Abstract: In this talk, I will try to shed new light on the mysterious Cayley—Dickson construction. We introduce a new class of polynomial rings with a “flipped” multiplication which all Cayley—Dickson algebras naturally appear as quotients of. In particular, this extends the classical construction of the complex numbers as a quotient of a polynomial ring to the quaternions, the octonions, and beyond. This is based on joint work with Masood Aryapoor (MDU).

Abstract: In this talk, we discuss ideas on discretisation of the classical octonionic analysis. In particular, we present the ideas related to a direct discretisation of partial derivatives by help of finite differences. We discuss the influence of the non-associativity of octonionic multiplication and present an explicit derivation of the associator in the discrete setting. Further, we present basics of discrete octonionic function theory, including Borel-Pompeiu formulae and Cauchy transforms. This is a joint work with Rolf Sören Kraußhar and Anastasiia Legatiuk. 

Abstract:  A linear map on a D-algebra A is also a linear map on A as a D-module. We use the notation f ◦ a = f(a) for the image of the linear map f.  We can identify a linear map f of a D-algebra A with a tensor f ∈ A2⊗ according to the equality  (a ⊗ b) ◦ c = acb. I considered solving the equation a ◦ x = b.  Let A be Banach D-algebra. A map f : A → A is called differentiable on the set U ⊂ A, if at every point x ∈ U, the increment of f can be represented as f(x + h) − f(x) = df(x)/ dx ◦ h + o(h), where df(x)/ dx : A → A is a linear map and o : A → A is a continuous map such that  lim h→0 ∥o(h)∥ /∥h∥ = 0. Newton's method for solving the equation f(x)=a, where f is a differentiable map, is an iterative method for finding the root. Let x0 be an initial approximation for the iteration step. The solution of linear equation  df(x)/dx ◦ x = −f(x0) + df(x) /dx ◦ x0  serves as the initial approximation for the next iteration step.

Abstract:  In this talk we will describe what is known about the arithmetic in the ring of Hurwitz integers, a very special subring of the quaternions, and the use of this arithmetic in solving some Diophantine problems, i.e. problems where one seeks integer solutions of some polynomial equations or systems of polynomials equations, as well as some open problems in this context. 

Abstract:  A Bernoulli Sequence is usually defined by

xi+1 = −an−1xi −...− a0xi−n+1, i ≥ n − 1

with the initial conditions

x0 = x1 = … = xn−2 = 0 and xn−1 = 1,

where  ai ∈ C, for i = 0, . . . , n − 1. In this talk, we consider the Octonion Bernoulli Sequence, in which the constants  ai ∈ O,  for i = 0, . . . , n − 1. We will consider some special cases where a closed formulae is possible to obtain. Furthermore, some properties are presented.
This is a joint work with José Vitória.

Abstract:  In this talk we explain the difficulties caused by the non-associativity that arise in attempting to properly define a Bergman kernel and a related Bergman projection of square integrable octonion valued functions satisfying the octonionic Cauchy-Riemann equation. Already in the Stokes formula there appears an associator term which in fact leads to two versions of the Cauchy and the Borel Pompeiu formula. In the definition of a Bergman projection these associator terms play again a crucial role and they lead to different possible versions. We shall see that a rather canonical version of the Bergman projection is obtained when implementing a certain intrinsic octonionic weight factor of unit length. In the cases of the unit ball, the right octonionic half-space and strip domains bounded in the real direction one then gets formally similarly looking formulas for the reproducing kernel that are compatible to the formulas for the corresponding kernel functions in the associative Clifford case.
This is a joint work with M. Ferreira, M.M. Rodrigues and N. Vieira.

Abstract: Function theory in the octonions is an important research topic nowadays. Starting from the octonionic Stokes formula we can derive important results such as Cauchy's integral formula and Borel Pompeiu's formula. These are important tools for a function theory as demonstrated in the hypercomplex setting. We will review them and define the Teodorescu and the Cauchy operators and give their main mapping properties. Next we will give new results concerning the Bergmann transform which plays a fundamental role in the L2-space decomposition for octonion-valued functions. Finally, we use the octonionic Teodorescu transform to establish a suitable octonionic generalization of the Ahlfors-Beurling or Π-operator. We prove an integral representation formula that presents a unified representation for the Π-operator arising in all hypercomplex function theories, and describe some of its mapping properties. Applications of the Π-operator associated with the octonionic Beltrami equation and the hyperbolic octonionic Dirac operator will be shown.