs (Rudin, 1969). Halmos’ treatment of complexification as well as Rudin’s introduction of holomorphic functions of several complex variables can still give rise to a better understanding of some aspects that receive little attention in the traditional definition of hyperholomorphic (monogenic) functions.