Schedule

In this talk, we discuss a new class of quaternions, namely, incomplete Horadam quaternions that are based on incomplete Horadam numbers which generalize the previously introduced incomplete Fibonacci and Lucas quaternions. Further, some identities including summation formulas and generating functions concerning these quaternions are also talk about.

The length of a finite system of generators for a finite-dimensional (not necessarily associative) algebra over a field is the least positive integer $k$ such that the products of generators of length not exceeding $k$ span this algebra as a vector space. The maximum length for the systems of generators of an algebra is called the length of the algebra [1]. The length function is an important invariant of algebras which is vital in fundamental studies as well as in practical applications, including mechanics of isotropic continua, numerical linear algebra, quantum physics [2]. We plan to discuss some general properties of this function on non-associative algebras and compute the lengths of quaternions, octonions, and several their generalizations, including composition algebras.
The talk is based on the joint works with K. Kudryavtsev and S. Zhilina.
[1] A. Paz, An application of the Cayley--Hamilton theorem to matrix polynomials in several variables, Linear Mult. Algebra, 15 (1984) 161-170.
[2] V. Futorny, R.A. Horn, V.V. Sergeichuk, Specht's criterion for systems of linear mappings, Linear Algebra Appl. 519 (2017) 278-295.

Discrete function theory in higher-dimensional setting has been in active development since many years. However, available results focus on studying discrete setting for such canonical domains as half-space, while the case of bounded domains generally remained unconsidered. Therefore, in this talk we discuss the extension of the higher-dimensional function theory to the case of arbitrary bounded domains in $\mathbb{R}^{n}$.
This is a joint work with Paula Cerejeiras, Uwe Kähler & Anastasiia Legatiuk

Quaternionic quantum mechanics (HQM) endures several difficulties. The usual anti-hermitian theory has several inconsistencies, and the breakdown of the Ehrenfest theorem is possibly the most important one. The recent development of HQM in the real Hilbert space was able to address these difficulties, and also several old open questions were accordingly solved. In this talk this theory will be briefly outlined.

Most people who work with quaternions know the story of Hamilton's breakthrough on Broome bridge in Dublin, Ireland where he carved the famous formula i^2 = j^2 = k^2 = ijk = -1, but some may not know much about Grassmann (Hermann Günther Grassmann ; April 15, 1809 – September 26, 1877) or Clifford (William Kingdon Clifford FRS ; 4 May 1845 – 3 March 1879) and the history of their contributions to this subject. Hamilton's approach was mainly connected to algebra, while Grassmann arrived at similar conclusions through geometry. Although Hamilton saw the adoption of quaternions in his lifetime, Grassmann was barely recognized during his. It wasn't until after he passed that other scientists and mathematicians began to recognize the depth of his work. One of these was Clifford. Although Clifford's life was relatively short due to tuberculosis his insight into how to connect Hamilton's algebra to Grassmann's geometry has endured.
In this talk Dr. Familton will give a brief discussion of Grassmann algebras, their history, and how Clifford connected the best of Grassmann’s and Hamilton’s work.

Based on studies around the quaternions and octonions, an application of these numbers to the Padovan numerical sequence is then performed. Thus, there is the process of complexification of this sequence, investigating mathematical properties of these numbers, from which their matrix form, generating function, and Binet's formula stand out. Therefore, identities inherent to these numbers are obtained, emphasizing the complexification process. For future work, the integration of this mathematical investigation with other contents is encouraged.
Joint work with Milena Carolina dos Santos Mangueira, Francisco Regis Vieira Alves and Paula Maria Machado Cruz Catarino.

The Leonardo sequence is a sequence that has recently been studied in the mathematical scope, this sequence is related to the Fibonacci and Lucas sequences. Thus, aiming to explore the Leonardo sequence, in this work, we will carry out the mathematical complexification process around this sequence presenting Leonardo's quaternions and octoniums, thus defining their recurrence, as well as their generating function, Binet's formula, matrix form and properties linked to those numbers. Leonardo's quaternions are presented from formal sums of scalars with usual vectors of three-dimensional space, existing four dimensions and, as for the octonium numbers, the terms of Leonardo's sequence in eight dimensions are presented. For future work, the exploration of this investigation with other mathematical content is encouraged.
I work together with Renata Passos Machado Vieira, Francisco Regis Vieira Alves and Paula Maria Machado Cruz Catarino.

The goal of this talk is to characterize the Christoffel pairs of timelike isothermic surfaces in the four-dimensional split-quaternions. When the ambient space is restricted to three-dimensional imaginary split-quaternions, we establish an equivalent condition for a timelike surface in $\mathbb R^3_2$ to be real or complex isothermic in terms of the existence of integrating factors. Our study was motivated by U. Hertrich-Jeromin works about Mobius differential geometry, London Math. Soc. Lectures 300 (2003), where an excelent study of the Christoffel pairs of isothermic surfaces in codimension-two was made using the quaternions $\mathbb H$, and the Riemann surface structure. So, this talk contributes to show the natural extension of the quaternion algebra to the split-quaternions algebra and to use that extension, together with the split-complex numbers to look at the complex or real isothermic timelike immersions and its associated pair.
The results to be presented in this talk form part of a joint work with Prof. M. Magid (Wellesley College).